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

  • P. V. Filevych Department of Mathematics, Lviv Polytechnic National University Lviv, Ukraine
  • O. B. Hrybel Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University Ivano-Frankivsk, Ukraine
Keywords: Dirichlet series, abscissa of absolute convergence, supremum modulus, maximal term, Young-conjugate function, generalized order, modified order

Abstract

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 $(R_{\alpha,\beta,\gamma}(F))^{1/\rho}=(R ^*_{\alpha,\beta,\gamma}(F))^{1/\rho}+\Delta_\Phi(\lambda)$.

Author Biographies

P. V. Filevych, Department of Mathematics, Lviv Polytechnic National University Lviv, Ukraine

Department of Mathematics,

Lviv Polytechnic National University

Lviv, Ukraine

O. B. Hrybel, Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University Ivano-Frankivsk, Ukraine

Faculty of Mathematics and Computer Science,

Vasyl Stefanyk Precarpathian National University

Ivano-Frankivsk, Ukraine

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Published
2024-06-19
How to Cite
Filevych, P. V., & Hrybel, O. B. (2024). Generalized and modified orders of growth for Dirichlet series absolutely convergent in a half-plane. Matematychni Studii, 61(2), 136-147. https://doi.org/10.30970/ms.61.2.136-147
Section
Articles