Global estimates for sums of absolutely convergent Dirichlet series in a half-plane
Abstract
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.
References
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
O.M. Mulyava, P.V. Filevych. On the growth of an entire Dirichlet series with nonnegative coefficients, Visnyk Lviv Univ. Ser. Mech. Math., 62 (2003), No6, 89–94. (in Ukrainian)
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
Copyright (c) 2023 P.V. Filevych, O.B. Grybel
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.