ON PSEUDOSTARLIKE AND PSEUDOCONVEX DIRICHLET SERIES

Abstract

The concepts of the pseudostarlikeness of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$ and the pseudoconvexity of the order $\alpha$ and type $\beta$ are introduced for Dirichlet series of the form $F(s)=e^{-sh}+\sum_{j=1}^{n}a_j\exp\{-sh_j\}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$, where $h>h_n>\dots>h_1\ge 1$ and $(\lambda_k)$ is an increasing to $+\infty$ sequence of positive numbers. Criteria for pseudostarlikeness and pseudoconvexity in terms of coefficients are proved. The obtained results are applied to the study of meromorphic starlikeness and convexity of the Laurent series \break $f(s)=1/z^p+\sum_{j=1}^{p-1}a_j/z^j+\sum_{k=1}^{\infty}f_kz^k$. Conditions, under which the differential equation $w''+\gamma w'+(\delta e^{2sh}+\tau)w=0$ has a pseudostarlike or pseudoconvex solution of the order $\alpha$ and the type $\beta=1$ are investigated.