Wiman's type inequality for analytic and entire functions and $h$-measure of an exceptional sets

We consider the class $\mathcal{A}(\varphi,\beta)$ of random analytic functions in the unit disk $\mathbb{C}=\{z\colon |z|<1\}$ of the form $f(z,\omega)=f(z,\omega_1,\omega_2)=\sum_{n=0}^{+\infty} R_n(\omega_1)\xi_n(\omega_2)a_nz^n,$ where $a_n\in\mathbb{C}\colon \lim\limits_{n\to+\infty}\sqrt[n]{|a_n|}=1,$ $\big(R_n(\omega)\big)$ is the Rademacher sequence, $\big(\xi_n(\omega)\big)$ is a sequence of complex-valued random variables (denote by $\Delta_{\varphi}$) such that there exists a constant $\beta>0$ and a function $\varphi(N,\beta)\colon\mathbb{N}\times\mathbb{R}_+\to[1;+\infty)$ non-decreasing by $N$ and $\beta$ for which $\displaystyle \Bigl(\mathbf{E}\Bigl(\max_{0\leq n\leq N}|\xi_n|^{\beta}\Bigl)\Bigl)^{1/\beta}\asymp\varphi(N,\beta),\ \ N\to+\infty,\ \ \alpha=\varlimsup_{N\to+\infty}\frac{\ln\varphi(N,\beta)}{\ln N}<+\infty,$ $\displaystyle (\exists \gamma>0)(\exists n_0\in\mathbb{N})\colon \sup\{ \mathbf{E}|\xi_n|^{-\gamma}\colon\ n\geq n_0\}<+\infty.$ By $\mathcal{A}_1(\varphi,\beta)$ we denote the class of random analytic functions in $\mathbb{D}$ of the form $f(z,\omega)=\sum_{n=0}^{+\infty} \xi_n(\omega)a_nz^n,$ where a sequence $\big(\xi_n(\omega)\big)\in\Delta_\varphi$ and, in particular, may be not sub-gaussian and not independent. In the paper, there are proved the following statements: Let $\delta>0.$ 1) Theorem 3: For $f\in\mathcal{A}(\varphi,\beta)$ there exist $r_0(\omega)>0$, a set $E(\delta)\subset(0;1)$ of finite logarithmic measure such that for all $r\in(r_0(\omega);1)\backslash E$ we have with probability $p\in(0;1)$ $\displaystyle M_f(r,\omega) \leq\frac{\mu_f(r)}{(1-p)^{1/\beta}}\varphi(N(r),\beta)\Big((1-r)^{-2}\cdot \ln\frac{\mu_f(r)\varphi(N(r),\beta)}{(1-p)(1-r)}\Big)^{1/4+\delta}. $ 2) Theorem 4: For a function $f\in\mathcal{A}_1(\varphi,\beta)$ there exist $r_0(\omega)>0$, a set $E(\delta)\subset(0;1)$ of finite logarithmic measure such that for all $r\in(r_0(\omega);1)\backslash E$ we get with probability $p\in(0;1)$ $\displaystyle M_f(r,\omega) \leq\frac{\mu_f(r)}{(1-p)^{1/\beta}}\varphi(N(r),\beta)\Big((1-r)^{-2}\cdot \ln\frac{\mu_f(r)\varphi(N(r),\beta)}{(1-p)(1-r)}\Big)^{1/2+\delta}. $

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Wiman’s Type Inequality for Some Double Power Series

Kuryliak

Скасків

2021

BMJ

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By $\mathcal{A}^2$ denote the class of analytic functions of the formBy $\mathcal{A}^2$ denote the class of analytic functions of the form$f(z)=\sum_{n+m=0}^{+\infty}a_{nm}z_1^nz_2^m,$with {the} domain of convergence $\mathbb{T}=\{z=(z_1,z_2)\in\mathbb C^2\colon|z_1|<1,\ |z_2|<+\infty\}=\mathbb{D}\times\mathbb{C}$ and$\frac{\partial}{\partial z_2}f(z_1,z_2)\not\equiv0$ in $\mathbb{T}.$ In this paper we prove some analogue of Wiman's inequalityfor analytic functions $f\in\mathcal{A}^2$. Let a function $h\colon \mathbb R^2_+\to \mathbb R_+$ be such that$h$ is nondecreasing with respect to each variables and $h(r)\geq 10$ for all $r\in T:=(0,1)\times (0,+\infty)$and $\iint_{\Delta_\varepsilon}\frac{h(r)dr_1dr_2}{(1-r_1)r_2}=+\infty$ for some $\varepsilon\in(0,1)$, where $\Delta_{\varepsilon}=\{(t_1, t_2)\in T\colon t_1>\varepsilon,\ t_2> \varepsilon\}$.We say that $E\subset T$ is a set of asymptotically finite $h$-measure on\ ${T}$if $\nu_{h}(E){:=}\iint\limits_{E\cap\Delta_{\varepsilon}}\frac{h(r)dr_1dr_2}{(1-r_1)r_2}<+\infty$ for some $\varepsilon>0$. For $r=(r_1,r_2)\in T$ and a function $f\in\mathcal{A}^2$ denote\begin{gather*}M_f(r)=\max \{|f(z)|\colon |z_1|\leq r_1,|z_2|\leq r_2\},\\mu_f(r)=\max\{|a_{nm}|r_1^{n} r_2^{m}\colon(n,m)\in{\mathbb{Z}}_+^2\}.\end{gather*}We prove the following theorem:{\sl Let $f\in\mathcal{A}^2$. For every $\delta>0$ there exists a set $E=E(\delta,f)$ of asymptotically finite $h$-measure on\ ${T}$ such that for all $r\in (T\cap\Delta_{\varepsilon})\backslash E$ we have \begin{equation*} M_f(r)\leq\frac{h^{3/2}(r)\mu_f(r)}{(1-r_1)^{1+\delta}}\ln^{1+\delta} \Bigl(\frac{h(r)\mu_f(r)}{1-r_1}\Bigl)\cdot\ln^{1/2+\delta}\frac{er_2}{\varepsilon}. \end{equation*}}

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Wiman's type inequality for analytic and entire functions and $h$-measure of an exceptional sets

Cited by 5 publications

References 5 publications

Wiman-Type Inequality in Multiple-Circular Domains: Lévy’s Phenomenon and Exceptional Sets

Wiman-Type Inequality in Multiple-Circular Domains: Lévy’s Phenomenon and Exceptional Sets

Arbitrary random variables and Wiman's inequality for analytic functions in the unit disc

Wiman’s Type Inequality for Some Double Power Series

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