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

Kuryliak, A. O.; Tsvigun, V. L.

doi:10.15330/ms.49.1.29-51

Cited by 5 publications

(3 citation statements)

References 14 publications

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“…In paper [25] some analogues of Wiman's inequality are proven for the analytic f (z) and random analytic f (z, t)…”

Section: Wiman's Type Inequality For Analytic Functions Of Several Variablesmentioning

confidence: 99%

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Wiman’s Type Inequality in Multiple-Circular Domain

Kuryliak¹,

Скасків²

2021

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In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions f∈A0p(G) in an arbitrary complete Reinhard domain G⊂Cp, p∈N represented by the power series of the form f(z)=f(z1,⋯,zp)=∑‖n‖=0+∞anzn with the domain of convergence G. We have proven the following statement: If f∈Ap(G) and h∈Hp, then for a given ε=(ε1,…,εp)∈R+p and arbitrary δ>0 there exists a set E⊂|G| such that ∫E∩Δεh(r)dr1⋯drpr1⋯rp<+∞ and for all r∈Δε∖E we have Mf(r)≤μf(r)(h(r))p+12lnp2+δh(r)lnp2+δ{μf(r)h(r)}∏j=1p(lnerjεj)p−12+δ. Note, that this assertion at p=1,G=C,h(r)≡const implies the classical Wiman–Valiron theorem for entire functions and at p=1, the G=D:={z∈C:|z|<1},h(r)≡1/(1−r) theorem about the Kővari-type inequality for analytic functions in the unit disc D; p>1 implies some Wiman’s type inequalities for analytic functions of several variables in Cn×Dk, n,k∈Z+,n+k∈N.

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“…In paper [25] some analogues of Wiman's inequality are proven for the analytic f (z) and random analytic f (z, t)…”

Section: Wiman's Type Inequality For Analytic Functions Of Several Variablesmentioning

confidence: 99%

“…The proof of the main result uses the probabilistic reasoning from [17,18] (see also [20]), which has already become traditional in this topic, and differs from the proofs of similar statements in [25].…”

Section: Auxiliary Lemmasmentioning

confidence: 99%

Wiman’s Type Inequality in Multiple-Circular Domain

Kuryliak¹,

Скасків²

2021

Axioms

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“…it is proved an analogue of the Wiman inequality for maximum modulus on bi-circle M (r, f ) = max{|f (z 1 , z 2 )| : |z 1 | = r 1 , |z 2 | = r 2 } and maximal term µ(r, f ) = max{|a n |r n : n ∈ Z 2 + }, r = (r 1 , r 2 ) ∈ R 2 + with 3/2 instead of 1/2 in (8). A. Schumitski ([8, 9]), P. Fenton ([10]), O. Skaskiv and O. Trakalo ([11]) and some others authors have improved Bitlyan and Goldberg's result as in the specification of inequality, and in the specification of describing exceptional set, and also established analogues Wiman's inequality for other classes analytic functions of several variables (see also[12]-[26]). …”

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confidence: 99%

Bitlyan-Gol'dberg type inequality for entire functions and diagonal maximal term

2020

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In the article is obtained an analogue of Wiman-Bitlyan-Gol'dberg type inequality for entire $f\colon\mathbb{C}^p\to \mathbb{C}$ from the class $\mathcal{E}^{p}(\lambda)$ of functions represented by gap power series of the form$$f(z)=\sum\limits_{k=0}^{+\infty} P_k(z),\quadz\in\mathbb{C}^p.$$Here $P_0(z)\equiv a_{0}\in\mathbb{C},$ $P_k(z)=\sum_{\|n\|=\lambda_k} a_{n}z^{n}$ is homogeneouspolynomial of degree $\lambda_k\in\mathbb{Z}_+,$ ànd $ 0=\lambda_0<\lambda_k\uparrow +\infty$\ $(1\leq k\uparrow +\infty ),$$\lambda=(\lambda_k)$.\ We consider the exhaustion of thespace\ $\mathbb{C}^{p}$\by the system $(\mathbf{G}_{r})_{r\geq 0}$ of a bounded complete multiple-circular domains $\mathbf{G}_{r}$with the center at the point $\mathbf{0}=(0,\ldots,0)\in \mathbb{C}^{p}$. Define $M(r,f)=\max\{|f(z)|\colon z\in\overline{G}_r\}$, $\mu(r,f)=\max\{|P_k(z))|\colon z\in\overline{G}_r\}$.Let $\mathcal{L}$ be the class of positive continuous functions $\psi\colon \mathbb{R}_{+}\to\mathbb{R}_{+}$ such that $\int_{0}^{+\infty}\frac{dx}{\psi(x)}<+\infty$, $n(t)=\sum_{\lambda_k\leq t}1$ counting function of the sequence $(\lambda_k)$ for $t\geq 0$. The following statement is proved:{\it If a sequence $\lambda=(\lambda_{k})$ satisfy the condition\begin{equation*}(\exists p_1\in (0,+\infty))(\exists t_0>0)(\forall t\geq t_0)\colon\quad n(t+\sqrt{\psi(t)})-n(t-\sqrt{\psi(t)})\leq t^{p_1}\end{equation*}for some function $\psi\in \mathcal{L}$,then for every entire function $f\in\mathcal{E}^{p}(\lambda)$, $p\geq 2$ and for any$\varepsilon>0$ there exist a constant $C=C(\varepsilon, f)>0$ and a set $E=E(\varepsilon, f)\subset [1,+\infty)$ of finite logarithmic measure such that the inequality\begin{equation*}M(r, f)\leq C m(r,f)(\ln m(r, f))^{p_1}(\ln\ln m(r, f))^{p_1+\varepsilon}\end{equation*}holds for all $ r\in[1,+\infty]\setminus E$.}The obtained inequality is sharp in general.At $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and the Bitlyan-Gol'dberg inequality (1959) it follows. In the case $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and from obtained statement we get the assertion on the Bitlyan-Gol'dberg inequality (1959), and at $p=1$ about the classical Wiman inequality it follows.

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

Kuryliak

Скасків

2022

Ukr Math J

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

Cited by 5 publications

References 14 publications

Wiman’s Type Inequality in Multiple-Circular Domain

Wiman’s Type Inequality in Multiple-Circular Domain

Bitlyan-Gol'dberg type inequality for entire functions and diagonal maximal term

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

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