ENTIRE DIRICHLET SERIES AND h − MEASURE OF EXCEPTIONAL SETS
Keywords:
Dirichlet series, maximal term, maximum modulus, minimum modulus, exceptional set, h-measure.Abstract
For entire Dirichlet series $F(z) =\sum_{n=0}^{+\infty} a_ne^{z\lambda_n},$ $0\le \lambda_n \uparrow +\infty$ $(n\to+\infty) ,$ we establish necessary conditions for the relation $\sup\{|F(x+iy)|: y \in\mathbb{R}\}= (1+o(1)) \max\{|a_n| e^{x\lambda_n}: n \ge 0\}$to hold when $x\to+\infty$ outside some set $E$ such that
$\int_E dh (x) <+\infty,$ where $h$ is positive continuous function increasing to $+\infty$ on $[0;+\infty)$ with non-increasing derivative.
References
1. Скаскив О.Б. Максимум модуля і максимальний член цілого ряду Діріхле / О.Б. Скаскив // Доп. АН УРСР, сер. А. – 1984. – 11. – С. 22-24.
2. Srivastava R.P. On the entire functions and their derivatives represented by Dirichlet series / R.P. Srivastava // Ganita. – 1958. – V.9, 2. – P.82-92.
3. Карлесон Л. Избранные проблемы теории исключительных множеств / Л. Карлесон. – М.: Мир, 1971. – 128 с.
4. Salo T.M. On the best possible description of exceptional set in Wiman–Valiron theory for entire functions / T.M. Salo, O.B. Skaskiv, O.M. Trakalo // Mat. Stud. – 2001. – V.16, No2. – P. 131-140.
5. Salo T.M. The minimum modulus of gap power series and h-measure of exceptional sets / T.M. Salo, O.B. Skaskiv // arXiv: 1512.05557v2 [math. CV] 21 Dec 2015. – 13 p.
6. Леонтьев А.Ф. Целые функции. Ряды експонент / А.Ф. Леонтьев. – М.: Наука, 1983. – 176 с.
2. Srivastava R.P. On the entire functions and their derivatives represented by Dirichlet series / R.P. Srivastava // Ganita. – 1958. – V.9, 2. – P.82-92.
3. Карлесон Л. Избранные проблемы теории исключительных множеств / Л. Карлесон. – М.: Мир, 1971. – 128 с.
4. Salo T.M. On the best possible description of exceptional set in Wiman–Valiron theory for entire functions / T.M. Salo, O.B. Skaskiv, O.M. Trakalo // Mat. Stud. – 2001. – V.16, No2. – P. 131-140.
5. Salo T.M. The minimum modulus of gap power series and h-measure of exceptional sets / T.M. Salo, O.B. Skaskiv // arXiv: 1512.05557v2 [math. CV] 21 Dec 2015. – 13 p.
6. Леонтьев А.Ф. Целые функции. Ряды експонент / А.Ф. Леонтьев. – М.: Наука, 1983. – 176 с.
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Published
2019-02-13
How to Cite
Скасків, О. Б., & Сало, Т. М. (2019). ENTIRE DIRICHLET SERIES AND h − MEASURE OF EXCEPTIONAL SETS. PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, (1(37), 37–41. Retrieved from https://pvntsh.nung.edu.ua/index.php/number/article/view/62
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Section
Mathematics and Mechanics
