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Filevich, P. V.; Sheremeta, М. М.

doi:10.1023/a:1025027418525

Cited by 4 publications

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“…In connection with inequalities (2), we also note that different estimates for the sum of a Dirichlet series by the maximal term of the same series have been obtained in many works (see, for example, [2,[7][8][9][10] and the bibliography therein). It is well known that such estimates can be correct for Dirichlet series only under additional conditions on their exponents.…”

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confidence: 88%

Global estimates for sums of absolutely convergent Dirichlet series in a half-plane

Filevych

Hrybel

2023

Mat. Stud.

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Let $(\lambda_n)_{n=0}^{+\infty}$ be a nonnegative sequence increasing to $+\infty$, $F(s)=\sum_{n=0}^{+\infty} a_ne^{s\lambda_n}$ be an absolutely convergent Dirichlet series in the half-plane $\{s\in\mathbb{C}\colon \operatorname{Re} s<0\}$, and let, for every $\sigma<0$, $\mathfrak{M}(\sigma,F)=\sum_{n=0}^{+\infty} |a_n|e^{\sigma\lambda_n}$. Suppose that $\Phi\colon (-\infty,0)\to\overline{\mathbb{R}}$ is a function, and let $\widetilde{\Phi}(x)$ be the Young-conjugate function of $\Phi(\sigma)$, i.e.$\widetilde{\Phi}(x)=\sup\{x\sigma-\alpha(\sigma)\colon \sigma<0\}$ for all $x\in\mathbb{R}$. In the article, the following two statements are proved: (i) There exist constants $\theta\in(0,1)$ and $C\in\mathbb{R}$ such that$\ln\mathfrak{M}(\sigma,F)\le\Phi(\theta\sigma)+C$ for all $\sigma<0$ if and only if there exist constants $\delta\in(0, 1)$ and $c\in\mathbb{R}$ such that $\ln\sum_{m=0}^n|a_m|\le-\widetilde{\Phi}(\lambda_n/\delta)+c$ for all integers $n\ge0$ (Theorem 2); (ii) For every $\theta\in(0,1)$ there exists a real constant $C=C(\delta)$ such that $\ln\mathfrak{M}(\sigma,F)\le\Phi( \theta\sigma)+C$ for all $\sigma<0$ if and only if for every $\delta\in(0,1)$ there exists a real constant $c=c(\delta)$ such that $\ln\sum_{m=0}^n|a_m|\le-\widetilde{\Phi}(\lambda_n/\delta)+c$ for all integers $n\ge0$ (Theorem 3).iii) Let $\Phi$ be a continuous positive increasing function on $\mathbb{R}$ such that $\Phi(\sigma)/\sigma\to+\infty$, $\sigma\to+ \infty$ and $F$ be a entire Dirichlet series. For every $q>1$ there exists a constant $C=C(q)\in\mathbb{R}$ such that $\ln\mathfrak{M}(\sigma,F)\le \Phi(q\sigma)+C,\quad \sigma\in\mathbb{R},$ holds if and only if for every $\delta \in(0,1)$ there exist constants $c=c(\delta)\in\mathbb{R}$ and $n_0=n_0(\delta)\in\mathbb{N}_0$ such that $\ln \sum_{m=n}^{+\infty}|a_m|\le-\widetilde{\Phi}(\delta\lambda_n)+c,\quad n\ge n_0$ Theorem 5. These results are analogous to some results previously obtained by M.M. Sheremeta for entire Dirichlet series.

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confidence: 88%

Global estimates for sums of absolutely convergent Dirichlet series in a half-plane

Filevych

Hrybel

2023

Mat. Stud.

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“…Examples of Dirichlet series F ∈ D of the form (1), for which ln µ(σ, F ) is a regularly varying function with given index ρ ≥ 1, and ln M (σ, F ) is not a regularly varying function with the same index, were also constructed in [4]. For the exponents of the constructed series we have λ n = ln ln n for all n ≥ n 0 in the case ρ = 1, and λ n ∼ (ln n) (ρ−1)/ρ as n → ∞ in the case ρ > 1.…”

Section: Theorem B ([4]mentioning

confidence: 99%

“…Problems of finding conditions for regular variation of the main characteristics of entire functions, presented by power series, Dirichlet series or Taylor-Dirichlet series, were considered, in particular, in the articles [2][3][4][5][6]. In this article, we give some addentum to the results from [4].…”

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confidence: 99%

“…Therefore, the function ln µ(σ, F ) cannot be regularly varying with index ρ < 1, because for each slowly varying function α(σ) we have ln α(σ) = o(ln σ) as σ → +∞ (see, for example, [1]). Necessary and sufficient conditions for the function ln µ(σ, F ) to be regularly varying with index ρ ≥ 1 was given in [4] (for series of the Taylor-Dirichlet type, see a similar statement in [5]).…”

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On regular variation of entire Dirichlet series

Filevych

Hrybel

2023

Mat. Stud.

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Consider an entire (absolutely convergent in $\mathbb{C}$) Dirichlet series $F$ with the exponents $\lambda_n$, i.e., of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, and, for all $\sigma\in\mathbb{R}$, put $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge0\}$ and $M(\sigma,F)=\sup\{|F(s)|:\operatorname{Re}s=\sigma\}$. Previously, the first of the authors and M.M.~Sheremeta proved that if $\omega(\lambda)<C(\rho)$, then the regular variation of the function $\ln\mu(\sigma,F)$ with index $\rho$ implies the regular variation of the function $\ln M(\sigma,F)$ with index $\rho$, and constructed examples of entire Dirichlet series $F$, for which $\ln\mu(\sigma,F)$ is a regularly varying function with index $\rho$, and $\ln M(\sigma,F)$ is not a regularly varying function with index $\rho$. For the exponents of the constructed series we have $\lambda_n=\ln\ln n$ for all $n\ge n_0$ in the case $\rho=1$, and $\lambda_n\sim(\ln n)^{(\rho-1)/\rho}$ as $n\to\infty$ in the case $\rho>1$. In the present article we prove that the exponents of entire Dirichlet series with the same property can form an arbitrary sequence $\lambda=(\lambda_n)_{n=0}^\infty$ not satisfying $\omega(\lambda)<C(\rho)$. More precisely, if $\omega(\lambda)\ge C(\rho)$, then there exists a regularly varying function $\Phi(\sigma)$ with index $\rho$ such that, for an arbitrary positive function $l(\sigma)$ on $[a,+\infty)$, there exists an entire Dirichlet series $F$ with the exponents $\lambda_n$, for which $\ln \mu(\sigma, F)\sim\Phi(\sigma)$ as $\sigma\to+\infty$ and $M(\sigma,F)\ge l(\sigma)$ for all $\sigma\ge\sigma_0$.

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“…So from another angle of view µ f (r1,r2,••• ,rn) µg(r1,r2,••• ,rn) , where r k → ∞, k = 1, 2, • • • , n is called growth of f with respect to g, in term of maximum terms, now in similar way we get growth of f with respect to g in terms of central index, [7], Biswas [1] and Pramanik [10] worked on central index. The details of the notations of maximum modulus, entire functions, growth, maximum term and central index for one variable appear in [1,2,3,4,6,7,8,9,11,13].…”

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Growth Properties of an Entire Function of Several Complex Variables on the Basis of Relative Order

Bhan¹,

Rastogi²

2023

jnanabha

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In this paper, we study some comparative growth properties of composite entire function of several complex variables, on the basis of relative order and relative lower order of an entire function with respect to an entire function. Here we are defining some definitions related to relative order and relative lower order in terms of central index. A Rastogi, Biswas, Pramanik, Somsundaram many authors work on central index.

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Cited by 4 publications

References 3 publications

Global estimates for sums of absolutely convergent Dirichlet series in a half-plane

Global estimates for sums of absolutely convergent Dirichlet series in a half-plane

On regular variation of entire Dirichlet series

Growth Properties of an Entire Function of Several Complex Variables on the Basis of Relative Order

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