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.