Some classes of entire functions in which the
Wiman-Valiron inequality can be almost certainly improved (in Ukrainian)

P. Filevych

For entire functions of the form $\Sigma \, a_n e^{i\theta_n t}z^n$ where the $\theta_n$ are integers satisfying condition $(\theta_{k+1}-\theta_n)/(\theta_k-\theta_n) \ge 1+1/\varphi(k-n), k>n,(\varphi(x)\nearrow +\infty, x\to +\infty, \varphi(0) \ge 1, \varphi(x) \le x (x\ge x_0) )$ it is proved that Wiman's inequality can be improved to $$ M_f(r,t)< \mu_f(r)\ln^{1/4}\mu_f(r)\ln^{1+\varepsilon}\ln\mu_f(r) \Bigl(v(e\mu_f(r))+ \varphi^{1/2}\Bigl(\frac{\ln^{5/8}\mu_f(r) \ln^{1+\varepsilon} \ln \mu_f(r)} {v^{1/2}(\mu_f(r))}\Bigr)\Bigr) $$ $(v(x) \nearrow +\infty, \; x \to +\infty, \; 0 < v(e^x) \le x^{1/4} \ln^{2/5} x)$, for almost every $t$ and all $r > 1$ excepting a set $E(\delta,t)$ of finite logarithmic measure.

1. Ж.-П. Кахан. *Случайные функциональные ряды*. — М.: Мир, 1973. 302 с.

2. P.C. Rosenbloom. *Probability and entire functions* // Studies Math. Analysis and Related Topics. Stanford: Calif. Univ. Press, 1962. P.325–332.

3. P. Lévy. *Sur la croissance des fonctions entières* // Bull. Soc. Math. France. 1930. V.58. P.29–59, 127–149.

4. P. Erdős, A. Rényi. *On random entire functions* // Zast. Matem. 1969. V.10. P.47–55.

5. P. Erdős. *On the uniform but not absolute convergence of power series with gaps* // Ann. Soc. Polon. Math. 1952. V.25. P.162–168.

6. J.M. Steele. *Sharper Wiman Inequality for Entire Functions with Rapidly Oscillating Coefficients* // J. of Math. Anal. and Appl. 1987. V.123. P.550–558.

7. J. Jakubowski, S. Kwapień. *On multiplicative systems of functions* // Bull. Acad. Pol. Sci. 1979. V.27. P.689–694.

Let me know if you'd like these added to your existing document or further standardized.

	Some classes of entire functions in which the Wiman-Valiron inequality can be almost certainly improved (in Ukrainian)
Author	P. Filevych Lviv University, Department of Mechanics and Mathematics
Abstract	For entire functions of the form $\Sigma \, a_n e^{i\theta_n t}z^n$ where the $\theta_n$ are integers satisfying condition $(\theta_{k+1}-\theta_n)/(\theta_k-\theta_n) \ge 1+1/\varphi(k-n), k>n,(\varphi(x)\nearrow +\infty, x\to +\infty, \varphi(0) \ge 1, \varphi(x) \le x (x\ge x_0) )$ it is proved that Wiman's inequality can be improved to $$ M_f(r,t)< \mu_f(r)\ln^{1/4}\mu_f(r)\ln^{1+\varepsilon}\ln\mu_f(r) \Bigl(v(e\mu_f(r))+ \varphi^{1/2}\Bigl(\frac{\ln^{5/8}\mu_f(r) \ln^{1+\varepsilon} \ln \mu_f(r)} {v^{1/2}(\mu_f(r))}\Bigr)\Bigr) $$ \((v(x) \nearrow +\infty, \; x \to +\infty, \; 0 < v(e^x) \le x^{1/4} \ln^{2/5} x)\), for almost every \(t\) and all \(r > 1\) excepting a set \(E(\delta,t)\) of finite logarithmic measure.
Keywords	entire functions, growth condition, Wiman's inequality, finite logarithmic measure
DOI	doi:10.30970/ms.6.1.59-66
Reference	1. Ж.-П. Кахан. Случайные функциональные ряды. — М.: Мир, 1973. 302 с. 2. P.C. Rosenbloom. Probability and entire functions // Studies Math. Analysis and Related Topics. Stanford: Calif. Univ. Press, 1962. P.325–332. 3. P. Lévy. Sur la croissance des fonctions entières // Bull. Soc. Math. France. 1930. V.58. P.29–59, 127–149. 4. P. Erdős, A. Rényi. On random entire functions // Zast. Matem. 1969. V.10. P.47–55. 5. P. Erdős. On the uniform but not absolute convergence of power series with gaps // Ann. Soc. Polon. Math. 1952. V.25. P.162–168. 6. J.M. Steele. Sharper Wiman Inequality for Entire Functions with Rapidly Oscillating Coefficients // J. of Math. Anal. and Appl. 1987. V.123. P.550–558. 7. J. Jakubowski, S. Kwapień. On multiplicative systems of functions // Bull. Acad. Pol. Sci. 1979. V.27. P.689–694. Let me know if you'd like these added to your existing document or further standardized.
Pages	59-66
Volume	6
Issue	1
Year	1996
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Some classes of entire functions in which the Wiman-Valiron inequality can be almost certainly improved (in Ukrainian)