Wiman's inequality for the Laplace integrals

Kuryliak, A. O.; Ye, I.; Скасків, О. Б.; Franko, Ivan

doi:10.12988/ijma.2014.4232

Cited by 8 publications

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“…Then we have that the mean Mξ = g ′ 1 (x) and the variance Dξ = g ′′ 1 (x), where g 1 (x) = ln g(x). By Bienayme-Chebyshev inequality we obtain (see also [4,8,9])…”

Section: Remarkmentioning

confidence: 99%

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

Скасків

Kuryliak

2020

Carpathian Math. Publ.

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Let $\mathcal{E}_R$ be the class of analytic functions $f$ represented by power series of the form $f(z)=\sum\limits\limits_{n=0}^{+\infty}a_n z^n$ with the radius of convergence $R:=R(f)\in(0;+\infty].$ For $r\in [0, R)$ we denote the maximum modulus by $M_f(r)=\max\{|f(z)|\colon$ $ |z|=r\}$ and the maximal term of the series by $\mu_f(r)=\max\{|a_n| r^n\colon n\geq 0\}$. We also denote by $\mathcal{H}_R$, $R\leq +\infty$, the class of continuous positive functions, which increase on $[0;R)$ to $+\infty$, such that $h(r)\geq2$ for all $r\in (0,R)$ and $ \int^R_{r_{0}} h(r) d\ln r =+\infty $ for some $r_0\in(0,R)$. In particular, the following statements are proved. $1^0.$ If $h\in \mathcal{H}_R$ and $f\in \mathcal{E}_R,$ then for any $\delta>0$ there exist $E(\delta,f,h):=E\subset(0,R)$, $r_0 \in (0,R)$ such that $$ \forall\ r\in (r_0,R)\backslash E\colon\ M_f(r)\leq h(r) \mu_f(r) \big\{\ln h(r)\ln(h(r)\mu_f(r))\big\}^{1/2+\delta}$$ and $$\int\nolimits_E h(r) dr < +\infty. $$ $2^0.$ If we additionally assume that the function $f\in \mathcal{E}_R$ is unbounded, then $$ \ln M_f(r)\leq(1+o(1))\ln (h(r)\mu_f(r)) $$ holds as $r\to R$, $r\notin E$. Remark, that assertion $1^0$ at $h(r)\equiv \text{const}$ implies the classical Wiman-Valiron theorem for entire functions and at $h(r)\equiv 1/(1-r)$ theorem about the Kövari-type inequality for analytic functions in the unit disc. From statement $2^0$ in the case that $\ln h(r)=o(\ln\mu_f(r))$, $r\to R$, it follows that $ \ln M_f(r)=(1+o(1))\ln \mu_f(r) $ holds as $r\to R$, $r\notin E$.

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“…Then we have that the mean Mξ = g ′ 1 (x) and the variance Dξ = g ′′ 1 (x), where g 1 (x) = ln g(x). By Bienayme-Chebyshev inequality we obtain (see also [4,8,9])…”

Section: Remarkmentioning

confidence: 99%

“…The proof of Lemma 1 is carried out by verbatim repetition of the arguments from article [8] (see also [4,7,9]). Indeed, denote Proof.…”

Section: Auxillary Propositionsmentioning

confidence: 99%

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

Скасків

Kuryliak

2020

Carpathian Math. Publ.

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“…A sharp Wiman's type inequality for integrals from the class I(ν) was obtained in [2], where I(ν) is the notation of the class of functions F (x) of the form (2). Formula (2) gives the notation of the Laplace-Stieltjes integral, where f (u) is a positive ν−measured function on R, ν -countable measure on σ-algebra B(R + ) of Borel's sets on R + , such that ν({x : 0 ≤ x ≤ b}) < +∞ for every b > 0. Let ν(E) = E ν(dx) be the ν−measure of a Borel's set E ⊂ R, ν(a, b] := ν((a, b]).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic estimates of Laplace-Stieltjes integrals

Ovchar¹

2016

nade

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“…λ k ≡ k (k ≥ 0), condition (5) holds with = 1. Therefore, inequality (4) holds with p 1 = 1/2, i.e., we have the classical Wiman's inequality (see [5,6,7]).…”

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confidence: 99%

“…Here H(r) is an entire function of one variable, for which condition (8) of Theorem 1 holds. Now we need one result from [7,Theorem 1]. Let I(ν) be the class of the functions F : R → R + defined by the integral of the form…”

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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's inequality for the Laplace integrals

Abstract: We find unimprovable conditions for the Laplace-Stieltjes type integrals under which the Wiman-type inequality holds.

Cited by 8 publications

References 6 publications

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

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

Asymptotic estimates of Laplace-Stieltjes integrals

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

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