On  Paley's effect for entire functions

P. V. Filevych

Let $M_f(r)$ be the maximum modulus of an entire function $f$, $T_f(r)$ be its Nevanlinna characteristic, $\Phi$ be a convex function such that $\Phi(x)/x\to+\infty$ as $x\to+\infty$, and $\cal A(\Phi)$ be the class of entire functions $f$ such that $\ln M_f(r)\le\Phi(\ln r)$ as $r\ge r_0(f)$. It is shown that for the relation $\ln M_f(r) \sim T_f(r)$ (or $\ln M_f(r) =O(T_f(r))$) as $r\to+\infty$ for each entire function $f\in\cal A(\Phi)$ it is necessary and sufficient that $\Phi(x)=O(x^2)$ as $x\to+\infty$.

1. Paley R. E. A. C. A note on integral functions, Proc. Cambridge Philos. Soc. 28 (1932), 262–265.

2. Гольдберг A. A., Островский И. В. Об эффекте Пейли для целых характеристических функций и целых функций, представленных рядами Дирихле, Теория функц., функц. анализ и их прил. 43 (1985), 18–23.

3. Zabolotskii N. V., Sheremeta M. N. On the slow growth of the main characteristics of entire functions, Math. Notes 65 (1999), No. 2, 168–174. Trans. from Mat. Zametki 65 (1999), No. 2, 206–214.

4. Filevych P. V. On the slow growth of power series convergent in the unit disk, Mat. Studii, 16 (2001), No. 2, 217–221.

5. Pólya G., Szeg o G. Aufgaben und Lehrsätze aus der Analysis, Zweiter Band, Springer-Verlag, Berlin-Göttingen-Heidelberg-New York (1964).

6. Hayman W. K. Meromorphic functions, Clarendon Press, Oxford (1964).

	On Paley's effect for entire functions
Author	P. V. Filevych Faculty of Mechanics and Mathematics, Lviv Ivan Franko National University
Abstract	Let $M_f(r)$ be the maximum modulus of an entire function $f$, $T_f(r)$ be its Nevanlinna characteristic, $\Phi$ be a convex function such that $\Phi(x)/x\to+\infty$ as $x\to+\infty$, and $\cal A(\Phi)$ be the class of entire functions $f$ such that $\ln M_f(r)\le\Phi(\ln r)$ as $r\ge r_0(f)$. It is shown that for the relation $\ln M_f(r) \sim T_f(r)$ (or $\ln M_f(r) =O(T_f(r))$) as $r\to+\infty$ for each entire function $f\in\cal A(\Phi)$ it is necessary and sufficient that $\Phi(x)=O(x^2)$ as $x\to+\infty$.
Keywords	maximum modulus, entire function $f$, convex function
DOI	doi:10.30970/ms.19.1.37-41
Reference	1. Paley R. E. A. C. A note on integral functions, Proc. Cambridge Philos. Soc. 28 (1932), 262–265. 2. Гольдберг A. A., Островский И. В. Об эффекте Пейли для целых характеристических функций и целых функций, представленных рядами Дирихле, Теория функц., функц. анализ и их прил. 43 (1985), 18–23. 3. Zabolotskii N. V., Sheremeta M. N. On the slow growth of the main characteristics of entire functions, Math. Notes 65 (1999), No. 2, 168–174. Trans. from Mat. Zametki 65 (1999), No. 2, 206–214. 4. Filevych P. V. On the slow growth of power series convergent in the unit disk, Mat. Studii, 16 (2001), No. 2, 217–221. 5. Pólya G., Szeg o G. Aufgaben und Lehrsätze aus der Analysis, Zweiter Band, Springer-Verlag, Berlin-Göttingen-Heidelberg-New York (1964). 6. Hayman W. K. Meromorphic functions, Clarendon Press, Oxford (1964).
Pages	37-41
Volume	19
Issue	1
Year	2003
Journal	Matematychni Studii
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