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

  • P.V. Filevych Lviv Politechnic National University, Lviv
  • O.B. Hrybel School of Mathematics, University of Bristol Bristol, United Kingdom Faculty of Mathematics and Computer Science Vasyl Stefanyk Precarpathian National University Ivano-Frankivsk, Ukraine

Анотація

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.

Посилання

S. Mandelbrojt, Dirichlet series: principles and methods, Dordrecht, D. Reidel Publishing Company, 1972.

T.Ya. Hlova, P.V. Filevych, Generalized types of the growth of Dirichlet series, Carpathian Math. Publ., 7 (2015), No2, 172–187. https://doi.org/10.15330/cmp.7.2.172-187

M.M. Sheremeta, On the growth of an entire Dirichlet series, Ukr. Math. J., 51 (1999), No8, 1296–1302. https://doi.org/10.1007/BF02592520

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T.Ya. Hlova, P.V. Filevych, Paley effect for entire Dirichlet series, Ukr. Math. J., 67 (2015), No6, 838–852. https://doi.org/10.1007/s11253-015-1117-x

P.V. Filevych, O.B. Hrybel. On regular variation of entire Dirichlet series, Mat. Stud., 58 (2022), No2, 174–181. https://doi.org/10.30970/ms.58.2.174-181

M.M. Sheremeta, Entire Dirichlet Series, Kyiv, ISDO, 1993. (in Ukrainian)

O.B. Skaskiv, Behavior of the maximum term of a Dirichlet series that defines an entire function, Math. Notes, 37 (1985), No1, 24–28. https://doi.org/10.1007/BF01652509

P.V. Filevych, M.M. Sheremeta, Regularly increasing entire Dirichlet series, Math. Notes, 74 (2003), No1, 110–122. https://doi.org/10.1023/A:1025027418525

T.Ya. Hlova, P.V. Filevych, The growth of entire Dirichlet series in terms of generalized orders, Sb. Math., 209 (2018), No2, 241–257. https://doi.org/10.1070/SM8644

Опубліковано
2023-03-29
Як цитувати
Filevych, P., & Hrybel, O. (2023). Global estimates for sums of absolutely convergent Dirichlet series in a half-plane. Математичні студії, 59(1), 60-67. https://doi.org/10.30970/ms.59.1.60-67
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