ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE


 Peter Harrington
 1 years ago
 Views:
Transcription
1 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau Itroductio ad prelimiaries We deote by U r the disc { C : < r}, < r, U U Let A be the class of fuctios f which are aalytic i U ad f ( f ( Let S be the class of the fuctios f A which are uivalet i U Defiitio Let f ad g be two aalytic fuctios i U We say that f is subordiate to g, f p g, if there exists a fuctio ϕ aalytic i U, which satisfies ϕ(, ϕ( < ad f( g(ϕ( i U Defiitio A fuctio L : U x I C, I [o,, L(, t is a Loewer chai, or a subordiatio chai if L is aalytic ad uivalet i U for all U ad for all t, t I, t < t, L(, t p L(, t Lemma [] Let r (, ] ad let L(, t a (t +, a (t be aalytic i for all t I ad locally absolutely cotiuous i I, locally uiform with respect to U r U r For almost all t I suppose: L,t L,t ( ( p(,t, t U r where p is aalytic i U ad satisfies Re p(, t >, U, t I L(,t If a (t for t ad forms a ormal family i a ( t U r the for each t I, L has a aalytic ad uivalet extesio to the whole disc U I the theory of uivalet fuctios a iterestig problem is to fid those itegral operators, which preserve the uivalece, respectively certai classes of 7
2 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia uivalet fuctios The itegral operators which trasform the class S ito S are preseted i the theorems A, B ad C which follow The itegral operators studied by Kim ad Meres is that from the theorem: Theorem A [] If f S, the for C, belogs to the class S ( F ( du the fuctio F defied by: u A similar result, for other itegral operator has bee obtaied by Pfaltgraff i: Theorem B [3] If f S, the for C, belogs to the class S ( G ( [ f ' ( u ] du, the fuctio G defied by: A itegral operator differet of ( ad ( is obtaied by Silvia Moldoveau ad NN Pascu i the ext theorem: Theorem C [] If f S, the for C,, the fuctio I defied by: (3 I ( f ( udu belogs to the class S I this ote, usig the subordiatio chais method, we obtai sufficiet coditios for the regularity ad uivalece of the itegral operator: ( ( H ( d u where f A,,,, 7
3 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia Mai results Theorem Let f, f,, f A, C, < ad,,, R,,,, If: f ( (5 ( (, ( U,,, f ( the the fuctio H defied by ( is aalytic ad uivalet i U Proof Because ( f f ( f ( + a + is aalytic i U, there ( f f ( f ( exists a umber r (, ] such that for ay f( f( f U r The for the fuctio choose the uiform brach equal to at the origi, aalytic i (6 + b +, ad we have: ' ( U r : we ca f( f( f( g ( + b + e t (7 ( u g u du g(,t, where: t b ( + t (8 g (,t e + + e + + Let we cosider the fuctio: t t t ( t 3( (9 g,t f (,t + ( e e e g ( e Sice < we have g 3(,t e ( + e for ay t I ad it results that there exists r (, r ] such that g 3 (, t i U for all t I For the t t r 73
4 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia fuctio [g 3 (, t] / we ca choose a uiform brach, aalytic i U r for ay t I It results that the fuctio: where: ( L(, t [g 3 (, t] / [g (, t] /, t e ( ( g (,t du + t + ( e e ( f ( e f ( e is aalytic i U r Usig Lemma we will prove that L is a subordiatio chai We observe that L(, t a (t +, where: ( a (t e t ( + e t / Because < we have a (t for all t I ad t t t ad < t a ( t e ( e t + (3 [ ] t Re ( lim a ( t lim e t t if Re > or r r It follows that Let p : U r I, L(,t a ( t forms a ormal family of aalytic fuctios i L (,t p (,t L (,t t U r, I order to prove that p has a aalytic extesio with positive real part i U, for all t I it is sufficiet to prove that the fuctio: 7
5 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia (5 p (,t w (,t is aalytic i U for t I ad p (,t + (6 w (, t <, ( U, t I But w (,t < max w (,t iθ w ( e,t, θ R ad it is sufficiet that θ t (7 w ( e i,, ( t >, or (8 ( u ( u f'( u + f'( u + + f'( u u f '( u ( ( u, from (5 where u e t e iθ, u U, u e t It results (6 Hece the fuctio L is a subordiatio chai ad L(, t H ( from ( is aalytic ad uivalet i U Theorem If f, f,, f S,,, the for C,, the fuctio H defied by ( belogs to the class S Proof Because f S,, we have: the ad f '( f ( +, ( U, f '( ( ( + < f ( f '( ( ( ( + < f ( because The, from Theorem we obtai that H S 75
6 Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia Example Let f (, f ( ad f 3 ( ( 5 The, for,, 3 we have: 3 3 f ( f3 ( ( ( 3 ( f + + ( ( + ( + ( H ( du ( For 3 H u ( 3 3 ( d 3 Refereces [] YJ, Kim, EP, Meres: O a itegral of powers of a spirallie fuctio Kyugpoo Math J, vol, r, (97, p 9 [] Silvia Moldoveau, NN, Pascu: Itegral operators which preserve the uivalece Mathematica (Cluj, 3 (55, r, (99, p [3] J Pfaltgraff: Uivalece of the itegral ( f '( c Bull Lodo Math Soc 7, (975, r 3, p 556 [] Ch, Pommeree: Uber die subordiactio aalytischer Fuctioe J Reie Agew Math 8 (965, p Authors: Maria E Gageoea ad Silvia Moldoveau  Trasilvaia Uiversity of Braşov Departamet of Mathematics Braşov 76
4. Trees. 4.1 Basics. Definition: A graph having no cycles is said to be acyclic. A forest is an acyclic graph.
4. Trees Oe of the importat classes of graphs is the trees. The importace of trees is evidet from their applicatios i various areas, especially theoretical computer sciece ad molecular evolutio. 4.1 Basics
More informationNo Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional Sample Covariance Matrices
No igevalues Outside the Support of the Limitig Spectral Distributio of Large Dimesioal Sample Covariace Matrices By Z.D. Bai ad Jack W. Silverstei 2 Natioal Uiversity of Sigapore ad North Carolia State
More informationThe Review of Economic Studies Ltd.
The Review of Ecoomic Studies Ltd. Walras' Tâtoemet i the Theory of Exchage Author(s): H. Uzawa Source: The Review of Ecoomic Studies, Vol. 27, No. 3 (Ju., 1960), pp. 182194 Published by: The Review of
More informationConsistency of Random Forests and Other Averaging Classifiers
Joural of Machie Learig Research 9 (2008) 20152033 Submitted 1/08; Revised 5/08; Published 9/08 Cosistecy of Radom Forests ad Other Averagig Classifiers Gérard Biau LSTA & LPMA Uiversité Pierre et Marie
More informationSOME GEOMETRY IN HIGHDIMENSIONAL SPACES
SOME GEOMETRY IN HIGHDIMENSIONAL SPACES MATH 57A. Itroductio Our geometric ituitio is derived from threedimesioal space. Three coordiates suffice. May objects of iterest i aalysis, however, require far
More informationHow Has the Literature on Gini s Index Evolved in the Past 80 Years?
How Has the Literature o Gii s Idex Evolved i the Past 80 Years? Kua Xu Departmet of Ecoomics Dalhousie Uiversity Halifax, Nova Scotia Caada B3H 3J5 Jauary 2004 The author started this survey paper whe
More informationWhich Extreme Values Are Really Extreme?
Which Extreme Values Are Really Extreme? JESÚS GONZALO Uiversidad Carlos III de Madrid JOSÉ OLMO Uiversidad Carlos III de Madrid abstract We defie the extreme values of ay radom sample of size from a distributio
More informationType Less, Find More: Fast Autocompletion Search with a Succinct Index
Type Less, Fid More: Fast Autocompletio Search with a Succict Idex Holger Bast MaxPlackIstitut für Iformatik Saarbrücke, Germay bast@mpiif.mpg.de Igmar Weber MaxPlackIstitut für Iformatik Saarbrücke,
More informationON THE EVOLUTION OF RANDOM GRAPHS by P. ERDŐS and A. RÉNYI. Introduction
ON THE EVOLUTION OF RANDOM GRAPHS by P. ERDŐS ad A. RÉNYI Itroductio Dedicated to Professor P. Turá at his 50th birthday. Our aim is to study the probable structure of a radom graph r N which has give
More informationStéphane Boucheron 1, Olivier Bousquet 2 and Gábor Lugosi 3
ESAIM: Probability ad Statistics URL: http://wwwemathfr/ps/ Will be set by the publisher THEORY OF CLASSIFICATION: A SURVEY OF SOME RECENT ADVANCES Stéphae Bouchero 1, Olivier Bousquet 2 ad Gábor Lugosi
More informationEverything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask
Everythig You Always Wated to Kow about Copula Modelig but Were Afraid to Ask Christia Geest ad AeCatherie Favre 2 Abstract: This paper presets a itroductio to iferece for copula models, based o rak methods.
More informationTesting for Welfare Comparisons when Populations Differ in Size
Cahier de recherche/workig Paper 039 Testig for Welfare Comparisos whe Populatios Differ i Size JeaYves Duclos Agès Zabsoré Septembre/September 200 Duclos: Départemet d écoomique, PEP ad CIRPÉE, Uiversité
More informationSolutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014
Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the
More informationBOUNDED GAPS BETWEEN PRIMES
BOUNDED GAPS BETWEEN PRIMES ANDREW GRANVILLE Abstract. Recetly, Yitag Zhag proved the existece of a fiite boud B such that there are ifiitely may pairs p, p of cosecutive primes for which p p B. This ca
More informationHOW MANY TIMES SHOULD YOU SHUFFLE A DECK OF CARDS? 1
1 HOW MANY TIMES SHOULD YOU SHUFFLE A DECK OF CARDS? 1 Brad Ma Departmet of Mathematics Harvard Uiversity ABSTRACT I this paper a mathematical model of card shufflig is costructed, ad used to determie
More informationSignal Reconstruction from Noisy Random Projections
Sigal Recostructio from Noisy Radom Projectios Jarvis Haut ad Robert Nowak Deartmet of Electrical ad Comuter Egieerig Uiversity of WiscosiMadiso March, 005; Revised February, 006 Abstract Recet results
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationSystemic Risk and Stability in Financial Networks
America Ecoomic Review 2015, 105(2): 564 608 http://dx.doi.org/10.1257/aer.20130456 Systemic Risk ad Stability i Fiacial Networks By Daro Acemoglu, Asuma Ozdaglar, ad Alireza TahbazSalehi * This paper
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationDoug Ravenel. October 15, 2008
Doug Ravenel University of Rochester October 15, 2008 s about Euclid s Some s about primes that every mathematician should know (Euclid, 300 BC) There are infinitely numbers. is very elementary, and we
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More informationA Kernel TwoSample Test
Joural of Machie Learig Research 3 0) 73773 Subitted 4/08; Revised /; Published 3/ Arthur Gretto MPI for Itelliget Systes Speastrasse 38 7076 Tübige, Geray A Kerel TwoSaple Test Karste M. Borgwardt Machie
More informationThe Arithmetic of Investment Expenses
Fiacial Aalysts Joural Volume 69 Number 2 2013 CFA Istitute The Arithmetic of Ivestmet Expeses William F. Sharpe Recet regulatory chages have brought a reewed focus o the impact of ivestmet expeses o ivestors
More informationThe Unicorn, The Normal Curve, and Other Improbable Creatures
Psychological Bulleti 1989, Vol. 105. No.1, 156166 The Uicor, The Normal Curve, ad Other Improbable Creatures Theodore Micceri 1 Departmet of Educatioal Leadership Uiversity of South Florida A ivestigatio
More informationSpinout Companies. A Researcher s Guide
Spiout Compaies A Researcher s Guide Cotets Itroductio 2 Sectio 1 Why create a spiout compay? 4 Sectio 2 Itellectual Property 10 Sectio 3 Compay Structure 15 Sectio 4 Shareholders ad Directors 19 Sectio
More informationA new continuous dependence result for impulsive retarded functional differential equations
CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas
More informationWhich Codes Have CycleFree Tanner Graphs?
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 6, SEPTEMBER 1999 173 Which Coes Have CycleFree Taer Graphs? Tuvi Etzio, Seior Member, IEEE, Ari Trachteberg, Stuet Member, IEEE, a Alexaer Vary,
More informationThings Your Next Firewall Must Do
10 Thigs Your Next Firewall Must Do Itroductio: 10 Thigs Your Next Firewall Must Do Much has bee made about brigig applicatio visibility ad cotrol ito etwork security. The reaso is obvious: applicatios
More informationCrowds: Anonymity for Web Transactions
Crowds: Aoymity for Web Trasactios Michael K. Reiter ad Aviel D. Rubi AT&T Labs Research I this paper we itroduce a system called Crowds for protectig users aoymity o the worldwideweb. Crowds, amed for
More informationJ. J. Kennedy, 1 N. A. Rayner, 1 R. O. Smith, 2 D. E. Parker, 1 and M. Saunby 1. 1. Introduction
Reassessig biases ad other ucertaities i seasurface temperature observatios measured i situ sice 85, part : measuremet ad samplig ucertaities J. J. Keedy, N. A. Rayer, R. O. Smith, D. E. Parker, ad M.
More information