WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS

Transcription

1 WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS A Thesis Submitted to the Graduate Faculty of the Louisiaa State Uiversity ad Agricultural ad Mechaical College i partial fulfillmet of the requiremets for the degree of Master of Sciece i The Departmet of Mathematics by Mabrouck K. Faradj B.S., L.S.U., 1986 M.P.A., L.S.U., 1997 August, 004

2 Ackowledgmets This work was motivated by a upublished paper writte by Dr. Madde i 000. This thesis would ot be possible without cotributios from may people. To every oe who cotributed to this project, my deepest gratitude. It is a pleasure to give special thaks to Professor James J. Madde for helpig me complete this work. This thesis is dedicated to my wife Mariaa for sacrificig so much of her self so that I may realize my dreams. It would ot have bee doe without her support. ii

3 Table of Cotets Ackowledgmets List of Tables List of Figures Abstract ii iv v vi Chapter 1. Itroductio The Origis of the Term Mea Atique Meas Geometric Iterpretatio of the Atique Meas Atique Meas Iequality Chapter. Classical Meas The History of Classical Meas The Developmet of Classical Meas Theory Nicomachus List of Meas Pappus List of Meas A Moder Recostructio of the Classical Meas Other Meas of the Aciet Greeks Chapter 3. Biary Meas The Theory of Biary Meas Classical Meas as Biary Mea Fuctios Biary Power Meas The Logarithmic Biary Mea Represetatio of Liks betwee Biary Meas Other Biary Meas Chapter 4. -ary Meas Historical Overview The Axiomatic Theory of -ary Meas Traslatio Ivariace Property of -ary Meas Iequality Amog -ary Meas Chapter 5. Coclusio Refereces Vita iii

4 List of Tables.1 Nicomachus Meas Pappus Equatios for Meas iv

5 List of Figures 1.1 Demostratio of Atique Meas usig a circle Proof Without Words: A Truly Algebraic Iequality Biary Meas as Parts of a Trapezoid Sigle Variable Fuctio Associated With Biary Meas v

6 Abstract The objective of this thesis is to give a brief expositio o the theory of meas. I Greek mathematics, meas are itermediate values betwee two extremes, while i moder mathematics, a mea is a measure of the cetral tedecy for a set of umbers. We begi by explorig the origi of the atique meas ad list the classical meas. Next, we preset a overview of the theories of biary meas ad -ary meas. We iclude a geeral discussio o axiomatic systems for meas ad preset theorems o properties that characterize the most commo types of meas. vi

7 Chapter 1. Itroductio I the this chapter we give a brief itroductio to the origis of the arithmetic, geometric, ad harmoic meas. 1.1 The Origis of the Term Mea Accordig to "Webster s New Uiversal Dictioary", the term mea is used to refer to a quatity that is betwee the values of two or more quatities. The term mea is derived from the Frech root word mie whose origi is the Lati word medius, a term used to refer to a place, time, quatity, value, kid, or quality which occupies a middle positio.the most commo usage of the term mea is to express the average of a set of values. The term average, from the Frech word averie, is itself rich i history ad has exteded usage. The term average was used i medieval Europe to refer to a taxig system levied by a liege lord o a vassal or a peasat. The word average is derived from the Arabic awariyah, which traslates as goods damaged i shippig. I the late middle ages, average was used i Frace ad Italy to refer to fiacial loss resultig from damaged goods, where it came to specify the portio of the loss bore by each of the may people who ivested i the ship or its cargo. I this usage, it is the amout idividually paid by each of the ivestors whe a loss is divided equally amog them. The otio of a average is very useful i commerce, sciece, ad legal pursuits; thus, it is ot surprisig that several possible kids of averages have bee iveted so that a wide array of choices of a itermediate value for a give set of values is available to the user to select from. 1. Atique Meas The earliest documeted usage of a mea was i coectio with arithmetic, geometry, ad music. I the 5th cetury B.C., the Greek mathematicia Archytas gave a defiitio of the three commoly used meas of his time i his treatise o music: we have the arithmetic mea whe, of three terms, the first exceeds the secod by the same amout as the secod exceeds the third; the geometric mea whe the first is to the 1

8 secod as the secod is to the third; the harmoic mea whe the three terms are such that by what ever part of itself the first exceeds the secod, the secod exceeds the third by the same part of the third. (Thomas, 1939, p. 36) This ca be traslated to moder terms as follows. Let a ad b be two whole umbers such that a > b ad A, G, ad H are the arithmetic, geometric, ad harmoic meas of a ad b respectively. The (i) a A = A b = A = a+b, (ii) a G = G b = G = ab, a H a (iii) = H b b = H = a+b ab. The origis of the ames give to the atique meas are obscured by time. The first of these meas, ad probably the oldest, is the arithmetic mea. To the aciet Greeks, the term αριθµητικς refers to the art of coutig, ad so, fittigly, they referred to what we commoly call the average as the arithmetic mea sice it pertais to fidig a umber that is itermediate to a give pair of atural umbers. As for the ame give to the geometric mea, it appears that the Pythagorea school coied the term mea proportioal, i.e., the geometric mea, to refer to the measure of a altitude draw from the right agle to the hypoteuse of a right triagle. The measure of such a altitude is betwee the measures of the two segmets of the hypoteuse. The source of the ame give to the harmoic mea ca oly be foud i legeds. The Roma Boethius (circa 5 A.D.) tells us of a leged about Pythagoras who o passig a blacksmith shop was struck by the fact that the souds caused by the beatig of differet hammers o the avil formed a fairly musical whole. This observatio motivated Pythagoras to ivestigate the relatio betwee the legth of a vibratig strig ad the musical toe it produced. He observed that differet harmoic musical toes are produced by particular ratios of the legth of the vibratig strig to its whole. He cocluded, accordig to the leged, that the musical harmoy produced was to be foud i particular ratios of the legth of the vibratig strig. Thus to the Pythagoreas, who believed that all kowledge ca be reduced to relatios betwee umbers, musical harmoy

9 occurred because certai ratio of umbers that lie betwee two extremes are harmoic, ad thus the term harmoic mea was give to that value. Propositio 1. Suppose 0 < a b. Let A := a+b, G := ab, ad H := a+b ab. The a A b, a G b, ad a H b Proof. Sice 0 < a b, the a + b b; therefore, a+b b. Similarly, a a + b; therefore a a+b a+b. Therefore, a b. Thus, a A b. Hece, b 1 a+b 1 ab a. Therefore, a a+b b, ad a H b. If a b, a ab b ; therefore, a ab b. Hece a G b. 1.3 Geometric Iterpretatio of the Atique Meas Sice geometry is the aciet Greeks preferred veue of scietific ivestigatio, Greek mathematicias produced umerous geometric treatises that related the three atique meas to each other by usig straight edge ad compass costructio. A excellet example ca be foud i Schild (1974) ad reproduced here: Example 1.1. Suppose a ad b are two whole umbers. Let A, G, ad H be the arithmetic, geometric, ad harmoic meas respectively of a ad b. The by usig a straight edge ad compass we ca illustrate that A = a+b, G = ab, ad H = a+b ab. Draw the lie segmet LMN with LM = a ad MN = b (see figure 1.1). With LN as diameter, draw a semi circle with ceter O ad fix P o its circumferece. Draw MQ perpedicular to OP ad MP perpedicular to LN. The OP = A, MP = G, ad QP = H. To show this is true, we give the followig argumet. Sice OP is the radius of the circle whose diameter is LN, the OP = 1 (a + b) = A, ad sice (MP) = (LM)(MN) = ab, the MP = ab = G. Let α = POM. Observe that QMP = α ad POM is similar to PMQ; thus, PQ PM = PM PO. Therefore, PQ = (PM) PO = ab a+b = ab a+b = H. I figure 1.1, observe what happes if (a + b) remais fixed, i.e., segmet LN is fixed, ad M is allowed to move. As M moves toward N, both G ad H decrease. As M moves towards O, both G ad H icrease. If M coicides with O, i.e., a = b, the A = G = H. This may have bee the motivatio for ivestigatig the iequality betwee the three meas. 3

10 P Q α α L O M N a b FIGURE 1.1. Demostratio of Atique Meas usig a circle. 1.4 Atique Meas Iequality I this sectio we will preset several proofs of the iequality: H G A (1.1) Of the umerous useful iequalities i mathematics, the arithmetic-geometric mea iequality occupies a special positio, ot oly from a historical stadpoit, but also o accout of its frequet usage i differet mathematical proofs. We will give a more i-depth discussio about this iequality i Chapters 3 ad 4. At this poit, it suffices to say that there have bee umerous proofs give for the above iequality over the ceturies. We begi our discussio by presetig a iformal argumet of the iequality. Referrig back to Figure 1.1, we ote that siα 1. From OPM, we have siα = MP OP, ad from PQM, we have siα = QP MP. Therefore MP = OPsiα G = Asiα. Hece G A, (1.) ad QP = MPsiα H = Gsiα. Hece H G. (1.3) From 1. ad 1.3, we get 1.1. However, sice the above argumet uses trigoometry, it does ot reflect the spirit of the aciet proofs for this iequality. I Figure 1., we preset a illustratio that captures the fudametal 4

11 character of this iequality i mathematics, which may have motivated the aciet mathematicias to establish proofs of the arithmetic-geometric mea iequality (Gallat 1977). The iequality as illustrated by Figure 1. requires oly rudimetary kowledge of geometry to prove. Now we give a more moder algebraic proof for the geometric-arithmetic mea iequality. ab a b FIGURE 1.. Proof Without Words: A Truly Algebraic Iequality. Theorem 1. For ay oegative umbers a ad b, ab a+b, with equality holdig if ad oly if a = b. Proof. Let a = c ad b = d. The a+b ab becomes c +d cd, or equivaletly, c +d cd 0. This is equivalet to c cd + d 0 which is i tur equivalet to (c d) 0. Sice the square of ay real umber is oegative, we see that the iequality stated i the theorem is ideed true. Equality holds if ad oly if c d = 0, that is c = d, or equivaletly, if ad oly if, a = b. We use the result from theorem 1 to establish a iequality betwee the harmoic ad geometric meas of ay two oegative umbers. Corollary 1.. For ay oegative umbers a ad b, ab a+b ab, with equality holdig if ad oly if a = b. Proof. Sice ab a+b, the ab (a + b). Therefore, ab (a + b) ab, ad ab a+b ab. 5

12 From theorem 1 ad corollary 1., we have H G A. (1.4) 6

13 Chapter. Classical Meas I this chapter we will explore the origis of the theory of biary meas. The chapter icludes two lists of the classical biary meas as give by Greek mathematicias. The followig list gives the ames of Greek mathematicia ad the approximate dates of their work o meas. It is helpful to the uderstadig of the historical developmet of the theory meas i the aciet Greek world (Smith 1951). Thales, 600 B.C. Pythagoras, 540 B.C. Archytas, 400 B.C. Plato, 380 B.C. Eudoxus, 370 B.C. Eudemus, 335 B.C. Euclid, 300 B.C. Archimedes, 30 B.C. Hero, 50 A.D. Nicomachus, 100 A.D. Theo, 15 A.D. Porphyrius, 75 A.D. Pappus, 300 A.D. Iamblichus, 35 A.D. Proclus, 460 A.D. Boethius, 510 A.D..1 The History of Classical Meas I this sectio we will give a brief discussio o what motivated Greek mathematicias to study ad develop a doctrie for meas by presetig the ratioale give by promiet Greek mathematicias who touched o the history of the theory of meas i their work ad the opiios of Greek mathematics scholars o this matter. Accordig to Gow (193), by Plato s time umbers were grouped ito two geeral categories. First, as sigle umbers categorized by their attributes such as odd, eve, triagular, perfect, excessive, defective, amicable etc. Secod, umbers were viewed as groups comprised of umbers that are either i series or proportios. The aciet Greeks viewed meas as a special case of proportios (Allema 1877, Thomas 1939, Gow 193). Smith (1951) writes, " Early [Greek] writers spoke of a arithmetic proportio, meaig b a = d c as i,3,4,5, ad of geometric proportio, meaig a : b = c : d as i, 4, 5, 10, ad a harmoic proportio, meaig 1 b a 1 = d 1 1 c as i 1, 1 3, 4 1; 1 5." I his commets o paradigms of aciet Greek mathematics, Allema (1877) says, "whe two quatities were compared [i Greek mathematics], the basis for the compariso seems to be either how much the oe is greater tha the other, i.e., a arithmetic 7

14 ratio, or how may times is the oe cotaied i the other, i.e., their geometrical ratio." Allema (1877) claims that this type of compariso of ratios would aturally lead to the theory of meas because for ay three positive magitudes, be it lies or umbers, a, b, ad c, if a b = b c, the three magitudes are i arithmetical proportio, but if a : b :: b : c, they are i geometrical proportio. Allema s claim seems to be supported by the work of Nicomachus i "Itroductio to Arithmetic". I this work, Nicomachus bega his discourse o meas by givig the defiitio that distiguished a ratio from a proportio. He referred to the latter as the compositio of two ratios. He the stated that whe oe term appears o both sides of a proportio, as i b a = b c, the proportio is kow as a cotiued proportio. The proportio is called disjuct whe the middle terms are differet. The highest term i a cotiued proportio is called the cosequet, the least is called the atecedet, ad the middle term is the mea, µεστητες, which is medius whe traslated ito Lati ad from which the word mea is derived (Gow 193). As we have oted above, Greek mathematics viewed meas as a special proportio ivolvig three magitudes; therefore, it is appropriate that we begi our review of the history of developmet of meas by metioig that Proclus attributed to Thales the begiig of the doctrie of proportios (Allema 1877). Thales established the theorem that equiagular triagles have proportioal sides (Allema 1877). I "Itroductio to Arithmetic", Nicomachus writes, "the kowledge of proportios is particularly importat for the study of aciet mathematicias." This ca be take to mea that the doctrie of proportios played a importat role i the developmet of Greek mathematics. Maziarz (1968) commets o the atural developmet of the theory of proportioals i Greek mathematics by sayig, "If a poit is a uit i a positio, the a lie is made of poits. Cosequetly, the ratio of two give segmets is merely the ratio of the umber of poits i each. Moreover, because ay magitude ivolves a ratio betwee the umber of uits it cotais ad the uit itself, ad, thus, the compariso of two magitudes implies either or 4 ratios." By poits, Maziarz seems to imply the tick marks that would be made if the segmets were divided ito may small equal uits. 8

15 From the historical perspective, the aciet sources of Greek mathematics history that we have refereced do ot metio whe the arithmetic mea was first developed. However, they offer various explaatios as to whe the geometric ad harmoic meas were first itroduced. Allema (1877) states that aciet sources (Iamblichus, Nicomachus, Proclus) poit to Eudoxus as the oe who established the harmoic mea ad to Pythagoras as the oe who established the otio of a mea proportioal betwee two give lies. It is iterestig to ote that some facets of the theory of meas appear i various aciet Greek texts. Some of these were iteded as mathematics treatises, such as the collectio of books that costitute Euclid s work kow as the "Elemets", but others did ot have a apparet mathematical purpose. Oe such example, oted by Maziarz (1968), ca be foud i passages of "Timaeus" kow as "The Costructio the world-soul." I this sectio of the book, Plato attempts to costruct the arithmetical cotiuum usig two geometric progressios 1,,4,8 ad 1,3,9,7; the fillig i the itervals betwee these umbers with the arithmetic ad harmoic meas. By successive duplicatio of the two progressios ad fillig i with the appropriate combiatio of arithmetic ad harmoic meas, all umbers ca be geerated, but ot i their atural order. Aother example ca be foud i Aristotle s "Metaphysics". I this work, Aristotle describes Plato s otio of distributive justice as, " The just i this sese is a mea betwee two extremes that are disproportioate, sice the proportioate is a mea, ad the just is proportioate. This kid of proportio is termed by mathematicias geometrical proportio." From the above examples, oe gets the sese that to the aciet Greeks, the theory of meas ad proportios may ot have bee just a mere mathematical cocept sice some aspects of the theory of meas was also reflected i their literature, philosophy, ad religio.. The Developmet of Classical Meas Theory It appears that the classical meas were developed over a log period of time by the gradual additio of seve more meas to the first three (Heath 1963). I all his work, Euclid oly uses the three atique meas (Allema 1887, Gow 193). However, by first cetury A.D., we kow that 9

16 Greek mathematicias referred to te meas. All the sources reviewed (Allema 1887, Bema 1910, Heath 191, Gow 193, Thomas 1939, Smith 1951) suggest that Greek mathematicias geerated these meas by cosiderig three quatities a, b, ad c, such that a > b > c. They assumed b to be the mea ad formed three positive differeces with the a, b, ad c: (a b), (b c), ad (a c). The they formed a proportio by equatig a ratio of two of these differeces to a ratio of two of the origial magitudes, a, b, ad c. For example, b is the harmoic mea of a ad c whe a b b c = a c. Nicomachus i "Itroductio to Arithmetic" (Gow 193) goes o to say: "Pythagoras, Plato, ad Aristotle kew oly six kids of [cotiued] proportios: the arithmetic, geometric, ad harmoic meas, ad their subcotraries, which have o ames. Later writers added four more." Greek mathematicias referred to certai classical meas as cotrary ad subcotrary meas because these meas were see to be i a cotrary (opposite) order from the arithmetic mea whe compared to the geometric or harmoic meas (Oxford Eglish Dictioary 004). I his work "I Nicomachus" (Heath 191), Iamblichus says, "the first three [atique meas] oly were kow to Pythagoras, the secod three were iveted by Eudoxus." The remaiig four, Iamblichus attributed to the later Pythagoreas. He adds that all te were treated i the Euclidea maer by Pappus. Gow (193) states that the umber of cotiued proportios was raised to te ad kept at that umber because the umber te was held by the aciet Greek mathematicias to be the most perfect umber. He adds, "how else ca we explai the fact that the golde mea, which Nicomachus calls the most perfect ad embracig of all proportios, was left out from the list of meas." All these testimoies poit to the coclusio that the theory of meas i Greek mathematics was well established by the First Cetury. Our mai complete source for aciet Greek mathematics theory of meas is Boethius commetary o the works of Pappus ad Nicomachus. I this work, 10

17 Boethius credits Nicomachus ad Pappus as the mai Greek mathematicias who dealt with meas from a theoretical perspective (Smith 1951)..3 Nicomachus List of Meas The earliest kow treatmet of classical meas as a idepedet body of kowledge was give by Nicomachus i "Itroductio to Arithmetic" (Allema 1887, Heath 191, Gow 193, Thomas 1939, Smith 1951). Allema, Gow, Heath, ad Thomas cocluded (seemigly idepedet of each other) that Nicomachus proceeded to develop his list as follows: He bega his list by commetig o the cotiued arithmetical proportio a b = b c. This suggests that a b : b c :: a : a, which allows us to make a coectio to other meas. Gow (193) remarks, "I a cotiued geometric proportio, a : b :: b : c, he otices that a b : b c :: a : b. Fially, the three magitudes, a, b, c, are i harmoic proportio if a b : b c :: a : c." A similar approach was used by Archytas (as cited by Porphyrius i his commetary o Ptolemy s "Harmoics") whe discussig the three atique meas i terms of three magitudes i cotiued arithmetic, geometric, ad harmoic proportios (Thomas 1939). Gow (193) also poits out that Nicomachus failed to metio that the arithmetic, geometric, ad harmoic meas of two umbers are i geometric proportio: a+b : ab : a+b ab. I Thomas traslatio of Nicomachus "Itroductio to Arithmetic" (Thomas 1939), Nicomachus itroduces the seve other meas usig the same treatmet as the oe metioed above. (The reader may wish to refer to Table.3 for a compact summary of the followig.) The fourth mea, which is also called the subcotrary by reaso of its beig reciprocal ad atithetical to the harmoic, comes about whe of the three terms the greatest bears the same ratio to the least as the differece of the lesser terms bears to the differece of the greater, as i the case of 3, 5; 6 (Thomas, 1939, p. 119). Nicomachus itroduces the fifth mea as the subcotrary mea to the geometric mea, The fifth [mea] exists whe of the three terms, the middle bears to the least the same ratio as their differece bears to the differece betwee the greatest ad the middle 11

18 terms, as i the case of, 4; 5, for 4 is double, the middle term is double the least, ad is double 1, that is the differece of the least terms is double the differece of the greatest. What makes it subcotrary to the geometric mea is this property, that i the case of the geometric mea the middle term bears to the lesser the same ratio as the excess of the greater term over the middle bears to that of the middle term over the lesser, while i the case of this mea a cotrary relatio holds (Thomas, 1939, p. 11). Nicomachus itroduces the sixth mea as, The sixth mea comes about whe of the three terms the greatest bears the same ratio to the middle as the excess of the middle term over the least bears to the excess of the greatest term over the middle as i the case of 1, 4; 6, for i each case the ratio is sesquialter [3 : ]. No doubt, it is called subcotrary to the geometric mea because the ratios are reversed, as i the case of the fifth mea (Thomas, 1939, p. 11). Nicomachus itroduces the last 4 meas by sayig, By playig about with the terms ad their differeces certai me discovered four other meas which do ot fid a place i the writigs of the aciets, but which evertheless ca be treated briefly i some fashio, although they are superfluous refiemets, i order ot to appear igorat. The first of these, or the seveth i the complete list, exists whe the greatest term bears the same relatio to the least as their differece bears to the differece of the lesser terms, as i the case of 6, 8; 9, for the ratio of each is see by compoudig the terms to be the sesquialter. The eighth mea, or the secod of these, comes about whe the greatest term bears to the least the same ratio as the differece of the extreme bears to the differece of the greater terms, as i the case of 6, 7; 9, for here the two ratios are the sesquialter. The ith mea i the complete series, ad the third i the umber of those more recetly discovered, comes about whe there are three terms ad the middle bears to the least the same ratio as the differece betwee the extremes bears to the differece betwee the least terms, as 4, 6; 7. Fially, the teth i the 1

19 complete series, ad the fourth i the list set out by the moders, is see whe i three terms the middle term bears to the least the same ratio as the differece betwee the extremes bears to the differece of the greater terms, as i the case of 3, 5; 8, for the ratio i each couple is the super-bi-partiet [5 : 3] (Thomas, 1939, p. 11). TABLE.1. Nicomachus Meas Mea Proportio Numbers Exhibitig the Mea Arithmetic a b : b c :: a : a, 4, 6 Geometric a b : b c :: a : b 4,, 1 Harmoic a b : b c :: a : c 6, 3, Cot. Harmoic b c : a b :: a : c 3, 5, 6 Cot. Geometric b c : a b :: b : c, 4, 5 Subco. Geometric b c : a b :: a : b 1,4,6 Seveth a c : b c :: a : c 6, 8, 9 Eighth a c : a b :: a : c 6, 7, 9 Nith a c : b c :: b : c 4, 6, 7 Teth a c : a b :: b : c 3, 5, 8 (Thomas 1939).4 Pappus List of Meas Pappus used a differet approach tha Nicomachus whe presetig his list of meas (Heath 191, Thomas 1939). Both Heath ad Thomas state that the meas o Pappus list are similar to those preseted by Nicomachus, but i a differet order after the sixth mea. Meas umber 8, 9, ad 10 i Nicomachus list are respectively umbers 9, 10, ad 7 o Pappus list. Moreover, Pappus omits mea umber 7 o Nicomachus list ad gives as umber 8 a additioal mea equivalet to the proportio c : b :: c a : c b. Therefore, the two lists combied give five additioal meas to the first six. I Thomas traslatio (1939) of Pappus work kow as "Collectios III", Pappus itroduces his discussio o meas as a respose to a questio posed by a uiformed geometer. He 13

20 demostrates his aswer by the costructio of the three meas i a semicircle (see figure 1.1). Pappus shows, i a series of propositios, that give three terms α, β, ad γ i geometrical progressio (Heath 191 uses "i geometric proportio"), it is possible to form from them three other terms a, b, ad c which are itegral liear combiatio of α, β, ad γ such that b is oe of the classical meas. The solutios to Pappus s equatios are show i Table.. The liear (Heath 191, Thomas 1939) TABLE.. Pappus Equatios for Meas Mea a, b, c Numbers exhibitig the mea Arithmetic a = α + 3β + γ 6, 4, b = α + β + γ c = β + γ Geometric a = α + β + γ 4,, 1 b = β + γ c = γ Harmoic a = α + 3β + γ 6, 3, b = β + γ c = β + γ Subcotrary a = α + 3β + γ 6, 5, b = α + β + γ c = β + γ Fifth a = α + 3β + γ 5, 4, b = α + β + γ c = β + γ Sixth a = α + 3β + γ 6, 4, 1 b = α + β + γ c = α + β γ Seveth a = α + β + γ 3,, 1 b = β + γ c = γ Eighth a = α + 3β + γ 6, 4, 3 b = α + β + γ c = β + γ Nith a = α + β + γ 4, 3, b = α + β + γ c = β + γ Teth a = α + β + γ 3,, 1 b = β + γ c = γ equatios show i Table. are moder equivalets of the literal traslatio of the Greek versio of Pappus. For example (Thomas 1939), i the case of the geometric mea metioed i Table., the literal traslatio of Pappus words would be, "To form a take α oce, β twice, ad γ oce; ad to form b we have to take β oce ad γ oce; ad to form c we take γ oce." Notice also that the examples give by Pappus for the proportios formed by his equatios sometimes differ 14

21 from those give by Nicomachus. For example for the fourth mea, Nicomachus gave 3, 5, ad 6 as a example for a solutio, while Pappus gave, 5, ad 6 as a solutio. Pappus expositio o meas by usig equatios may be better uderstood from the perspective that proportios were used i those days to solve equatios. Usig Proclus commetary o Euclid as a referece, Klei (1966) states, " Greek mathematics usage of proportios ca be compared to the moder sese of costructio of a equatio, ad a equatio may be viewed as a solutio of a proportio. This may be due to the uderstadig of ratios, proportios, ad harmoy o the basis of a commo mathematical property." Bema (1910) claims that the mathematicias of Alexadria uderstood equatios of secod degree mostly i the form of proportios. If we express Pappus method i moder terms, Pappus is parmeterizig meas by quadratics ad, equivaletly, givig quadratic polyomials to illustrate the relatio amog terms i the various meas. For example, to calculate the harmoic mea, usig three quatities i geometric progressio is equivalet to usig α = 1, β = x, ad γ = x ; thus, give a = + 3x + x, b = x + x, ad c = x + x, we have ac a+c = (+3x+x )(x+x ) = x(x+1)(x +3x+) = x(x +3x+) +3x+x +x+x (x+1) x+1 = x(x + ) = x + x = b..5 A Moder Recostructio of the Classical Meas I this sectio, we will use a similar approach to the oe used by Nicomachus to geerate the classical meas by cosiderig three positive quatities a, b, ad c such that a > b > c, ad we wish to make b the mea of a ad c. We will form three positive differeces with these quatities: (a b), (b c), ad (a c). The we will form a proportio by equatig a ratio of two of these differeces to a ratio of two of the origial quatities (ot ecessarily distict). For example, if we set the ratio a b b c equal to the ratio a b, the result is b = ac, which represets the geometric mea. If you look at all the possible ways of doig this, several of them are automatically ruled out by the assumed iequality of a, b, ad c. The oes that are ot (ecessarily) ruled out are the eleve meas summarized below (Madde 000, Heath 1963): 1. (a b) (b c) = a a = b b = c c (a+c), we have the arithmetic mea b =. 15

22 . (a b) (b c) = b c = a b, we have the geometric mea b = ac (a b) (b c) = a c ; we have the harmoic mea b = 1 a +. 1 c (a b) (b c) = c a ; we have the cotra-harmoic mea b = a +c a+c. (a b) (b c) = c b ; we have the first cotra-geometric mea b = a c+ a ac+5c. (a b) (b c) = b a ; we have the secod cotra-geometric mea b = c a+ 5a ac+c. (b c) (a c) = a c; b = ac c a. This mea is o Nicomachus list but ot Pappus list. (b c) (a c) = c b ; b = c+ 4ac 3c. (a b) (a c) = c a ; b = a ac+c a. (a b) (a c) = b a ; b = a c a. This mea is o Pappus list but ot Nicomachus list. (a b) (a c) = c b ; b = a c. Note that some of these meas are ot very robust defiitios of meas. For example, if oe uses the 11 th mea o our list to fid the mea of 5 ad 4, the M(5,4) = 1, which is ot betwee 5 ad 4. Note also that usig the 5 th mea o our list to fid the mea of 1 ad, we obtai the celebrated golde umber Φ = However, as we will show i the ext sectio, the above list does ot exhaust all the meas kow to the aciet Greek world..6 Other Meas of the Aciet Greeks I this sectio, we poit out that Greek mathematicias cotiued to develop ew meas which were ever icluded amog the classical meas. Nicomachus referred to a special mea obtaied by the divisio of a segmet ito what he called "the most perfect proportios". This mea, which we will call b, ca be expressed by the divisio of a segmet of magitude a ito two parts: A greater part, b, ad a lesser part, a b, i such a fashio that the ratio of a to b is equal to the ratio 16

23 of b to a b. Hece the proportio: a : b = b : (a b) This, i tur, leads to the quadratic equatio b + ab a = 0, The positive root of which is b = 1 a 5 1. A special solutio of this equatio is whe a = 1, b is the celebrated umber Φ. The mathematicias of Alexadria referred to other quatities as meas. For example Hero s mea. Defiitio.3. Suppose a ad c are positive umbers. The Hero s mea is b = a + ac + c. 3 To check that Hero s mea of ay two positive values is always betwee these two values, let 0 < a < c. The by the arithmetic-geometric mea iequality, a < ac < a+c < c. Thus a = 3a 3 < a + ac 3 < a + ac + c 3 < c + ac 3 < 3c 3 = c. Hero s mea is used i calculatig the volume of a pyramidal frustum (a prismatoid figure formed by choppig off the top of a pyramid), where a ad c are the bottom ad top areas respectively of the pyramidal frustum. The cetroidal mea is aother example of a mea produced by aciet Greek mathematics which was ot icluded i the list of classical meas. This mea was developed by Archimedes for his work o cetroids. Defiitio.4. Let a ad c be two atural umbers. The the cetroidal mea of a ad c is b = (a + ac + c ). 3(a + c) 17

24 We shall demostrate that the cetroidal mea of two positive values is always betwee these two values. Let 0 < a < c. Usig the iequality from propositio 1, we have a a+c ac c; therefore, a 3 + (a + c ) 3(a + c) ac 3(a + c) + (a + c ) 3(a + c) c 3 + (a + c ) 3(a + c). Now, a < c a + ac < a + c a a +c a+c a < 3 a + (a +c ). A similar argumet ca be used to show c c 3 + (a +c ) 3(a+c). Hece, a (a +c ) 3(a+c) c. 3(a+c) 18

25 Chapter 3. Biary Meas I this chapter we will give a cotemporary defiitio for biary meas ad preset o overview of the developmet of the theory of biary meas. The chapter icludes a expositio of the most commo types of biary meas. The chapter cocludes with a summary o iequalities amog biary meas. 3.1 The Theory of Biary Meas Based upo the sources reviewed, aciet Greek mathematics treatmet of meas teded to be limited to fidig the mea of two magitudes, be it lie segmets, areas, or volumes. Berlighoff (00) claims that this limited view o meas i Greek mathematics may have stemmed from their iterest i geometry, where meas of magitudes of segmets, areas, ad volumes are itermediate value betwee the two extremes. Therefore, fidig the mea of more tha two such magitudes was a problem that was ot ecoutered because such a mea would ot represet itermediate value betwee two extremes. I moder times, this outlook has chaged. The arithmetic mea, geometric mea, ad the harmoic mea came to be viewed as specific cases of a geeral fuctio ot of just two variables but also of -variables. I this chapter we will limit our discussio to biary meas ad postpoe our dealig with -ary meas to the ext chapter. Hutigto (197) cites work published by R. Schimmack i 1909 which treats meas as a cotiuous fuctio that satisfies give restrictios. Dodd (1933) credits B. de Fietti s 1931 work with formulatig specific criteria that a mea fuctio must satisfy. Although i both cases, the fuctio referred to is a variable fuctio, a similar view may be exteded to two-variable mea fuctios. Borwei (1987) lists postulates for a mea fuctio of two variables, f (a,b), similar to the restrictios cited by Hutigto ad Dodd for a mea fuctio of -variables. We will use Borwei s (1987) defiitio ad criteria for biary meas to develop a defiitio for a geeralized biary mea. Next we will subject the classical meas to the criteria we have 19

26 developed for biary meas to coclude whether or ot these meas ca be cosidered meas i our refied sese. We will also itroduce the moder otio of power meas, ad show the various iequalities that relate biary meas to each other. We will coclude the chapter with examples of other fuctios that geerate biary meas. We begi by itroducig the term isotoe (Borwei 1987), which we will subsequetly use to idetify a specific property for fuctios of two variables. Defiitio 3.5. Let f : R + R + R + be a fuctio. f is isotoe if for each a R + ad b R +, f (a,x) ad f (x,b) are mootoe icreasig fuctios of x. To demostrate that f (a,b) is isotoe, we fix oe variable, say a, ad show that f (a,x) is mootoe icreasig as a fuctio of x. The we appeal to the same argumet for b. Defiitio 3.6. A biary mea fuctio, f, is a positive real valued fuctio, f (a,b), of two strictly positive real variables a ad b that satisfies the followig postulates: CR f is a cotiuous ad real valued fuctio. IS IN f is a isotoe. f is iteral, i.e., mi(a,b) f (a,b) max(a,b). DI f is diagoal, i.e., f (a,b) = a or f (a,b) = b if ad oly if a = b. HO f is homogeeous, i.e., f (λa,λb) = λ f (a,b), where λ 0. SY f is symmetric, i.e., f (a,b) = f (b,a). Remark: Note that HO permits us to write M(a,b) = am(1, b a ), a useful result utilized i proofs of may theorems o meas. 3. Classical Meas as Biary Mea Fuctios We will revisit our eleve classical meas to explore which of these satisfy the biary mea postulates listed i defiitio 3.6. For the followig argumets, we will assume that a ad b are positive real umbers such that a < b ad M is the mea. Propositio. If M is equal to A, G, or H, the M is a biary mea fuctio. 0

27 Proof. Suppose 0 < a b ad M(a,b) is A, G, or H. Clearly, M satisfies CR. To show M satisfies IS, fix a ad let 0 < x 1 < x. Now we show M(a,x 1 ) < M(a,x ). If M = A or M = H, the M(a,x 1 ) < M(a,x ) because a+x 1 < a+x ad ax 1 a+x 1 < ax a+x by properties of additio ad multiplicatio of positive umbers. If M = G, we have x 1 < x ad ax 1 < ax. Thus, ax1 < ax, sice the square root fuctio is mootoe icreasig. Thus M(a,x 1 ) < M(a,x ). A similar argumet ca be used to show M(x,b) is mootoe icreasig. M satisfies IN by propositio 1. M satisfies DI. This ca be checked by substitutig b = a i the defiitio of A, G, ad H. M satisfies HO by the distributive property of multiplicatio over additio. M satisfies SY by the commutative properties of additio ad multiplicatio. The remaiig eight classical meas are ot biary mea fuctios accordig to defiitio 3.6. To substatiate this claim, we take each i turs. 1. The cotra-harmoic mea, M = a +b a+b, fails to satisfy IS. For example, M(6,) = 5 = M(6,3).. The cotra-geometric mea, M = (a b + a ab + 5b )/ fails to satisfy SY. For example, M(1,) = = O the other had, M(,1) = = The subcotra-geometric mea, M = (b a + 5a ab + b )/, fails to satisfy SY. For a example, M(1,4) = = O the other had, M(4,1) = = M = ab b a. M fails to satisfy SY. For example, M(1,) = 4 4 = 0. O the other had, M(,1) = 4 1 = M = b+ 4ab 3b also fails SY. 6. M = a ab+b a. Clearly, this mea fails to satisfy SY. 7. M = a b a. M fails to satisfy SY. 1

28 8. M = b a. Clearly, M fails to satisfy SY. Therefore, of the eleve classical meas, oly the atique meas are cosidered biary mea fuctios accordig to defiitio 3.6for a mea. 3.3 Biary Power Meas Aother represetatio for biary mea fuctios is kow as power meas. The a root-mea-square (also kow as the Euclidea mea), R(a,b) = +b, may have bee the first example of this ew class of meas (Li 1974). Defiitio 3.7. Suppose r > 0, a > 0, ad b > 0. The the r th power mea of a ad b, deoted M r (a,b), is ( ar +b r ) 1 r. Theorem. Let a > 0, b > 0, ad r 0. The fuctio M r (a,b) = ( ar +b r ) 1 r is a biary mea fuctio. Proof. M r (a,b) statisfies CR. Clearly M r (a,b) is cotiuous for r > 0, a > 0 ad b > 0 sice it is a compositio of cotiuous fuctios. M r (a,b) satisfies IS. Fix a. Let x 1 ad x be ay positive umbers such that 0 < x 1 < x. If r > 0, the x r 1 < xr ad ar +x r 1 < ar +x r ( ) a r +x r r ( ) 1 a < r +x r r. If r < 0, the x r 1 > x r ad ar +x r 1 > ar +x r. Therefore,. Therefore, ( a r +x r 1 M r (a,b) satisfies IN. Suppose a < b. If r > 0, the we have a < ( ar +b r ) 1 r < b sice ) r < ( a r +x r a r < ar +b r < b r. Similarly, if r < 0, we have a r > ar +b r > b r ad a < ( ar +b r ) 1 r < b. M r (a,b) satisfiesdi. Suppose ( ar +b r ) 1 r = a. The we have ar +b r = a r which implies b = a. Similarly, if b = a, the ( ar +a r ) 1 r = a. Therefore, M r (a,b) = a if ad oly if b = a. M r (a,b) satisfies HO. Fix λ > 0. The M r (λa,λb) = ( (λa)r +(λb) r ) 1 r = λ( ar +b r ) 1 r. M r (a,b) satisfies SY, sice M r (a,b) = ( ar +b r ) 1 r = ( br +a r ) 1 r = M r (b,a). ) r. Note that the arithmetic mea, the harmoic mea, ad the root-mea-square are power mea fuctios by direct substitutio i M r = ( ar +b r ) 1 r with the appropriate value for r: 1. r = 1, the M 1 (a,b) yields the arithmetic mea A = a+b.

29 . r = 1, the M 1 (a,b) yields the harmoic mea H = ab a+b. 3. r =, the M (a,b) yields the root-mea-square R = a +b. We ow show that the geometric mea is a limit of power mea fuctios. Theorem 3. lim r 0 M r (a,b) = ab. { Proof. Observe that lim( ar +b r r 0 ) 1 r = lim exp ( 1 r 0 r )l( ar +b r Applyig L Hopital rule, lim ( = lim ) ( ) a r la+b r lb r 0 a r +b r = ab. ( l(a r +b r )/ r 0 r ) = lim r 0 = la+lb. Therefore, exp } { = exp ) ) ( ddr ((l(a r +b r ))/) 1 { (lim r 0 ( 1 r )l( ar +b r ) lim ( 1 r 0 r )l( ar +b r ) } }. = exp { } la+lb Defiitio 3.8. Let a ad b be ay positive umbers. The M 0 (a,b) := ab. With the developmet of this represetatio for meas, ways had to be foud to compare these meas to each other ad to the already established oes. This led to the to the establishmet of some of the most well-kow iequalities i mathematics. Theorem 4. If a, b, ad r are positive umbers, the M 0 (a,b) M r (a,b). With equality holdig if ad oly if a = b. Proof. Note that a = b ( ) ab = a r +b r 1 r. Suppose a < b, the M 0 (a,b) = ab ad ( ) M r (a,b) = a r +b r 1 r. Observe that ab = (ab) 1. The (ab) r = (a r b r ) 1 ad [ ( ) a r +b r 1 ] r r = ar +b r. By the arithmetic-geometric mea iequality, (a r b r ) 1 < ar +b r. Therefore, M 0 (a,b) < M r (a,b). Theorem 5. If a, b, r, ad s are positive umbers such that r < s, the M r (a,b) < M s (a,b). The proof we preset is a modified versio of the proof give i Schaumberger (1988) for -ary power meas. 3

30 Proof. Let x > 0 ad f (x) = rx s + (s r) sx r. We ote that f (x) has a absolute miimum oly at x = 1 (sice f (x) = rsx s 1 rsx r 1 = 0 oly at x = 1 ad f (1) = rs(s r) > 0). Observe that f (1) = 0; therefore, f (x) = rx s + (s r) sx r 0. Hece, rx s + (s r) sx r, (3.5) with equality holdig if ad oly if x = 1. Let T = ( a r +b r ) 1 r. Put x 1 = a T ad x = b T. By substitutig for x 1 ad x i equatio 3.5 successively for x ad addig, we obtai [ ( r at ) s ( + bt ) s ] [ ( + s r s at ) r ( + bt ) r ] [ ] [. Hece, r a s +b s T s + s r s a r +b r T ]. r But [ ] ( ) T r = ar +b r. Therefore, a s +b s T s. Hece as +b s T s, ad this implies as +b s a r +b r s r, which ( ) leads to a s +b s 1 ( ) s a r +b r 1 r. Therefore, M r (a,b) M s (a,b). 3.4 The Logarithmic Biary Mea The logarithmic mea is ecoutered i various applicatios such as i ivestigatio of heat trasfer, fluid mechaics (Li 1974), ad the distributio of electrical charge o a coductor (Stolarsky 1975). Defiitio 3.9. Let a > 0 ad b > 0. The L(a,b) = a a b la lb if a b if a = b Theorem 6. L is a biary mea. Proof. First we prove that L(a,b) satisfies CR. Clearly, L(a,b) is cotiuous o (0, ) (0, ) except maybe o the lie a = b. To show that L(a,b) is cotiuous whe a = b, we ote first that ) ) = 1 by L Hopital rule. Thus lim = a; therefore, by substitutig x y for u, lim u 1 ( u 1 lu we have lim (x,y) (a,a) ( y( x y 1) l x y a > 0 ad let x (0, ). Let g(x) := L(a,x) = ) (y,u) (a,1) y( u 1 lu = a. So, lim L(x,y) = a. Now we show L(a,b) satisfies IS. Fix (x,y) (a,a) x a. We must show that g(x) is mootoe lx la = x a l a x 4

31 icreasig. It suffices to show g (x) > 0, except possibly at fiitely may poits. Whe x a, g (x) = l a x (1 a x ) (l a x ; so, it suffices to show l a x (1 a x ) ) (l a x > 0, except at fiitely may poits. Let ) h(u) = lu 1 + u. We eed to show h(u) > 0, except at fiitely may poits. By examiig h, we see that h is decreasig o (0,1) ad icreasig o (1, ). Sice h(1) = 0, we have proved what is eeded. We show that L(a,b) satisfies IN, i.e., mi(a,b) L(a,b) max(a,b). Let f (x) = l(x). The for ay 0 < a < b, by the Mea Value Theorem, there exists a t i [a,b] such that f (t) = lb la b a. Therefore, 1 t = la lb a b. Hece t = la lb a b ad a t b. That L(a,b) satisfies DI is evidet from the defiitio of L(a,b). We show that L(a,b) satisfies HO. Let λ > 0. The L(λa,λb) = λa λb logλa lλb = λ(a b) We show that L(a,b) satisfies SY. L(a,b) = a b la+lλ la lλ) = λ( la lb ) = λl(a,b). a b la lb = (b a) (lb la) = The followig theorem establishes a iequality betwee L, A ad G. b a lb la = L(b,a). Theorem 7. If a > 0 ad b > 0 such that a b,the G(a,b) < L(a,b) < A(a,b). The followig proof was give by Carlso (197) Proof. If t > 0, the iequality of the arithmetic ad geometric mea implies that t +t(a + b) + ( a+b ) > t +t(a + b) + ab > t + t(ab) 1 + ab. Thus R dt < R (t+ 0 a+b ) 0 fractios, we fid ab < dt (t+a)(t+b) < R a b la lb < a+b 0 a+b < a b 1 lim dt (t+ ab). Evaluatig the middle itegral by the method of partial [l(t + b) l(t + R a)]r 0 < 1. This implies ab Based upo the results obtaied above, we have the followig iequality that relates the harmoic mea, geometric mea, logarithmic mea, arithmetic mea, ad the root-mea-square. Corollary Let a > 0 ad b > 0 such that a > b, the H G L A R. It is iterestig to ote that the logarithmic mea does ot quite led itself to a atural geeralizatio to variables (Pitteger 1985). This mea fails a particular axiom (amely the associativity axiom) for -ary meas. However, due to the use of this -ary mea i various applicatios such as i defiig average temperatures ad aalysis of idex umbers i 5

32 ecoomics, a theoretical framework for the geeralizatio of the logarithmic mea of variables has bee recetly developed. We refer the iterested reader to the work of Pitteger (1985) for more iformatio o -ary logarithmic meas. 3.5 Represetatio of Liks betwee Biary Meas Eves (003) gives a excellet geometric lik betwee various biary meas usig a trapezoid. Let a > b > 0. Suppose a trapezoid has parallel sides a ad b as show i figure 3. The various a b ab C R T N G H FIGURE 3.3. Biary Meas as Parts of a Trapezoid meas ca be raked i size relative to each other as the legths of vertical segmets. The segmet whose legth is: The harmoic mea, H, passes through the itersectio of the diagoals. The geometric mea, G, divides the trapezoid ito two similar trapezoids. The Heroia mea, N, is oe third of the way from the arithmetic mea to the geometric mea. The arithmetic mea, A, bisects the sides of the trapezoid. The cetroidal mea, T, passes through the cetroid of the trapezoid. The root-mea-square, R, bisects the area of the trapezoid. 6

33 The cotra-harmoic mea, C, is as far to the right of the arithmetic mea as the harmoic mea is to the left of it. 3.6 Other Biary Meas Iterest i geeratig differet biary meas fuctios cotiued to grow ito the late 0 th cetury as other fuctios of two variables were foud that satisfy give criteria for a desired mea fuctio. Borwei (1987) defied a class of biary mea fuctios, M p (a,b), that is derived from a mea fuctio, M(a,b), that satisfies the postulates give i Sectio 3.1. This class of biary meas is determied by the formula M p (a,b) := M(ap,b p ) M(a p 1,b p 1 ). where p R. We refer the reader to Borwei (1987) for the proof that M p (a,b) satisfies the postulates give i Sectio 3.1. Example of such biary meas iclude (Borwei 1987): Lehmer meas. Let a,b > 0 ad p R. The Lehmer meas, L p, is defied as L p (a,b) = ap + b p a p 1 + b p 1. Observe that L 1 = A ad L 1 = G. Gii meas. Let a,b > 0 ad r s. The Gii mea, G (s,r) (a,b), is defied as ( a s + b s ) ( 1 s r) G (s,r) (a,b) = a r + b r Stolarsky s Meas. Let a,b > 0 ad p 0,1. The the Stolarsky s Mea, S p (a,b), is defied as ( a p + b p S p (a,b) = p(a b) ) ( ) 1 p 1 7

34 Observe that S 0 (a,b) = lim S p (a,b) = b a p 0 lb la, which is the logarithmic mea. Ad S 1 (a,b) = lim S p (a,b) = e 1 (a a b b ) a b 1, which is also kow as the idetric mea. p 1 We refer the iterested reader to Borwei (1987) for more iformatio o the meas listed above. Mays (1983) ivestigated coditios uder which biary meas ca be associated with a sigle variable fuctio. I this work, Mays developed a idea preseted by Moskovitz (1933). Mays also poited out some errors cotaied i Moskovitz (1933). Give a fuctio f from (0, ) ito R, Mays (1983) defies M f (a,b) to be the X-itercept of the lie coectig (a, f (a)) ad (b, f (b). See Figure 3.4 (reproduced from Mays (1983)). Clearly M f (a,b) satisfies IN ad SY. We fid a formula for M f (a,b) by calculatig the slope of the lie through (a, f (a)) ad (b, f (b) i two ways: Solvig for M f (a,b), we get: f (a) a M f (a,b) = f (b) b M f (a,b). M f (a,b) = a f (b) + b f (a) f (a) + f (b). (3.6) Theorem 8. M f = M g g = k f for some k > 0 Proof. If g = k f, k cacels i the right had side of M k f (a,b) = ak f (b)+bk f (a) k f (a)+k f (b) = M g = M f. If g k f, pick a, b, ad k so that g(a) = k f (a) but g(b) k f (b). The if M f (a,b) = M g (a,b), we ] ] have =. Therefore, a f (a)(k f (b) g(b)) = b f (a)(k f (b) g(b)). [ a f (b)+b f (a) f (a)+ f (b) [ ag(b)+bk f (a) k f (a)+g(b) Sice k f (b) g(b) 0 ad f (a) 0, the a = b, a cotradictio. Corollary If M = M f, the there exists f such that M = M f ad f = 1. Proof. Let f = f (x) f (1). Corollary 3.11 allows us to assume, without loss of geerality, whe associatig M f with a give fuctio f that f (1) = 1. Defiitio 3.1. Let f be a fuctio i oe variable. f is multiplicative if the domai of f is closed uder multiplicatio ad f (xy) = f (x) f (y) for every x, y i the domai of f. 8

35 FIGURE 3.4. Sigle Variable Fuctio Associated With Biary Meas Lemma Suppose f : (0, ) = R ad f (1) = 1. The f is multiplicative if ad oly if f (a) f (λb) = f (λa) f (b) (3.7) for all a, b, ad λ (0, ). Proof. Note that the coditio 3.7 implies f (λb) = f (λ) f (b) for all λ, b (0, ). Coversely, if f is multiplicative, the f (a) f (λb) = f (a) f (λ) f (b) = f (λa) f (b). Theorem 9. Suppose f (1) = 1. The M f is homogeeous if ad oly if f is multiplicative. Proof. By equatio 3.6, a, b, λ > 0 λm f (a,b) = M f (λa,λb) λ ( a f (b)+b f (a) f (a)+ f (b) ) = a f (b) f (λa) + a f (b) f (λb) + b f (a) f (λa) + b f (a) f (λb) = λa f (λb)+λb f (λa) f (λa)+ f (λb) af(a)f(λb) + b f (a) f (λa) + a f (b) f (λb) + b f (b) f (λa) a f (b) f (λa) + b f (a) f (λb) = a f (a) f (λb) + b f (b) f (λa) (a b) f (b) f (λa) = (a b) f (a) f (λb) f (b) f (λa) = f (λb) f is multiplicative. 9

36 We ow explore some ideas that are motivated by Mays (1983). Let M be a fuctio of two variables (ot ecessarily a mea). We defie F M (x,y) := M(x,y) y x M(x,y). Similarly, if F is a fuctio of two variables, we defie M F (x,y) := xf(x,y) + y 1 + F(x,y). Note that if M is a mea, F is the kid of ratio that was cosidered by the Greeks i developig the classical meas. Theorem 10. F MF = F ad M FM = M as fuctios o {(x,y) x y}. If we choose M ad F such that M F = M ad F M = F, the: 1. M is homogeeous if ad oly if F is "projective", i.e., F(x,y) = F(λx,λy).. M is itermediate if ad oly if F is positive. 3. M is symmetric if ad oly if F(x,y)F(y,x) = 1. Proof. We show F MF = F. F MF = M F y x M F = xf+y 1+F y x xf+y 1+F = xf+y y yf x+xf xf y = (x y)f (x y) = F. We show M FM = M. M FM = xf M y M+y x( x M )+y 1+F M = 1+( M y x M ) = x(m y)+xy My x M+M y = (x y)m (x y) = M. Now we show that M is homogeeous if ad oly if F(x,y) = F(λx,λy). Suppose M is homogeeous. The F(λx,λy) = M(λx,λy) λy λx M(λx,λy) = λm(x,y) λy λx λm(x,y) = λ(m(x,y) y) F(x,y) = F(λx,λy). M(λx,λy) = λxf(λx,λy)+λy 1+F(λx,λy) λ(x M(x,y)) = M(x,y) y ( x+f(x,y)+y = λ 1+F(x,y) x M(x,y) = F M(x,y). Now suppose ) = λm(x, y). We ow show M is itermediate if ad oly if F > 0. x(1 + F) > xf + y > y ad M y x M > 0 m is betwee x ad y. To show M is symmetric if ad oly if F(x, y)f(y, x) = 1, let F = F(x, y) ad F = F(y, x). Observe that M F is symmetric xf+y 1+F = yf+x xf + y + xff + yf = yf + x + yff + Fx 1+F x + yff = y + xff (x y) = (x y)ff FF = 1. 30

37 Mays addressed the problem of whe a give biary mea m ca be expressed as M f for some f (x). We ow use F ad M to give a more geeral solutio to the problem tha Mays. Propositio 3. Suppose M is ay fuctio of two variables x ad y such that x y. The M = M f if ad oly if F M (x,y) = f (y) f (x). Proof. Suppose M = M f. The m = xf(x,y)+y 1+F(x,y) = x f (y)+y f (x) f (x)+ f (y). Therefore, x f (x)f(x,y) + y f (x) + x f (y)f(x,y) + y f (y) = y f (x)f(x,y) + y f (x) + x f (y)f(x,y) + x f (y), ad x f (x)f(x,y) + y f (y) = y f (x)f(x,y) + x f (y). Hece, (x y) f (x)f(x,y) = (x y) f (y), which implies F(x,y) = f (y) f (y) f (x). Coversely, we ca show if F(x,y) = f (x), the M f = xf(x,y)+y 1+F(x,y) = x f (y)+y f (x) f (x)+ f (y) let M = M f, we have proved what is eeded. by usig a similar argumet to the oe give above. Therefore, if we 31

38 Chapter 4. -ary Meas The chapter begis with a brief discussio of the historical developmet of -ary meas. Next we preset a overview of postulates of -ary meas, startig with postulates for the arithmetic mea ad follow their evolutio ito postulates for geeralized meas. We discuss the traslatio ivariace property for -ary meas. We preset examples of various types of -ary meas. We coclude the chapter with a brief discussio of the theory of iequalities amog -ary meas. 4.1 Historical Overview Oe of the earliest kow refereces cocerig the arithmetic mea of several umbers is give by Iamblichus i a treatise o what we call ow umber theory. I this work, "The Theology of Arithmetic," Iamblichus outlies a example ivolvig fidig the arithmetic mea of the umbers 1 to 9: I the first place, we must set out i a row the sequece of umbers from the moad up to ie: 1,, 3, 4, 5, 6, 7, 8, 9. The we must add up the amout of all of them together, ad sice the row cotais ie terms, we must look for the ith part of the total to see if it is already aturally preset amog the umbers i the row; ad we will fid that the property of beig [oe] ith [of the sum] oly belogs to the [arithmetic] mea itself (Heath, 191, p.8). As we have metioed i Chapter 1, i the middle ages the term average referred to the equal apportiomet of a loss or expese icurred by a ship (or its cargo), i which case the idividual compesatio made by the owers (or isurers) of a ship or its cargo is i proportio to the value of their respective iterests. This otio of average represets the most documeted usage of the arithmetic mea durig that period. I the 17 th cetury, astroomers were makig several observatios of specific cosmic evets for cofirmatio purposes. They were faced with the problem of combiig observatios to come up with a sigle value that best represeted the true value of the quatity beig measured. Hald 3

39 (1998) states that for a log period of time the usual practice [by astroomers] for estimatig the true value was to select the best amog several observatios of the same object, the best beig defied by such criteria as the occurrece of good observatioal coditios, the exertio of special care, ad so o. Gradually, however, it became commo practice to use the arithmetical mea as a estimate of the true value. No theoretical foudatio for this practice seems to have existed before the works of Simpso ad Lagrage i the mid 18 th cetury. The problem of fidig the best estimate of a ukow parameter from a set of direct observatios of that parameter may be very difficult. It depeds o the distributio of the parameter, ad (as Gauss showed) oly whe the distributio is ormal is the arithmetic mea i every case the best estimate. (The precise sese of "best" is itself a complex problem that we shall avoid addressig). Therefore, the arithmetic mea is ot always the best choice for averagig a set of observatioal data. Steves (1955) states that i choosig a method of averagig physical magitudes, oe fudametal issue to be cosidered is the atural method of combiig them. Where magitudes are aturally combied by takig sums, the arithmetic mea is meaigful ad may be useful. However, where positive magitudes are aturally combied by takig products, the geometric average may be the most appropriate to use. To establish a geeral framework for the presetatio of the various meas ad the postulates o -ary meas, we itroduce the followig: Covetio Let X R, where R is the set of real umbers ad i N. We cosider a sequece of fuctios f i : X i R. For coveiece, we sometimes write f (a,b,...,c) lettig f represet the appropriate f i. We call X the domai of f. It will be useful for us at this poit to defie the aalogues for the most commo meas ecoutered: Defiitio The arithmetic mea, A = {A i } i N is: A(x 1,x,...,x ) = A = x 1 + x x. 33

40 The geometric mea, G is: G(x 1,x...,x ) = G = (x 1 x...x ) 1. The harmoic mea, H is: H(x 1,x...,x ) = H = 1 x 1 + x x 1 = 1 A(x 1 1,x 1,...,x 1 ). The root-mea-square, R, of the sequece is: R(x 1,x...,x ) = R = ( 1 (x 1 + x x )) 1. Remark Note that A ad R are defied for X = R. While G is defied oly for X = R 0 := {x R x 0}, ad H is defied for X = R >0 := {x R x 0}. The first three meas show above are clearly similar to the classical meas, while the root-mea-square is a obvious geeralizatio of the biary root-mea-square. The arithmetic mea ad the root-mea-square are widely used i mechaics (as i the defiitios of the ceter of gravity ad radius of gyratio), ad i the moder theory of statistics. The root-mea-square of the differeces of some variable from its arithmetic mea is the stadard deviatio (x i x) i=1. The geometric mea is used i the costructio of idex umbers i ecoomics. The harmoic mea is little used, except i special ivestigatios (Hutigto 197). A atural geeralizatio of these meas is referred to as power meas, sometimes kow as Cauchy meas (Bulle 198): 34

41 Defiitio Let r R \ {0}. The r th power mea is: M r = [ 1 i=1 x r i ] 1r. Remark Note that the domai of M r always cotais {x R x > 0}, but i some cases it is larger. Uless specified otherwise, we will take M r to refer to a sequece of fuctios with domai {x R x > 0}. Thus M r satisfies axiom PO (see below). I the cases r = -1, 1, ad, M r is the harmoic mea, the arithmetic mea ad the root-mea-square respectively. Although r is defied to be ozero, the followig theorem establishes that the geometric mea is a limitig case of M r as r teds to 0. Theorem 11. Let x 1,...,x be positive real umbers. The lim r 0 M (x 1,x,x 3,...,x ) = (x 1 x x 3...x ) 1 r. The followig proof is from Burrows (1986). Proof. Let y(r) = 1 x r i, where r 0. The y (r) = 1 x r i lx i. By the Mea Value Theorem i=1 i=1 y(r) = y(0) + ry (θ), where 0 < θ < r. Hece y(r) = 1 + ry (θ). Now, 1 r ly(r) = 1 r l(1 + ry (θ) = y (θ) + o(r) = 1 r (ry (θ) + +o(r ) 1 = lim r 0 r ly(r) = y (0) = 1 i=1 lx i. Sice y (r) is cotiuous. Therefore, takig the atilogarithm of this last result we get lim y(r) = (x 1x...x ) 1 r. r 0 35

42 4. The Axiomatic Theory of -ary Meas Begiig aroud 1900, several authors took up the problem of fidig ad aalyzig axiomatic characteristics for various -ary meas. The followig postulates appear i the various postulate systems we have reviewed, ad, thus, we give them special labels for coveiece. We demad the equatios to be true whe the terms are defied, i.e. whe all argumets belog to X. PO The domai of f is X = {x R x > 0} ad f i (x 1,x...,x i ) > 0. SY f is symmetric, i.e., it is idepedet of the order i which the quatities x 1,x,...,x X, are take, i.e., f (x 1,x...,x i,x j,...,x ) = f (x 1,x...,x j,x i,...,x ). DI f is diagoal, i.e., f (a,a,a,...,a) = a. IN f is iteral, i.e., a f (x 1,...,x ) b if a x i b for all i. HO f is homogeeous, i.e., for all k, f (kx 1,kx,kx 3,kx 4,...,kx ) = k M(x 1,x,...,x ), where x i X. OD f is odd, i.e., f ( x 1,..., x ) = f (x 1...x ) (Note that this a special case of HO). TR f is traslatio ivariat, i.e, f (k + x 1,...,k + x ) = k + f (x 1...x ) for ay k. AS f is "associative" i the sese that f (x 1,x...,x ) = f ( f i,..., f } {{ } i,x i+1,...x ), i times where f i = f (x 1,...x i ). AS f (x 1,x,x 3,x 4,...,x ) = M(m,m,x 3,x 4,...,x ), where m = f (x 1,x ). The earliest approach to the theory of meas by usig the postulatio method is Schimmack (1909 p. 18). He gave a set of axioms that completely characterize the arithmetic mea of positive umbers. Specifically, he proved the followig theorem (Schimmack 1909): Theorem 1. Let f be a sequece of fuctios such that f satisfies T R, OD, SY, ad AS. The f is the arithmetic mea. We refer the reader Schimmack (1909) for a elegat proof of the above theorem. Beetle (1915) established the complete idepedece of Schimmack s postulates. Beetle (1915) states, "The 36

43 otio of complete idepedece is much more restrictive tha the requiremet of idepedece. The requiremet for the latter is that o oe property is a logical cosequece of ay of the others. However, these properties are ot ecessarily devoid of iterrelatios. For example, it may well be that o-possessio of oe property implies possessio of aother. Complete idepedece implies either ay oe of them, or its egative, is a logical cosequece of ay combiatio formed by the others ad their egatives." To show the complete idepedece of Schimmack s four postulates, Beetle (1915) proved the existece of 4 types of systems, f, each defied ad real valued for all real values of its argumets, i which at least oe system possess ay give combiatio of the properties but does ot possess the remaiig properties. Gratta-Guiess (000) refers to Hutigto as oe of the major America postulatioists whose mai mathematical iterest was developig axiomatic systems for various mathematical cocepts ad establishig their cosistecy, idepedece, completeess, ad equivalece. Hutigto (197) exteded Schimmack s work by cosiderig fuctios f that satisfy the geeral postulates give below. He established the idepedece of these postulates i a maer similar Beetle s (1915) method i establishig the idepedece of Schimmack s postulates. Defiitio Hutigto s geeral postulates are: PO, HO, DI, SY, ad AS. We call f a Hutigto mea if it satisfies Hutigto s geeral postulates. Hutigto cocered himself with the arithmetic mea, geometric mea, the harmoic mea ad the root-mea-square. We will geeralize some of his results to power meas, which we defied earlier. Theorem 13. Suppose M r is the r th power mea, where r is a ozero real umber. The M r is a Hutigto mea. Proof. M r satisfies PO, sice each x i is positive. M r satisfies SY by the commutative law of additio. 37

44 [ ] k M r satisfies HO. Let k > 0. The M r (kx 1,kx,...,kx ) = r x r 1 +kr x r +...+kr x r 1 r = [ x r ] k 1 +x r xr r = km r. M r satisfies DI, sice M r (a,a,...,a) = a. M r satisfies AS. Suppose M r (x 1,x ) = m.the m = xp 1 +xp. Therefore, [ x r ] M r (x 1,x,x 3,x 4,...,x ) = 1 +x r 1 [ x r +...+xr r 1 +x r ] + xr 1 +xr 1 +x = x r r = [ m+m+x r ] x r 1 r = M r. Theorem 14. G is a Hutigto mea. Proof. G = (x 1 x...x ) 1. The G satisfies PO, sice the product ad powers of positive umbers are positive. G satisfies SY, sice multiplicatio is commutative. To show G satisfies HO, let k > 0. The ((kx 1 )(kx )...(kx )) 1 = (k (x 1 x x 3...x )) 1 = k(x 1 x x 3...x ) 1 = kg. G satisfies DI, sice G(a,a,...,a) = a. G satisfies AS. Suppose G = (x 1 x ) 1 = m x 1 x = m. The G = (x 1 x x 3...x ) 1 = G = (x 1 x x 3...x ) 1 = (m x 3...x ) 1 As we have metioed earlier, Hutigto (197) established several other properties that completely characterize each of the four meas, A, G, H, ad R. The followig theorem summarizes some of the results that Hutigto preseted. Theorem 15. Let f be a Hutigto mea. The: a) f = A if ad oly if f (1 x 1,1 x,...,1 x ) = 1 f (x 1,x,...,x ). b) f = H if ad oly if x 1 f ( x 1 1, x x 1,..., x x 1 ) = f (x 1,x,...,x ) f (x 1,x,...,x ) 1. 38

45 c) f = G if ad oly if f ( 1 x 1, 1 x,..., 1 x ) = 1 f (x 1,x,...,x ). d) f = R if ad oly if f ((1 x 1) 1,(1 x ) 1,...,(1 x ) 1 ) = (1 ( f (x1,x,...,x )) ) 1. The followig theorem geeralizes parts (a), (b), ad (d) of theorem 15. Theorem 16. Let f be a Hutigto mea. Suppose r 0. The f = M r if ad oly if f ((1 x r 1 ) 1 r,(1 x r ) 1 r,...,(1 x r ) 1 r ) = (1 ( f (x1,x,...,x )) r ) 1 r. (4.8) for all 0 < x i < 1. Proof. First we prove that if f = M r, the M r satisfies equatio 4.8. We have ( ) 1 M r ((1 x r 1 ) 1 r,(1 x r ) 1 r,...,(1 x r ) 1 r r ) = = ( 1 xr 1 +xr +...+xr ) 1 r = (1 (M r (x 1,x,...,x ) r ) 1 r. ((1 x 1 ) 1 r) r +((1 x r ) 1 r ) r +...+((1 x r ) 1 r ) r To prove the coverse, assume equatio 4.8. First, we show f (a,b) = M r (a,b). ( ) f (a,b) = (a r + b r ) 1 a b r f, (a r + b r ) 1 r (a r + b r ) 1 r b r (by HO) = (a r + b r ) 1 r f ((1,(1 (a r + b r ) 1 r (a r + b r ) 1 r (by algebraic idetities) ( { ( )} r ) 1 r = (a r + b r ) 1 b a r 1 f, (a r + b r ) 1 r (a r + b r ) 1 r (by 4.8)3.6. a r = ((a r + b r ) f (b,a) r ) 1 r (by HO & simple maipulatio). ) 39

46 Thus f (a,b) r = (a r + b r ) f (b,a) r. So f (a,b) r = (a r + b r ), ad f (a,b) r = ar +b r. Therefore, ( a r + b r ) 1 r f (a,b) =. Now we prove that f = M r for ay umber of argumets. Suppose that x 1,x,...x are give. We claim that for k = 1,..., there is a q such that x r 1 + xr xr k (k 1)qr > 0. To prove the claim, ote that x r 1 + xr xr k (k 1)qr > 0 for k =,..., if ad oly if x r 1 + xr xr k > (k 1)qr if ad oly if xr 1 +xr +...+xr k k 1 > q r, for k =,..., if ad oly if ( x r 1 +x r +...+xr k k 1 ( x r 1 +x r +...+xr k k 1 ) 1 r > q whe r > 0 ) 1 r < q whe r < 0 for k =,...,. So it is oly ecessary to pick q satisfyig fiitely may iequalities. The claim is thus proved. Now, M r (q,(x r 1 + xr qr ) 1 r ) = Mr (x 1,x ). Let Z k = (x r 1 + xr xr k (k 1)qr ) 1 r. Our choice of q, esures that Z k is the r th root of a positive umber. Also, from f (a,b) = M r (a,b), we have f (x 1,x ) = f (q,z ), ad i geeral f (Z k,x k+1 ) = f (q,z k+1 ), for k = 1,..., 1. So, from AS, we have 40

47 f (x 1,x,...x ) = f (q,z,x 3,...x ) = f (q,q,z 3,x 4,...x ) =... (4.9) = f (q,q,q,...,q,z ) Now, put a = M r (x 1,...,x ). The a = f (a,a,...,a) by DI = f (q,q,...,q,(a r ( 1)q r ) 1 r )) by 4.9 ad xi = a = f (x 1,x,...,x ) by 4.9 Now we give a proof part (c) of theorem 15. Proof. First, we show that G( 1 x 1, 1 x,..., 1 x ) = 1 G(x 1,x,...,x ). (4.10) We have G(x 1,x,x 3,...,x ) = (x 1 x x 3...x ) 1. Therefore, G( 1 x 1, 1 x,..., 1 x ) = ( 1 x 1 1 x... 1 x ) 1 1 = ( ) 1 x 1 x...x 1 = (x 1 x...x ) 1 1 = G(x 1,x,x 3,...,x ). 41

48 Now we prove the coverse. Suppose f ( 1 x 1, 1 x,..., 1 x ) = 1 f (x 1,x,...,x ). (4.11) Let a ad b be positive umbers. Usig equatio 4.11 for =, we have f ( 1 a, 1 b ) = 1 f (a,b) Multiplyig by ab ad usig HO, we get f (b,a) = f ( ab a, ab ) = ab b f (a,b) But by SY ad HO, the left had side is f (a,b) = f (b,a). Thus f (a,b) = ab ad f (a,b) = (ab) 1. (4.1) f (x 1,x,x 3,...,x ) = f (1,x 1 x,x 3,...,x ) by 4.9 ad AS = f (1,1,x 1 x x 3,x 4,...,x ) by 4.9 ad AS =... by 4.9 ad AS = f (1,1,...,1,x 1 x...x ). Set a = (x 1 x x 3...x ) 1 ad let each of the x i = a, we see that f (a,a,...,a) = f (1,1,...,1,a ) = a, by DI. Hece f (x 1,x,x 3,...,x ) = (x 1 x x 3...x ) 1. 4

49 Aother milestoe work o the properties of geeralized -ary meas that is cotemporary with Hutigto (197) is Nagumo (199). Nagumo proved the followig theorem: Theorem 17. Suppose M is a sequece of fuctios of real umbers satisfyig the postulates: SY, DI, AS, ad IN i additio, the property: For ay x 1 < x, we demad x 1 < M(x 1,x ) < x (a stregtheig of IN). The M is of the form: M(x 1,x,...,x ) = ϕ 1 ( i=1 ϕ(x i ) ), where ϕ(x) is a cotiuous mootoe icreasig fuctio with iverse ϕ 1. Dodd(1934) proved the complete idepedece of the postulates for meas of the type give by Nagumo (199). 4.3 Traslatio Ivariace Property of -ary Meas Hoeh & Nive (1985) proved that certai meas other tha A, while failig to satisfy T R, satisfy "traslatioal iequalities". Their theorem is : Theorem 18. Let A be the arithmetic mea, G be the geometric mea, H be the harmoic mea, ad R be the root-mea-square. Let a 1,a...,a,x be positive umbers, where the a i s are ot all equal. The: (i) A(x + a 1,x + a,...,x + a ) = x + A(a 1,a,...,a ). (ii) G(x + a 1,x + a,...,x + a ) > x + G(a 1,a,...,a ). (iii) H(x + a 1,x + a,...,x + a ) > x + H(a 1,a,...,a ). (iv) R(x + a 1,x + a,...,x + a ) < x + R(a 1,a,...,a ). First Hoeh & Nive proved lemma 4.0, i which they also proved part (i) of theorem 18 ad used the results from the lemma to prove the parts (ii) of theorem

50 We will preset the proof from Hoeh & Nive (1985) for lemma 4.0 ad part (ii) of theorem 18. After this, we shall state ad prove a theorem about M r that geeralizes parts (iii) ad (iv) of theorem 18. Proof. I part (ii) of theorem 18, we must show: G(x + a 1,x + a,...,x + a ) > x + G(a 1,a,...,a ). First apply the mea value theorem to the differetiable fuctio G(x) o the iterval [0,c]. Thus, G(a 1 +c,a +c,...,a +c) G(a 1,a,...,a ) c 0 = G (a 1 + θ,a + θ,...,a + θ) for 0 < θ < c. But G > 1 by part (b) of lemma 4.0. Therefore, G(a 1 +c,a +c,...,a +c) G(a 1,a,...,a ) c 0 > 1 ad G(a 1 + c,a + c,...,a + c) G(a 1,a,...,a ) > c + G (a 1,a,...,a ). Lemma 4.0. Let a 1,a...,a,x be positive real umbers, where the a i s are ot all equal. The: (a) da dx = 1, ad (b) dg dx > 1. Proof. Part (a): First, we will establish that A(x) = A(x + a 1,x + a,...,x + a ) = x + A(a 1,a,...,a ). A(x + a 1,x + a,...,x + a ) = x + a 1 + x + a x + a = x + a 1 + a a = x + a 1 + a a, = x + A(a 1,a,...,a ). Therefore, differetiatig A(x) with respect to the variable x we have da dx = d dx A(x + a 1,x + a,...,x + a ) = x + A(a 1,a,...,a ) = 1. 44

51 Part (b): For the geometric mea, ote that the derivative of the product i=1 (a i + x) is the ( 1) elemetary symmetric polyomial of the (a i + x), deoted here by S 1. For example, i the case = 3 the polyomial S would be S = (a 1 + x)(a + x) + (a 1 + x)(a 3 + x) + (a + x)(a 3 + x). I geeral, S 1 = ( j i (a j + x). i=1 Let G(x) = G(x + a 1,x + a,...,x + a ); therefore G = i=1 (a i + x). Thus, d dx G = G 1 dg dx = S 1. To prove that dg dx > 1, it suffices to show S 1 > G 1. This is othig more tha the arithmetic-geometric mea iequality applied to the terms of S 1 = [ j i (a j + x) ]. The geometric mea of these terms is the th root of their product, i=1 which is [(a 1 + x)(a + x)...(a + x)] 1, or G ( 1), because each (a j + x) appears i exactly 1 terms of S 1. Theorem 19. Suppose M r (x 1,x,...,x ) = ( xr 1 +xr +...+xr ) 1 r, where r 0. Also, suppose a 1,a,...,a are positive real umbers (ot all the same). The M r (x + a 1,x + a,...,x + a ) > M r (a 1,a,...,a ) if r < 1, ad M r (x + a 1,x + a,...,x + a ) < M r (a 1,a,...,a ) if r > 1. To prove the above theorem 19, we will use lemma 4.1 (the proof of lemma 4.1 will follow below). Proof. Whe r < 1. By lemma 4.1, d dx m r(x) > 1. The M r is a mootoe icreasig fuctio, ad together with a 1,a...,a,x beig positive real umbers, we have: M r (x + a 1,x + a,...,x + a ) > M r (a 1,a,...,a ). Whe r > 1. Similarly, by lemma 4.1, d dx m r (x) < 1. The M r is mootoe a decreasig fuctio, ad together with a 1,a...,a,x beig positive real umbers, we have: 45

52 M r (x + a 1,x + a,...,x + a ) < M r (a 1,a,...,a ) if r > 1. Lemma 4.1. Let a 1,a...,a,x be positive real umbers, where the a i s are ot all equal, ad r 0. Suppose m r (x) = M r (x + a 1,x + a,...,x + a ): The d dx m r(x) > 1 whe r < 1, ad d dx m r(x) < 1 whe r > 1. Proof. Suppose 0 < r < s. The M r (y 1,y...,y ) = ( yr 1 +yr +...+yr ) 1 r, ad usig the iequality, we have ( yr 1 + yr yr ) 1 y s r 1 + ys ys ) 1 s. Hece, m r (x) > m r 1 (x). This implies that (m r (x)) r 1 > 1 < ((x + a 1) r 1 + (x + a ) r (x + a ) r 1 ) r > 1 as r < 1 Now (m r (x)) r 1 ((x + a 1) r + (x + a ) r (x + a ) r ). So r(m r (x)) r 1 d dx m r(x) = r ((x + a 1) r 1 + (x + a ) r (x + a ) r 1 ); Thus, d dx m r(x) = 1 (x + a 1 ) r 1 + (x + a ) r (x + a ) r 1 (m r (x)) r 1 > < 1 as r > 1 r < 1. Based upo the theorem 18 ad lemma 4.0, Hoeh & Nive (1985) wet o to ote that for every positive value x, (G(x) A(x)) < 0 although (G(x) A(x)) is a icreasig fuctio, ad (R(x) A(x)) > 0 although (R(x) A(x)) is a decreasig fuctio. This observatio motivated 46

53 Hoeh & Nive (1985) to look at the limits of these fuctios as x teds to ifiity. Hoeh & Nive (1985) oted that the equatio: lim [F(x + a 1,x + a,...,x + a ) A(x + a 1,x + a,...,x + a ) = 0 x holds with ay oe of A, G, H, or R i place of F. Hoeh & Nive (1985) established the result by the followig theorem: Theorem 0. lim x [F(x + a 1,x + a,...,x + a ) x] = A(a 1,a,...,a ) holds with ay oe of A, G, H, or R i place of F. Proof. We begi with the case of F(x) = A(x). We have lim [F(x + a 1,x + a,...,x + a ) x] = lim [A(x + a 1,x + a,...,x + a ) x]. x x But x = A(x + a 1,x + a,...,x + a ) A(a 1,a,...,a ) by lemma 4.1. Therefore, lim [A(x + a 1,x + a,...,x + a ) x] = A(a 1,a,...,a ). x I the other three cases, it suffices to show F = H(x) ad F = R(x) ad usig the iequality H < G < R for the case F = G. To prove F = H, we use the result from calculus: c m x m + c m 1 x m c 0 lim x k m x m + c m 1 x m 1 = c m. (4.13) k 0 k m assumig k 0 0, we write H = H(x + a 1,x + a,...,x + a ) i the form H = G S 1 where G is the product give above ad S 1 is as above. Thus we write G xs 1 lim (H x) = lim. (4.14) x x S 1 Now S 1 is a polyomial i x of degree 1. The coefficiet of x 1 is, ad the coefficiet of x is ( 1)(a 1 + a a ). The coefficiet of x i G is 1, ad x 1 is (a 1 + a a ). Hece, i G xs 1, the terms of degree cacel, ad by applyig 4.13 to 4.14, we have lim x (H x) = lim G xs 1 x S 1, ad we get lim x (H x) = (a 1+a +...+a ) = A. 47

54 To prove the case F = R, we write lim x [R(x + a 1,x + a,...,x + a ) x] = lim(r x) = lim R x R+x. Expadig R x as a quadratic polyomial i x, we otice that the x terms cacel, so that R x = (a 1+a +...+a ) + (a 1 +a +...+a ). It follows that R x R+x (a 1+a +...+a) = + (a 1 +a +...+a ) x R x+1 we have divided the umerator ad deomiator by x. As x we ote that lim R x calculatio R x = 1 [( a 1 x + 1) + ( a x + 1) ( a x + 1) ]. Hece we have lim [R(x + a 1,x + a,...,x + a ) x] = lim(r x) = lim R x x R+x = (a 1+a +...+a) = A., where = 1 from the 4.4 Iequality Amog -ary Meas Oe of the most promiet property of -ary meas, at least from the theoretical stad poit, is of course the iequalities betwee the various meas. This topic has attracted the attetio of may mathematicia as evidet by the richess of the research i that area. We begi the sectio with the arithmetic-geometric mea iequality for -ary meas. Cauchy appears to have bee the first to state the theorem i its most geeral form. The theorem is listed i Cauchy s Cours d Aalyse (p ), which appeared i 181. We will preset two differet proofs for this iequality. The we give some commetary o this iequality. Next we exted this iequality to iclude the harmoic mea, ad we close the sectio with a theorem o iequalities betwee power meas. Theorem 1. Let x 1,x,x 3...,x be o-egative real umbers. The 1 (x 1 + x x ) (x 1 x...x ) 1, ad equality occurs if ad oly if all the x i s are equal. The first proof we preset is by Chrystal (1916). We chose this proof because it was used i a umber of college freshma algebra textbooks aroud the tur of the 0th cetury ad represets a elemetary approach to provig the iequality. Proof. Let a,b,...k be a sequece of o-egative real umbers. Cosider their geometric mea, (ab...k) 1. If a,b,...k are ot all equal, replace their greatest ad least of them, say a ad k, by a+k. 48

55 The ( a + k ) > ak, ad the result has bee to icrease the geometric mea of the sequece of umbers, while the arithmetic mea of the quatities, a+k a+k,b,c,..., is clearly the same as the arithmetic mea of a,b,c,...k. If the ew set of quatities are ot equal, replace the greatest ad the least as before, ad so o. By repeatig the process sufficietly ofte, we ca make all the quatities as early equal as we please, ad the the geometric mea of the sequece of umbers becomes equal to the arithmetic mea. But, sice the arithmetic mea remaied ualtered throughout, ad the geometric mea has bee icreased at each step, it follows that the first geometric mea, amely (abc...k) 1, is less tha the arithmetic mea, a+b+...+k. The ext proof we have selected is the oe give by Thacker (1851). We chose this proof because it is cocurret with Chrystal s proof, but it shows a approach that is more mathematically balaced yet still straight forward. Proof. If x > 0 ad is a iteger. The usig the biomial theorem, we ote the followig: (1 + x ) = 1 + x x + 1 ( 1)(1 ) x ,ad (4.15) 3 1 (1 + x 1 ) 1 = 1 + x x + 1 ( 1 1 )(1 3 Note that for ay positive iteger, we have the followig iequalities 1 ) x (4.16) 1 1 > 1 1 1, 1 > 1 1,... >... 49

56 Hece every term ivolvig i the series expasio i 4.15 is greater tha the correspodig term i the series expasio i 4.16, ad sice the terms of series i 4.15 ad 4.16 are positive, we have (1 + x ) > (1 + x 1 ) 1. (4.17) Now, let there be positive quatities a 1,a,...,a arraged i order of magitude with a 1 beig the least. The ( a1 + a + a a ) ( = a a 1+a +...+a a 1 a 1 ) ( ) 1 > a 1+a1 +a +...+a a 1 1 a 1 ( 1) by 4.17 Note that a 1 ( 1+a1 +a +...+a a 1 a 1 ( 1) ) 1 ( = a +a a a ) 1. 1 Therefore, we have ( ) a1 + a + a a > a 1 ( a + a a 1 ) 1 > ( a + a a ) 1 > 1 a ( a 3 + a a ) > a 3 ( a 4 + a a ) 3 > 3... a 1 + a > a 1 a Hece, by multiplicatio we get ( ) a1 + a +...a > (a 1 a...a ), or ( ) a1 + a +...a > (a 1 a...a ) 1. 50

57 We have give two proofs for this iequality for several reasos. I the first place, the iequality is iterestig i that it ca established i a large umber of ways. There are literally dozes of differet proofs for the arithmetic-geometric iequality that are based o ideas represetig a great variety of sources. I the secod place, it has a fudametal role i the theory of iequalities ad is the keystoe o which may other very importat results rest. I the third place, we ca use some of its cosequeces to solve a umber of maximizatio ad miimizatio problems (Beckebach 1961). Therefore, from the historical prospective ad its ubiquitousess i mathematics, this iequality truly deserve to be listed first. Just about every aspect of this iequality has bee ivestigated. Tug (1975) proved that the upper ad lower bouds of the differece betwee the arithmetic ad the geometric meas of quatities ca be give as follows: Theorem. Let x 1,x,...,x be positive real umbers. Suppose t ad T are the smallest ad largest values respectively of the give quatities, ad c = l[ T (T t) ]l T t l T t. The ( 1 ( T t) ) A G (ct + (1 c)t t c T 1 c ) We refer the reader to Tug (1975) for the proof of this theorem. Others mathematicias sought more exotic iequalities betwee the betwee the arithmetic ad geometric meas. For example, Kedlaya (1994) proved a cojecture made F. Hollad i 199 which states the followig: Theorem 3. Let x 1,x,...,x be positive real umbers. The arithmetic mea of the umbers x 1, x 1 x, 3 x 1 x x 3,..., x 1 x...x 51

58 does ot exceed the geometric mea of the umbers x 1, x 1x, x 1x x 3,..., x 1x...x 3 Now we establish the iequality betwee G ad H. Theorem 4. Let x 1,x,...,x be positive real umbers. The H G, ad H = G if ad oly if all the x i s are equal. Proof. Pick a 1,a,...,a to be positive real umbers. For the umbers a 1,a,...,a, by the arithmetic-geometric meas iequality, we have ( ) 1 a1 + a a 1 a 1 a a The 1 a 1 + a a 1 (a 1 a,...,a ) 1. Equality holds here if ad oly if a 1 1 = a 1 =... = a 1. Note by theorems 1 ad 4, we have: H G A, ad H = G = A if ad oly if all the x i s are equal. The followig theorem establishes a geeralized iequality betwee power meas. Theorem 5. Suppose p ad q are ozero real umbers such that p > q. The for ay positive umbers a 1,a,...,a, ( ap 1 + ap ap ) 1 p ( aq 1 + aq aq ) 1 q, 5

59 with equality holdig if ad oly if a 1 = a... = a. We preset the proof give by Schaumberger (1988). Proof. Suppose p > q > 0 or 0 > p > q, ad let f (x) = qx p + (p q) px q. The for x > 0: qx p + p q px q (4.18) with equality holdig if ad oly if x = 1. This follows from f (x) = qx p + (p q) px q has a absolute miimum at x = 1 (because f (x) = qpx p 1 qpx q 1 vaishes if ad oly if x = 1, ad f (1) = (qp(p q) > 0). Now cosider ay positive umbers a 1,a,...,a ad let A = ( aq 1 + aq aq ) 1 q. (4.19) Substitutig x i = a i A for 1 i i equatio 4.19 ad addig we get q( ap 1 +ap +...+ap A p ) + p q p( aq 1 +aq +...+aq A q ). Sice a q 1 + aq aq = A q, it follows that q( ap 1 + ap ap A p ) q (4.0) If p > q > 0, the equatio 4.19 gives a p 1 + ap ap A p, Which ca be writte as ( ap 1 + ap ap ) 1 p ( aq 1 + aq aq ) 1 q. If 0 > p > q, the divisio by q reverses the iequality ad equatio 4.19reduces to 53

60 a p 1 + ap ap A p. Raisig both sides to the 1 p power reverses the iequality agai, ad we have ( ap 1 + ap ap ) 1 p ( aq 1 + aq aq ) 1 q. Note, furthermore, that equality holds if ad oly if each of the substituted values x i = a i A which is equivalet to a 1 = a... = a. equals 1, Letp > 0 > q. The for x > 0, f (x) = qx p +(p q) px q has a absolute maximum at x = 1. Thus qx p + (p q) px q (4.1) with equality holdig if ad oly if x = 1. Agai let A = ( aq 1 + aq aq ) 1 q. Substitutig x i = a i A successively i i equatio 4.1, ad addig the iequalities, we obtai q( ap 1 + ap ap A p ) + p q p( aq 1 + aq aq A q ) = p. Thus, q( ap 1 + ap ap A p ) q. Sice 0 > q, divisio by q reverses the iequality ad leads to ( ap 1 + ap ap ) 1 p ( aq 1 + aq aq ) 1 q. 54

61 Equality holds if ad oly if each of the substituted values x i = a i A a 1 = a... = a equals 1, which is equivalet to 55

62 Chapter 5. Coclusio The history of meas is log ad lade with details as meas are used by ordiary people, experts, academicias, ad scietists to express a represetative umber that typifies a set of values. The atique meas appear to have bee well kow by the daw of Greek mathematics. After the 4th cetury A.D., the theory of meas appears to have come to a stadstill. With Pappus s documetatio of the te kow meas of his time i the 4th cetury A.D., it seems that the iterest i the theory of meas had waed. Boethius is the last kow aciet writer to metio the eleve classical meas as part of his work "Arithmetic", which is a commetary o works of Nicomachus ad Pappus (Gow 193). The ext referece to the classical meas after Boethius appeared as a quotatio by Ocreatus i his work "Prolgus I Helceph," which was writte i the 1 th or 13 th cetury A.D. (Heath 191). Thus, it seems reasoable to speculate that the oly meas that may have draw ay iterest to be metioed i mathematics literature betwee the 5 th cetury ad the 16 th cetury are the three atique meas which were treated as the oly types of meas. That the atique meas cotiued to be of iterest from the 5 th cetury to the 16 th cetury comes as o surprise, sice the arithmetic mea is ofte useful i commercial trasactios, the geometric mea was preserved by the use of the mea proportioal i geometry, ad the harmoic mea is closely tied to music theory. I Middle Ages, the golde sectio became a favorite topic of theological speculatio. May leared people, ispired by the argumets of the Pythagorias ad Platoists, sought ad foud i this proportio a key to the mystery of creatio, declarig that extreme ad mea ratios were the very priciple which the Supreme Architect had adopted i the cosmic ad global desig; hece, the title divie proportio bestowed upo this ratio (Datzig 1955). The late reaissace iterest i sciece ad aciet Greek mathematics brought about a ew iterest i the theory of meas as particular aspects of some physical pheomea ca best be expressed by usig their mea values. Advaces i the theory of statistics has show that the 56

63 arithmetic mea ca be used as a represetative of a set of observatio data (Buhler 1944). This outlook carried over quite aturally to fidig ew ways to express a mea of a set of values that best fits the purpose at had, ad thus theory of meas was rebor. This rich history motivates us to dig deeper ito the cocept of meas to uderstad the uderlyig foudatio i which it is couched. We close by otig that there is a proliferatio of meas i mathematics of which we oly touched o a few. This proliferatio may be attributed to the fact that ew ways to express a mea of a set of umbers arise i applicatios cotiuously. Ufortuately, our discussio did ot iclude may other types of meas that play a pivotal role i mathematics research such as the rich area of iterated meas, e.g. Gauss arithmetic-geometric mea, o-symmetric meas, ad weighted meas. Meas are coected with diverse areas of mathematics research from Fourier Series, error measuremets, to aggregatio ad social choice. With each ew mea developed, ways have to be foud that will relate this mea to the oes already kow. 57

64 Refereces [1] Allema, G.j.,Greek Geometry from Thales to Euclid, Hodges ad Figgs, [] Almkvist,G.ad Berdt,B. Gauss, Lade, the Arithmetic Mea, Ellipses, π, ad the Ladies Diaries.Ameri. math. Mothly, 95(1988),p [3] Beckebach E.F.A Class of Mea Value Fuctios. Ameri. math. Mothly, 57 (1950),p [4] Beckebach,E.F. ad Bellma, R., A itroductio to Iequalities, New York, Radom House, [5] Beetle, R. D. O the complete idepedece of Schimmack s postulate for the arithmetic mea, Mathematische Aale, Vol.76, 1915, p [6] Beetle, R. D. Bulleti of the America Mathematical Society, Vol., 1916, p [7] Bellma, R., Iequalities. Mathematics Magazie, Vol.8, 1954, p.1-6. [8] Bellma, R., O the arithmetic-geometric mea iequality. Mathematics Studet,Vol.4,1956/57, p [9] Bema, W. W. ad Smith, D.G., A Brief History of Mathematics, Chicago: Ope Court Publishig co., [10] Berlighoff, W. ad Gouvea, F., Math Through the Ages, Farmigto: Oxto House Publishers, 00. [11] Borwei, J. ad Borwei, P., Pi ad the AGM, New York: Wiley Itersciece, [1] Buhler W.K., Gauss: A Biographical Study. New York: Spriger, 1944,p [13] Bulle, P.S., Mitriovic, D.S., ad Vasic, P.M.,Meas ad Their Iequalities, Bosto: D. Reidel Publishig Compay,198. [14] Burrows, B.L. ad Talbot, R.F. Which Mea Do you Mea. It. J. Math. Educ. Sci. Techol., 1986, vol. 3, p [15] Carlso, B.C., Algorithms Ivolvig Arithmetic Ad Geometric Meas, Ameri. math. Mothly, 78(1971),p [16] Carlso, B.C., The Logarithmic Mea,Ameri. math. Mothly, 79(197),p [17] Chrystal, G., Algebra, Lodo: A ad C Black, 1916, p [18] Cox, D.A. The Arithmetic-Geometric Mea Of Gauss, L Eseigemet Mathematique, t.30 (1984), p [19] Datzig, T., Number The Laguage of Sciece Mathematics, New York: Macmilla,

65 [0] Datzig, T., The Bequest of the Greeks, Lodo: George Alle & Uwi LTD., [1] De Fietti, B., Sul Cocetto di media, Giorale dell istituto Italiao degli Attuari, Vol., 1931, p [] Dodd, E.L., The Covergece of Geeral Meas ad the Ivariace of Form of Certai Frequecy Fuctios. The America Joural of Mathematics, Vol. 49, No., (Apr., 197), p [3] Dodd, E.L., The complete Idepedece of Certai Properties of Meas. Aals of Mathematics, vol. 35,No.4,October, [4] Ercolao, J.L., Geometrical Iterpretatio of some classical iequalities. Math. Magazie. Vol. 45, 197, p [5] Ercolao, J.L., Remarks o the Neglected Mea. Mathematics Teacher. Vol. 66, 1973, p [6] Eves, H., Meas appearig i geometric figures. Math. Magazie, Vol. 76, No. 4, Oct. 003, p [7] Gallat, C., Proof Without Words:A truly geometric iequality. Mathematics Magazie, vol. 50,1977,p. 98. [8] Gow, J., A short History of Greek Mathematics, New York: G.E. Stechert & co., 193. [9] Gratta-Guiess, I, The Search for Mathematical Roots, , Piceto: Priceto Uiversity Press, 000. [30] Hald, A., A History of Mathematical Statistics from 1750 to 1930, New York: Joh Wiley & Sos, Publicatio [31] Heath,T., A History of Greek Mathematics, Vol. 1, Lodo: Oxford Uiversity Press, 191. [3] Heath, T. L., A Maual of Greek Mathematics, Dover Publicatio, New York, [33] Hoeh, L. ad Nive, I., Averages o the Move, Mathematics Magazie, Vol. 58, No. 3, (May, 1985),p [34] Hubbard, J. ad Moll, V., A geometric view of ratioal Lade trasformatio, Bull. Lodo Math. Soc., Vol. 35, 003, p [35] Hutigto, E.V., Sets of Idepedet Postulates for the Arithmetic Mea, the Geometric Mea, the Harmoic Mea, ad the the Root-Mea-Square. Trasactios of the America Mathematical Society, Vol. 9,No.1 (Ja. 197), p 1-. [36] Kedlaya, K., Proof of a mixed Arithmetic-mea, Geometric-Mea Iequality. America Mathematical Mothly, Vol. 101, No. 4 (Apr., 1994),p

66 [37] Klei, J. Greek Mathematical Thought ad the Origi of Algebra, Cambridge: M. I. T. Press, [38] Koold, C. ad Pollatsek, A., Data Aalysis as the Search for sigals i oisy processes. Joural for research i mathematics educatio, 00, vol. 33,p [39] Lagto, S., Gauss, recurrece relatio, ad the AGM, Talk delivered at sessio Mathematics i the Age of Euler. Joit Mathematics Meetigs, New Orleas, Louisiaa, 1/11/001. [40] Leach, E. b. ad Sholader, M.C., Exteded Mea Values, America Mathematical Mothly, Vol. 85, No. (Feb., 1978),p [41] Lewiter, M. ad Widulski, W., The Saga of Mathematics,Upper Saddle River: Pretice Hall, 00. [4] Li, Tug-Po The Power Mea ad the Logarithmic Mea, Ameri. math. Mothly, 81(1974),p [43] Madde, J., Some Notes o Meas, Upublished paper, 6/3/000. [44] Mays, M.E., Fuctios Which Parametrize Meas. The America Mathematical Mothly, Vol. 90, No.10, (Dec., 1983), p [45] Maziarz, E.A. ad Greewood, T., Greek Mathematical Philosophy, New York: Frederick Ugar Publishig co., [46] Moroey, M.J., O the Average ad Scatter. The World of Mathematics. Newma, J.R., Editor, Simo ad Schuster,New York, 1956, p [47] Moskovitz, D., A Aligmet Chart for Various Meas. The America Mathematical Mothly, Vol. 40, No.10, (Dec., 1933), p [48] Mosteller, F. ad Rourke, R., Probability with Statistical Applicatios, Lodo:Addiso-Wesley Publishig Co [49] Nagumo, M., Uber eie Klasse der Mittelwerte, Japaese Joural of Mathematics, Vol. 7, 1930, p [50] Oxford Eglish Dictioary, Oxford Uiversity Press, Oxford: 004. [51] Pitteger, A.O., The Logarithmic Mea i Variables, America Mathematical Mothly, Vol. 9, No. (Feb., 1985),p [5] Polya, G., O the Harmoic mea of two umbers, America Math. Moth., vol.57, 1960, p.6-8. Polya, G., The miimum fractio of the popular vote that ca elect the presidet of the Uited States. Mathematics Teacher, vol.54, 1961, p

67 [53] Ruthig, D., Proofs of the arithmetic-geometric mea iequality, It. J. Math. Educ. Sci. Techol., 198, vol. 13, p [54] Shaughsey, J.,ad Bergma, B., Thikig about Ucertaity, Wilso, P. editor,research ideas for the classroom, Macmilla New York, 1993,p [55] Schaumberger, N., Aother Proof of the Iequality betwee Power Meas. The College Mathematics Joural, Vol. 19, No. 1 (Ja., 1988) p [56] Schild, A., Geometry of the Meas. Mathematics Teacher. Vol. 67, 1974, p [57] Schimmack, R., Der Satz vom arithmetische Mittel i axiomatischer Begrudug. Mathematische Aale, vol. 68, 1909, p [58] Schoeberg, I. J., O the Arithmetic-Geometric Mea, Delta,vol.7, Fall 1977, p [59] Schoeberg, I. J., O the Arithmetic-Geometric Mea ad similar iterative algorithms, Mathematical Time Exposures, Mathematical Associatio of America, 198, p [60] Scott,W.R., Meas i Groups,America Joural of Mathematics, Vol. 74, No.3, (Jul., 195), p [61] Slay, J.C. ad Solomo, J.L.,A Mea Geeratig Fuctio, The Two-Year College Mathematics Joural, Vol. 1, No.1, (Ja., 1981),7-9. [6] Smith, D.E., History of Mathematics, Vol., New York: Dover,1951. [63] Steves, S.S., O the Averagig of Data, Sciece, Vol.11, No.3135, (Ja. 8, 1955), p [64] Sulliva, J.J., The electio of a presidet. Mathematics Teacher, vol.65, 197, p [65] Tag, J. O the costructio ad iterpretatio of meas, It. J. Math. Educ. Sci. Techol., 1983, Vol. 14, No.1, p [66] Thacker, A., Demostratio of the kow Theorem, that the Arithmetic Mea betwee ay umber of positive quatities is greater tha their geometric. Cambridge ad Dubli mathematical Joural, vol. 6, 1851, p.81=-83. [67] Thomas, I., Selectios Illustratig the History of Greek Mathematics, Vol.1;,Cambridge, Harvard Press, [68] Tug, S. H., O the lower ad upper bouds of the differece betwee the Arithmetic-Geometric Mea. Math. of Computatio,vol.9,July 1975, p [69] Vamaamurthy, M.K., Iequalities for Meas. Joural of Mathematical Aalysis ad Applicatios, Vol. 183, (1994), p [70] Watso, G.N., The Marquis ad the lad-aget. Math. Gaz., vol.17,1933, p

68 [71] Yu, Li-We, Some Geeralizatios of the Iequalities about Mea, Tamsui Oxford Jour. of Math. Sc.,Vol.18, 00, p

69 Vita Mabrouck Khalifa Faradj was bor o February 5, 1958, i Fort Lamy (ow N Djamea), Chad. He fiished his udergraduate studies at Louisiaa State Uiversity i May, He eared the degree of Master of Public Admiistratio from Louisiaa State Uiversity i August, He is curretly a cadidate for the degree of Master of Sciece i mathematics, which will be awarded i August