M f (r) = (1 + o(1)) f (r), M f (r) = (1 + o(1))m f (r), (1) (де M f (r) = max{ f(z) : z = r}, m f (r) = min{ f(z) : z = r}, f (r) = max{ a k r n k

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1 202 Математичний вiсник НТШ, т. 5, 2008 p. ПРО РЯДИ ДIРIХЛЕ З МОНОТОННИМИ КОЕФIЦI НТАМИ I ОСТАТОЧНIСТЬ ОПИСУ ВИНЯТКОВО МНОЖИНИ c2008 р. Ярослав СТАСЮК Львiвський нацiональний унiверситет iменi Iвана Франка, вул. Унiверситетська, 1, Львiв Редакцiя отримала статтю 12 липня 2008 р. Нехай S клас цiлих рядiв Дiрiхле F (z) = a n e z n з показниками, що задовольняють умову 0 n < sup{ j : j 0}, а n = ln a n, при цьому a n > a n1 (n 0). Вiдомо, що у випадку, коли ( n1 n ) 1 <, спiввiдношення sup{ F ( it) : t R} (, F ) inf{ F ( it) : t R}, де (, F ) = max{ a n e n : n 0}, виконуються при 0 зовнi деякоё винятковоё множини E скiнченноё логарифмiчноё мiри. У статтi доведено, що вказаний опис винятковоё множини E в загальному покращити не можна. Вiдомо, що в теорiё Вiмана-Валiрона отримуванi там асимптотичнi спiввiдношення виконуються зовнi виняткових множин.так у випадку цiлих функцiй f, зображуваних лакунарними степеневими рядами f(z) = a k z n k, асимптотичнi спiввiдношення k=0 M f (r) = (1 o(1)) f (r), M f (r) = (1 o(1))m f (r), (1) (де M f (r) = max{ f(z) : z = r}, m f (r) = min{ f(z) : z = r}, f (r) = max{ a k r n k : k 0} максимальний член ряду) виконуються при r зовнi деякоё множини E [1, ) скiнченноё логарифмiчноё мiри УДК ; MSC 2000: 30B50

2 Про ряди Дiрiхле ln -meas E = E[1,) d ln r<, як тiльки викону ться умова k=0 1 n k1 n k <. Цей результат належить Фентону (P. Fenton, 1978 [2]). У слабшiй формi (спiввiдношення (1) викону ться лише при r = r j ) доведення дано в [1]. У 2001 р. Т.М. Сало i О.Б. Скаскiв [3] довели, що опис винятковоё множини у теоремi Фентона покращити не можна. Цей факт вони отримали з подiбного твердження, встановленого для цiлих рядiв Дiрiхле F (z) = a n e z n (2) з невiд' мними зростаючими до показниками ( n ). При цьому непокращуваним описом винятковоё множини E у спiввiдношеннях якi за умови M(x, F ) = (1 o(1))m(x, F ), M(x, F ) = (1 o(1))(x, F ), (3) 1 n1 n < виконуються при x (x / E), скiнченнiсть ЁЁ мiри Лебега [5], тобто E[0,) dx < (в [4] асимптотичнi спiввiдношення (3) отримано при x = x k ). Тут M(x, F ) = sup{ F (x iy) : y R}, m(x, F ) = inf{ F (x iy) : y R} (x, F ) = max{ a n e x n : n 0}. Через (x) =(x, F ) = max{n : a n e xn = (x, F )} позначатимемо центральний iндекс ряду Дiрiхле (2). У 1994 р. О.Б. Скаскiв [6] отримав аналог теореми Фентона для цiлих рядiв Дiрiхле, послiдовнiсть дiйсних показникiв яких допуска як наявнiсть будь-якоё кiлькостi скiнченних точок скупчення, так i обмеженiсть. Власне, припуска ться лише, щоб послiдовнiсть ( n ) не досягала сво Ё точноё верхньоё межi, тобто: (j 0) : 0 j < sup{ k : k 0} =. (4)

3 204 Я.Стасюк Теорема 1 ([6]). Нехай F зобража ться цiлим рядом Дiрiхле (2), показники якого задовольняють умову (4), а { n : n 0} = { ln a j : j 0} впорядкована за неспаданням послiдовнiсть ( ln a j ). Якщо викону- ться умова 1 <, (5) n1 n то спiввiдношення F (x iy) = (1 o(1))a (x) e (xiy) (x) (6) викону ться при x (x / E 1, де E 1 множина скiнченноё логарифмiчноё мiри, тобто ln -meas E 1 = E 1 [1,) d ln x< ) рiвномiрно по y R. Подiбне твердження отримано для рядiв Дiрiхле з нульовою абсцисою абсолютноё збiжностi. Теорема 2 ([6]). Нехай F функцiя, яка зобража ться абсолютно збiжним у пiвплощинi {z :Rez < 0} рядом Дiрiхле (2), показники якого задовольняють умову (4) з =, а{ n : n 0} = {ln a j : j 0} впорядкована за неспаданням послiдовнiсть (ln a j ). Якщо викону ться умова (5), то спiввiдношення (6) викону ться при x 0 (x / E 2, де E 2 множина скiнченноё логарифмiчноё мiри на промiжку [1, 0), тобто ln 0 -meas E 2 = d ln(1/ x ) < ) рiвномiрно по y R. E 2 [1,0) Ми доводимо наступнi теореми, якi вказують на те, що опис виняткових множин, отриманих утеоремах 1 i 2, непокращуваним. Теорема 3. Для будь-якоё зростаючоё послiдовностi ( n ), яка задовольня умову (5), i для кожноё додатноё функцiё h(x), такоё що h(x) (x ), iснують цiлий ряд Дiрiхле вигляду (2) з коефiцi нтами a n = exp{ n }, показники якого задовольняють умову (4), множина E i стала >0, такi що F (x) (1 )(x, F ) (7) для всiх x E i h-meas E = E[1,) h(x)d ln x =. Теорема 4. Для будь-якоё зростаючоё послiдовностi ( n ), яка задовольня умову (5), i для кожноё додатноё функцiё h(x), такоё, що h(x) (x 0) iснують функцiя F, яка зобража ться абсолютно збiжним у пiвплощинi {z :Rez<0} рядом Дiрiхле з коефiцi нтами a n = exp{ n },

4 Про ряди Дiрiхле множина E i стала >0б такi, що (7) викону ться для всiх x E i h-meas E = h(x)d ln(1/ x ) =. E[1,0) Доведення теореми 3. Як i в [6] визначимо послiдовнiсть C n = max{ k k1 :1kn}. Позаяк з умови (5) виплива, що k k1 (k ), то зрозумiло, що C n (n ), атакож C n n=1 n n1 =. Виберемо зростаючу послiдовнiсть (n ), { яка задовольня умови: } 1) h( n ) C n (n 1); 2) n1 n max 1 n n1 ;1 n1 n при n 1, де довiльне фiксоване додатне число. Оскiльки функцiя h(x) зроста до, то умови вибору ( n ) несуперечливi. Виберемо тепер показники ( n ) з рекурентних спiввiдношень n = n1 ( n n1 )/ n (n 1), 0 =0. Очевидно, що n < n1 (n 0). Розглянемо ряд Дiрiхле (2) з щойно вибраними показниками та з коефiцi нтами a n = exp{ n }. За побудовою ма мо, що ln a n1 ln a n n (n ). Добре вiдомо, що n n1 у цьому випадку (x, F )=a n e x n ( x [ n ; n1 ]). (8) За теоремою Штольца ([7, с.25]) отриму мо 1 n ln a n (n ) i, отже ([8, с.85]), визначена в (2) функцiя F належить до класу цiлих рядiв Дiрiхле з послiдовнiстю показникiв ( n ). Для n 1 позначимо t n = ( ) 1 1 n1 n = n1 n n1 n. Зрозумiло, що 0 <t n < 1 i t n n1 > n (n 1). Тодi для x [ n1 t n, n1 ] I n ма мо a n1 e x n1 (x, F ) exp { ( 1 = exp{( n n1 )x( n1 n )} n1 n ) 1 } e. Отже, нерiвнiсть (7) викону тьсядляx n=1 I n. Розглянемо тепер {n1 h-meas I n = h(x)d ln x h( n )ln 1 = h( n )ln ( ) 1. { n1 t n t n ( ) n1 n Оскiльки n1 n (n ), тоln 1 n1 n 1 2 длявсiх n n 0. Отже, h-meas I n 2 n=n 0 Теорему 3 доведено. n=n 0 C n n1 n =. n1 n

5 206 Я.Стасюк Доведення теореми 4. Як i в доведеннi теореми 3, виберемо за послiдовнiстю ( n ) зростаючу послiдовнiсть C n (n ), для якоё n=1 C d n/( n1 n )=. Нехай t n =1 n1 n, де d довiльне фiксоване додатне число. Визначимо строго зростаючу до нуля послiдовнiсть вiд' мнихчисел ( n ) так, щоб вона задовольняла умови: 1) h(t n n1 ) C n ;2) n1 n /t n для всiх n 1. Неважко зрозумiти, що умови вибору послiдовностi ( n ) несуперечливi. Виберемо тепер показники ( n ) з рекурентнихспiввiдношень n = n1 ( n1 n )/ n (n 1), 0 =0. Зрозумiло, що n < n1 (n 0). Розглянемо ряд Дiрiхле вигляду (2). За побудовою i вибором послiдовностi n ма мо ln a n1 ln a n n n1 = n1 n n n1 n 0 (n ). (9) За теоремою Штольца ([7, с.25]), iсну границя lim n n / n =0. Звiдси, для фiксованого x<0 ма мо n x n =(1o(1))x n (n ). З умови (5) виплива, що n1 n (n ), а тому n1 n c 1 ( n 0), де c 1 деяке додатне число. Крiм того, n (n ), тому 1/ n1 c 2 > 0(n 0). Отже, n1 n = 1 { n1 ( n1 n ) c 1 c 2. Звiдки, n c 1 c 2 n. Тому exp{ n x n }exp{ x 2 nc 1c 2 } ( n n 0 ). Звiдси виплива, що ряд (2) абсолютно збiжний у пiвплощинi {z :Rez<0}. Позаяк з (9) виплива, що викону ться (8), то для всiх x [t n n1, n1 ] I n отриму мо a n1 e x n1 = exp{( n n1 )x( n1 n )} = (x, F ) = exp { ( n1 n ) ( x n1 1)} exp{( n1 n )(t n 1)} = e d, а отже, (7) викону ться з = e d для всiх x I n. Розглянемо h-meas I n = = h(x) d ln(1/ x ) h(t n n1 )ln t n n1 I n n1 ( ) ( d d = h(t n n1 )ln 1 C n ln 1 n1 n n1 n Оскiльки ln ( d ) d 1 =(1o(1)) (n ), то iз n1 n n1 n розбiжностi ряду C n /( n1 n ) отриму мо h-meas ( ) I n =. n=1 Теорему 4 доведено. ).

6 Про ряди Дiрiхле [1] Erdos P., Macintyre A.J. Integral functions with gap power series // Proc. Edinburgh Math. Soc. (2) P [2] Fenton P.C. The minimum modulus of gap power series // Proc. Edinburgh Math. Soc P [3] Salo T.M., Skaskiv O.B., Trakalo O.M. On the best possible description of exeptional set in Wiman-Valiron theory for entire functions // Матем. студiё , 2. С [4] Srivastava R.P. On the entire functions and their derivatives represented by Dirichlet series // Ganita , 2. P [5] Скаскiв О.Б. Максимум модуля i максимальний член цiлого ряду Дiрiхле // Доп АН УРСР, сер. А С [6] Скаскив О.Б. О минимуме модуля суммы ряда Дирихле с ограниченной последовательностью показателей // Мат. заметки. Т.56, С [7] Демидович Б.П. Сборник задач и упражнений по математическому анализу. М.: Наука, с. [8] Леонтьев А.Ф. Целые функции. Ряды експонент. М.: Наука, ON THE DIRICHLET SERIES WITH MONOTONOUS COEFFICIENTS AND THE FINALITY OF DESCRIPTION OF THE EXCEPTIONAL SET Yaroslav STASYUK Ivan Franko National University of Lviv 1 Universytetska Str., Lviv 79000, Ukraine Let S be the class of entire Dirichlet series F (z) = a n e zn with exponents and coecients satisfying the conditions 0 n < sup{ j : j 0}, a n > a n1 (n 0); n = ln a n. It is known that if ( n1 n ) 1 <, then the relations sup{ F ( it) : t R} (, F ) inf{ F ( it) : t R}, hold as outside a certain exceptional set E of the nite logarithmic measure where (, F ) = max{ a n e n : n 0}. In this paper we prove that the above description of exceptional set E in general cannot be improved.

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