Wiman’s inequality for entire functions of several complex
variables with rapidly oscillating coefficients

A. O. Kuryliak; L. O. Shapovalovska

doi:10.15330/ms.43.1.16-26

Let $\mathcal{E}^p$ be a class of entire functions of the form $ f(z)=\sum_{\|n\|=0}^{+\infty}a_nz^n$, $ \|n\|=n_1+\ldots+n_p$ ($p\geq 2$), $z=(z_1,\ldots,z_p)\in\mathbb{C}^p$, and $\mathcal{K}(f,\theta)=\{f(z,t)=\sum_{\|n\|=0}^{+\infty}a_n \exp\{2\pi i\theta_n t\}r^n\colon t\in\mathbb R\}$, where $\{\theta_n\}$ is a sequence of positive {integers} such that its arrangement $\{\theta_k^*\}$ by increasing, i.e. $\{\theta_n\colon n\in \mathbb{Z}_+^p\}=\{\theta_k^*\colon k\geq 0\}$, $\theta_{k+1}^*>\theta_k^*$, satisfies the condition $\theta_{k+1}^*/\theta_k^*\geq q>1$ $(k\geq0)$. In this paper it is established that for $f\in \mathcal{E}^p$ almost surely for $t\in\mathbb R$ there {exists} a set $E(t)\subset\mathbb{R}_+^p$, such that for all $r\in\mathbb{R}_+^p \setminus E(t)$ the inequality $$ \mathfrak{M}_f(r,t)=\max \limits_{|z|\leq r}|f(z,t)|\leq \mu_f(r)(\Lambda_f(r))^{1/4}\ln^{3} \Lambda_f(r) $$ holds, where $E(t)$ is a set of finite asymptotically logarithmic measure and $M_f(r)=\max\{|f(z,t)|\colon|z_i|=r_i,\ i\in \{1,\ldots,p\}\},$\ $\mu_f(r)=\max_{n\in\mathbb{Z}_+^p}\{|a_n|r^n\colon r=(r_1,\ldots,r_p)\in\mathbb{R}_+^p\},$ $\Lambda_f(r)=\ln^p\mu_f(r)\cdot\prod_{i=1}^p\ln^{p-1}r_i.$

1. Gopala Krishna J., Nagaraja Rao I.H. Generalised inverse and probability techniques and some fundamental growth theorems in Ck// Jour. of the Indian Math. Soc. – 1977. – V.41. – P. 203–219.

2. Bitlyan I.F., Goldberg A.A. Wiman-Valiron’s theorem for entire functions os several complex variables// Vestn. Leningrad univ. ser. mat., mech. and astr. – 1959. – V.2, №13. – P. 27–41. (in Russian)

3. Shumitzky A. A probabilistic approach to the Wiman-Valiron’s theory for entire functions of several complex variables// Complex Variables. – 1989. – V.13. – P. 85–98.

4. Skaskiv O.B., Trakalo O.M. On classical Wiman’s inequality for entire Dirichlet series// Mat. metods and fys.-mekh. polya. – V.43, №3. – P. 34–39. (in Ukrainian)

5. Zrum O.V., Kuryliak A.O., Skaskiv O.B. Levy’s phenomenon for entire functions of several variables arXiv: 1307.6164v1. – 14 p.

6. Kuryliak A.O., Skaskiv O.B., Zrum O.V. Levy’s phenomenon for entire functions of several variables// Ufa Math. Journ. – 2014. – V.6, №2. – P. 118–127.

7. Kuryliak A.O., Skaskiv O.B. Wiman’s type inequalities without exceptional sets for random entire functions of several variables// Mat. Stud. – 2012. – V38, №1. – P. 35–50.

8. Fenton P.C. Wiman-Valiron theory in two variables// Trans. Amer. Math. Soc. – 1995. – V.347, №11. – P. 4403–4412.

9. Zrum O.V., Skaskiv O.B. On Wiman’s inequality for random entire functions of two variables// Mat. Stud. – 2005. – V.23, №2. – P. 149–160. (in Ukrainian)

10. Skaskiv O.B., Zrum O.V. Wiman’s type inequality for entire functions of two complex variables with rapidly oscillating coefficient// Mat. metods and fys.-mekh. polya. – 2005. – V.48, №4. – P. 78–87. (in Ukrainian)

11. Skaskiv O.B., Zrum O.V. On inprovement of Fenton’s inequality for entire functions of two complex variables// Math. Bull. Shevchenko Sci. Soc. – 2006. – V.3. – P. 56–68. (in Ukrainian)

12. Filevych P.V. The Baire categories and Wiman’s inequality for entire functions// Mat. Stud. – 2003. – V.20. – №2. – P. 215–221.

13. Kuryliak A.O., Skaskiv O.B., Chyzhykov I.E. Baire categories and classes of analytic functions in which the Wiman-Valiron type inequality can be almost surely improved// Bulletin de la societe des sciences et des lettres de Lodz. – 2012. – V.62, №3. – P. 17–33.

14. Skaskiv O.B., Zrum O.V. Adjustment of Fenton’s inequalities for entire functions of two complex variables// Mat. visn. NTSh. – 2006. – V.3. – P. 56–68. (in Ukrainian)

15. Kuryliak A.O., Shapovalovska L.O., Skaskiv O.B. Wiman’s type inequality for some double power series// Mat. Stud. – 2013. – V.39, №2. – P. 134–141.

16. Steele J.M., Sharper Wiman inequality for entire functions with rapidly oscillating coefficients// J. of Math. Anal. and Appl. – 1987. – V.123. – P. 550–558. 17. Polya G., Szego G. Aufgaben und Lehrsatze aus der Analysis. V.2. – Berlin: Springer, 1925.

	Wiman’s inequality for entire functions of several complex variables with rapidly oscillating coefficients
Author	A. O. Kuryliak, L. O. Shapovalovska kurylyak88@gmail.com, shap.ludmila@gmail.com Ivan Franko National University of Lviv
Abstract	Let $\mathcal{E}^p$ be a class of entire functions of the form $ f(z)=\sum_{\\|n\\|=0}^{+\infty}a_nz^n$, $ \\|n\\|=n_1+\ldots+n_p$ ($p\geq 2$), $z=(z_1,\ldots,z_p)\in\mathbb{C}^p$, and $\mathcal{K}(f,\theta)=\{f(z,t)=\sum_{\\|n\\|=0}^{+\infty}a_n \exp\{2\pi i\theta_n t\}r^n\colon t\in\mathbb R\}$, where $\{\theta_n\}$ is a sequence of positive {integers} such that its arrangement $\{\theta_k^\}$ by increasing, i.e. $\{\theta_n\colon n\in \mathbb{Z}_+^p\}=\{\theta_k^\colon k\geq 0\}$, $\theta_{k+1}^>\theta_k^$, satisfies the condition $\theta_{k+1}^/\theta_k^\geq q>1$ $(k\geq0)$. In this paper it is established that for $f\in \mathcal{E}^p$ almost surely for $t\in\mathbb R$ there {exists} a set $E(t)\subset\mathbb{R}_+^p$, such that for all $r\in\mathbb{R}_+^p \setminus E(t)$ the inequality $$ \mathfrak{M}_f(r,t)=\max \limits_{\|z\|\leq r}\|f(z,t)\|\leq \mu_f(r)(\Lambda_f(r))^{1/4}\ln^{3} \Lambda_f(r) $$ holds, where $E(t)$ is a set of finite asymptotically logarithmic measure and $M_f(r)=\max\{\|f(z,t)\|\colon\|z_i\|=r_i,\ i\in \{1,\ldots,p\}\},$\ $\mu_f(r)=\max_{n\in\mathbb{Z}_+^p}\{\|a_n\|r^n\colon r=(r_1,\ldots,r_p)\in\mathbb{R}_+^p\},$ $\Lambda_f(r)=\ln^p\mu_f(r)\cdot\prod_{i=1}^p\ln^{p-1}r_i.$
Keywords	Wiman-Valiron’s inequality; random entire function of several variables
DOI	doi:10.15330/ms.43.1.16-26
Reference	1. Gopala Krishna J., Nagaraja Rao I.H. Generalised inverse and probability techniques and some fundamental growth theorems in Ck// Jour. of the Indian Math. Soc. – 1977. – V.41. – P. 203–219. 2. Bitlyan I.F., Goldberg A.A. Wiman-Valiron’s theorem for entire functions os several complex variables// Vestn. Leningrad univ. ser. mat., mech. and astr. – 1959. – V.2, №13. – P. 27–41. (in Russian) 3. Shumitzky A. A probabilistic approach to the Wiman-Valiron’s theory for entire functions of several complex variables// Complex Variables. – 1989. – V.13. – P. 85–98. 4. Skaskiv O.B., Trakalo O.M. On classical Wiman’s inequality for entire Dirichlet series// Mat. metods and fys.-mekh. polya. – V.43, №3. – P. 34–39. (in Ukrainian) 5. Zrum O.V., Kuryliak A.O., Skaskiv O.B. Levy’s phenomenon for entire functions of several variables arXiv: 1307.6164v1. – 14 p. 6. Kuryliak A.O., Skaskiv O.B., Zrum O.V. Levy’s phenomenon for entire functions of several variables// Ufa Math. Journ. – 2014. – V.6, №2. – P. 118–127. 7. Kuryliak A.O., Skaskiv O.B. Wiman’s type inequalities without exceptional sets for random entire functions of several variables// Mat. Stud. – 2012. – V38, №1. – P. 35–50. 8. Fenton P.C. Wiman-Valiron theory in two variables// Trans. Amer. Math. Soc. – 1995. – V.347, №11. – P. 4403–4412. 9. Zrum O.V., Skaskiv O.B. On Wiman’s inequality for random entire functions of two variables// Mat. Stud. – 2005. – V.23, №2. – P. 149–160. (in Ukrainian) 10. Skaskiv O.B., Zrum O.V. Wiman’s type inequality for entire functions of two complex variables with rapidly oscillating coefficient// Mat. metods and fys.-mekh. polya. – 2005. – V.48, №4. – P. 78–87. (in Ukrainian) 11. Skaskiv O.B., Zrum O.V. On inprovement of Fenton’s inequality for entire functions of two complex variables// Math. Bull. Shevchenko Sci. Soc. – 2006. – V.3. – P. 56–68. (in Ukrainian) 12. Filevych P.V. The Baire categories and Wiman’s inequality for entire functions// Mat. Stud. – 2003. – V.20. – №2. – P. 215–221. 13. Kuryliak A.O., Skaskiv O.B., Chyzhykov I.E. Baire categories and classes of analytic functions in which the Wiman-Valiron type inequality can be almost surely improved// Bulletin de la societe des sciences et des lettres de Lodz. – 2012. – V.62, №3. – P. 17–33. 14. Skaskiv O.B., Zrum O.V. Adjustment of Fenton’s inequalities for entire functions of two complex variables// Mat. visn. NTSh. – 2006. – V.3. – P. 56–68. (in Ukrainian) 15. Kuryliak A.O., Shapovalovska L.O., Skaskiv O.B. Wiman’s type inequality for some double power series// Mat. Stud. – 2013. – V.39, №2. – P. 134–141. 16. Steele J.M., Sharper Wiman inequality for entire functions with rapidly oscillating coefficients// J. of Math. Anal. and Appl. – 1987. – V.123. – P. 550–558. 17. Polya G., Szego G. Aufgaben und Lehrsatze aus der Analysis. V.2. – Berlin: Springer, 1925.
Pages	16-26
Volume	43
Issue	1
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Wiman’s inequality for entire functions of several complex variables with rapidly oscillating coefficients