On regular variation of entire Dirichlet series

Filevych, P. V.; Hrybel, O. B.

doi:10.30970/ms.58.2.174-181

Mat. Stud.

2023

DOI: 10.30970/ms.58.2.174-181

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On regular variation of entire Dirichlet series

P. V. Filevych

O. B. Hrybel

Abstract: 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 $\rho$ impli… Show more

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Cited by 2 publications

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“…Auxiliary results. The following two lemmas, which we will need later, are well known (see, for example, [2,13]).…”

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confidence: 99%

Generalized and modified orders of growth for Dirichlet series absolutely convergent in a half-plane

Filevych,

Hrybel

2024

Mat. Stud.

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Let $\lambda=(\lambda_n)_{n\in\mathbb{N}_0}$ be a non-negative sequence increasing to $+\infty$, $\tau(\lambda)=\varlimsup_{n\to\infty}(\ln n/\lambda_n)$, and $\mathcal{D}_0(\lambda) $ be the class of all Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_n(F)e^{s\lambda_n}$ absolutely convergent in the half-plane $\operatorname{Re}s<0$ with $a_n(F)\not=0$ for at least one integer $n\ge0$. Also, let $\alpha$ be a continuous function on $[x_0,+\infty)$ increasing to $+\infty$, $\beta$ be a continuous function on $[a,0)$ such that $\beta(\sigma)\to+\infty$ as $\sigma\uparrow0$, and $\gamma$ be a continuous positive function on $[b,0)$. In the article, we investigate the growth of a Dirichlet series $F\in\mathcal{D}_0(\lambda)$ depending on the behavior of the sequence $(|a_n(F)|)$ in terms of its $\alpha,\beta,\gamma$-orders determined by the equalities$$R^*_{\alpha,\beta,\gamma}(F)=\varlimsup_{\sigma\uparrow0}\frac{\alpha(\max\{x_0,\gamma(\sigma)\ln\mu(\sigma)\})}{\beta(\sigma)},$$$$R_{\alpha,\beta,\gamma}(F)=\varlimsup_{\sigma\uparrow0}\frac{\alpha(\max\{x_0,\gamma(\sigma)\ln M(\sigma)\})}{\beta(\sigma)},$$where $\mu(\sigma)=\max\{|a_n(F)|e^{\sigma\lambda_n}\colon n\ge0\}$ and $M(\sigma)=\sup\{|F(s )|\colon \operatorname{Re}s=\sigma\}$ are the maximal term and the supremum modulus of the series $F$, respectively. In particular, if for every fixed $t>0$ we have $\alpha(tx)\sim \alpha(x)$ as $x\to+\infty$, $\beta(t\sigma)\sim t^{-\rho}\beta(\sigma)$ as $\sigma\uparrow0$ for some fixed $\rho>0$, $0<\varliminf_{\sigma\uparrow0}\gamma(t\sigma)/\gamma(\sigma) \le\varlimsup_{\sigma\uparrow0}\gamma(t\sigma)/\gamma(\sigma)<+\infty$,$\Phi(\sigma)=\alpha^{-1}(\beta(\sigma))/\gamma(\sigma)$ for all $\sigma\in[\sigma_0,0)$,$\widetilde{\Phi}(x)=\max\{x\sigma-\Phi(\sigma)\colon \sigma\in[\sigma_0,0)\}$ for all $x\in\mathbb{R}$, and $\Delta_\Phi(\lambda)=\varlimsup_{n\to\infty}( -\ln n/\widetilde{\Phi}(\lambda_n))$, then: (a) for each Dirichlet series $F\in\mathcal{D}_0(\lambda)$ we have$$R^*_{\alpha,\beta,\gamma}(F)=\varlimsup_{n\to +\infty}\left(\frac{\ln^+|a_n(F)|}{-\widetilde{\Phi }(\lambda_n)}\right)^\rho;$$ (b) if $\tau(\lambda)>0$, then for each $p_0\in[0,+\infty]$ and any positive function $\Psi$ on $[c,0)$ there exists a Dirichlet series $F\in\mathcal{D}_0(\lambda)$ such that $R^*_{\alpha,\beta,\gamma}(F)=p_0$ and $M(\sigma,F)\ge \Psi(\sigma)$ for all $\sigma\in[\sigma_0,0)$; (c) if $\tau(\lambda)=0$, then $(R_{\alpha,\beta,\gamma} (F))^{1/\rho}\le (R^*_{\alpha,\beta,\gamma}(F))^{1/\rho}+\Delta_\Phi(\lambda)$ for every Dirichlet series\linebreak $F\in\mathcal{D}_0(\lambda)$; (d) if $\tau(\lambda)=0$, then for each $p_0\in[0,+\infty]$ there exists a Dirichlet series $F\in\mathcal{D}_0(\lambda)$ such that $R^*_{\alpha,\beta,\gamma}(F)=p_0$ and $(R_{\alpha,\beta,\gamma}(F))^{1/\rho}=(R ^*_{\alpha,\beta,\gamma}(F))^{1/\rho}+\Delta_\Phi(\lambda)$.

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“…Auxiliary results. The following two lemmas, which we will need later, are well known (see, for example, [2,13]).…”

mentioning

confidence: 99%

Generalized and modified orders of growth for Dirichlet series absolutely convergent in a half-plane

Filevych,

Hrybel

2024

Mat. Stud.

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“…Note that inequalities (2) can also be effectively applied to study other properties of entire Dirichlet series (see [4][5][6]).…”

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confidence: 99%

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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On regular variation of entire Dirichlet series

Cited by 2 publications

References 8 publications

Generalized and modified orders of growth for Dirichlet series absolutely convergent in a half-plane

Generalized and modified orders of growth for Dirichlet series absolutely convergent in a half-plane

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

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