On certain classes of Dirichlet series with real coefficients absolute convergent in a half-plane

Abstract

For $h>0$, $\alpha\in [0,h)$ and $\mu\in {\mathbb R}$ denote by $SD_h(\mu, \alpha)$ a class
 of absolutely convergent in the half-plane $\Pi_0=\{s:\, \text{Re}\,s<0\}$ Dirichlet series
 $F(s)=e^{sh}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$ such that
 
 \smallskip\centerline{$\text{Re}\left\{\frac{(\mu-1)F'(s)-\mu F''(s)/h}{(\mu-1)F(s)-\mu F'(s)/h}\right\}>\alpha$ for all $s\in \Pi_0$,}
 
 \smallskip\noi and
 let $\Sigma D_h(\mu, \alpha)$ be a class of absolutely convergent in half-plane $\Pi_0$ Dirichlet series
 $F(s)=e^{-sh}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$ such that
 
 \smallskip\centerline{$\text{Re}\left\{\frac{(\mu-1)F'(s)+\mu F''(s)/h}{(\mu-1)F(s)+\mu F'(s)/h}\right\}<-\alpha$ for all $s\in \Pi_0$.}
 
 \smallskip\noi
 Then $SD_h(0, \alpha)$ consists of pseudostarlike functions of order $\alpha$ and $SD_h(1, \alpha)$ consists of pseudoconvex functions of order $\alpha$.
 
 For functions from the classes $SD_h(\mu, \alpha)$ and $\Sigma D_h(\mu, \alpha)$, estimates for the coefficients and growth estimates are obtained. {In particular, it is proved the following statements: 1) In order that function $F(s)=e^{sh}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$ belongs to
 $SD_h(\mu, \alpha)$, it is
 sufficient, and in the case when $f_k(\mu\lambda_k/h-\mu+1)\le 0$ for all $k\ge 1$, it is necessary that}
 
 \smallskip\centerline{$
 \sum\limits_{k=1}^{\infty}\big|f_k\big(\frac{\mu\lambda_k}{h}-\mu+1\big)\big|(\lambda_k-\alpha)\le h-\alpha,$}
 
 \noi {where $h>0, \alpha\in [0, h)$ (Theorem 1).}
 
 \noi 2) {In order that function $F(s)=e^{-sh}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$ belongs to $\Sigma D_h(\mu, \alpha)$, it is
 sufficient, and in the case when $f_k(\mu\lambda_k/h+\mu-1)\le 0$ for all $k\ge 1$, it is necessary that
 
 \smallskip\centerline{$\sum\limits_{k=1}^{\infty}\big|f_k\big(\frac{\mu\lambda_k}{h}+\mu-1\big)\big|(\lambda_k+\alpha)\le h-\alpha,$}
 
 \noi where $h>0, \alpha\in [0, h)$ (Theorem~2).} Neighborhoods of such functions are investigated. Ordinary Hadamard compositions and Hadamard compositions of the genus $m$ were also studied.