Levy�s phenomenon for analytic functions in DxC

Kuryliak, A. O.; Скасків, О. Б.; Tsvihun, V.

doi:10.15330/ms.46.2.121-129

Cited by 3 publications

(7 citation statements)

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“…One can find similar inequalities for entire functions of one complex variable in [9,10], for analytic functions in the unit disc in [13], for entire functions of several variables in [18], [19], [23], [26], [29], and for analytic functions without exceptional sets in [16,32].…”

Section: It Remains To Remark That ν(Gmentioning

confidence: 89%

“…In the papers [18,19] one can find an affirmative answer to this question on Fenton's inequality ( [15]) for random entire functions of two complex variables, in [23] about an inequality from [14] for random entire functions of several complex variables, in [24] in the case of analytic functions in the polydisc.…”

Section: ) Does Levy's Fenomenon Take Place For Analogues Of Wiman'smentioning

confidence: 99%

“…But in the case q > 1 the sequence of random variables (cos θ n t) n∈Z + need not be a MS (see [21] A theorem analogous to Theorem 3 is also valid for analytic functions with rapidly oscillating coefficients. , θ) be an analytic function of the from (24) and a sequence of positive integers (θ n ) n∈Z p + satisfy condition (25). Then almost surely for t ∈ R there exists E(δ, t) ∈ Υ such that for all r ∈ T \ E we have…”

Section: It Remains To Remark That ν(Gmentioning

confidence: 99%

“…Some analogues of Wiman's inequality for entire functions of several complex variables can be found in [14]- [23], for analytic functions in the polydisc D p , p ≥ 2, in [24,25,26], for analytic functions in the unbounded cylinder D × C in [27,29], respectively.…”

mentioning

confidence: 99%

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Wiman's type inequality for multiple power series in the unbounded cylinder domain

Kuryliak

Tsvigun

2017

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In this paper we prove some analogues of Wiman's inequality for analytic f (z) and random analytic functionsis a multiplicative system of random variables on the Steinhaus probability space, uniformly bounded by the number 1. In particular, we prove the following statements: For every δ > 0 there exist sets.hold for all r ∈ T \ E 1 and for all r ∈ T \ E 2 a.s. in t, respectively. Also sharpness of the obtained inequalities is proved.1. Introduction and the main result. Let f be an analytic function in the disc D R = {z : |z| < R}, 0 < R ≤ +∞, represented by the power seriesPut [6,7,8]) that for each nonconstant entire function f (z) and for every ε > 0 there exists a set E(ε, f ) ⊂ [1, +∞) such that Wiman's inequality M f (r) ≤ µ f (r)(ln µ f (r))2010 Mathematics Subject Classification: 30B20, 30D20.

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Section: It Remains To Remark That ν(Gmentioning

confidence: 89%

Section: ) Does Levy's Fenomenon Take Place For Analogues Of Wiman'smentioning

confidence: 99%

Section: It Remains To Remark That ν(Gmentioning

confidence: 99%

mentioning

confidence: 99%

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Wiman's type inequality for multiple power series in the unbounded cylinder domain

Kuryliak

Tsvigun

2017

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“…In the papers [16,17] we have found an affirmative answer to this question about Fenton's inequality [18] for random entire functions of two complex variables, in [19] about a inequality from [21] for random entire functions of several complex variables, in [27] in the case of analytic functions in the polydisc.…”

Section: Analog Of Wiman's Inequality Is Not Considered Only In the Casementioning

confidence: 97%

Wiman's inequality for analytic functions in $\mathbb{D}\times\mathbb{C}$ with rapidly oscillating coefficients

Kuryliak¹,

Tsvigun²

2018

Carpathian Math. Publ.

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Let $\mathcal{A}^2$ be a class of analytic functions $f$ represented by power series of the from $$ f(z)=f(z_1,z_2)=\sum^{+\infty}_{n+m=0}a_{nm}z_1^nz^m_2$$ with the domain of convergence $\mathbb{T}=\{ z\in \mathbb{C}^2 \colon |z_1|<1, |z_2|<+\infty \} $ such that $\frac{\partial}{\partial z_2}f(z_1,z_2)\not\equiv0$ in $\mathbb{T}$ and there exists $r_0=(r^0_1, r^0_2)\in [0,1)\times[0,+\infty)$ such that for all $r\in(r^0_1,1)\times(r^0_2,+\infty)$ we have $ r_1\frac{\partial}{\partial r_1}\ln M_f(r)+\ln r_1>1, \ $ where $M_f(r)=\sum_{n+m=0}^{+\infty}|a_{nm}|r_1^nr_2^m.$ Let $K(f,\theta)=\{f(z,t)=\sum_{n+m=0}^{+\infty}a_{nm}e^{2\pi it(\theta_n+\theta_m)}:t\in \mathbb{R}\}$ be class of analytic functions, where $(\theta_{nm})$ is a sequence of positive integer such that its arrangement $(\theta^*_k)$ by increasing satisfies the condition $$ \theta^*_{k+1}/\theta^*_{k}\geq q>1, k>0. $$ For analytic functions from the class $\mathcal{K}(f,\theta)$ Wiman's inequality is improved.

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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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Levy�s phenomenon for analytic functions in DxC

Cited by 3 publications

References 2 publications

Wiman's type inequality for multiple power series in the unbounded cylinder domain

Wiman's type inequality for multiple power series in the unbounded cylinder domain

Wiman's inequality for analytic functions in $\mathbb{D}\times\mathbb{C}$ with rapidly oscillating coefficients

Wiman’s Type Inequality for Some Double Power Series

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