Pseudostarlike and pseudoconvex Dirichlet series of the order $\alpha$ and the type $\beta$

The concepts of the pseudostarlikeness of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$ and the pseudoconvexity of order $\alpha$ and type $\beta$ are introduced for Dirichlet series with null abscissa of absolute convergence. In terms of coefficients, the pseudostarlikeness and the pseudoconvexity criteria of order $\alpha$ and type $\beta$ are proved.Let $h\ge 1$, $\Lambda=(\lambda_k)$ be an increasing to $+\infty$ sequence of positive numbers ($\lambda_1>h$. We call a conformal function of the form $F(s)=e^{sh}+\sum\nolimits_{k=1}^{\infty}f_k\exp\{s\lambda_k\}, \ s=\sigma+it,$in $\Pi_0=\{s\colon \, \text{Re}\,s<0\}$ pseudostarlike of order $\alpha\in [0,\,1)$ and type$\beta \in (0,\,1]$ if\begin{equation*}\left|\frac{F'(s)}{F(s)}-h\right|<\beta\left|\frac{F'(s)}{F(s)}-(2\alpha-h)\right|,\quad s\in \Pi_0.\end{equation*}The main results of the article are contained in Theorems 1 and 2. Theorem 1 states: \textit{If $\alpha \in [0, \, 1)$ and $\beta \in (0, \, 1]$ such that\begin{equation*}\sum\limits_{k=1}^{\infty}\{(1+\beta)\lambda_k -2\beta\alpha -h(1-\beta)\}|f_k|\le 2\beta (h-\alpha)\label{t7}\end{equation*}then the function $F$ is pseudostarlike of order $\alpha$ and type $\beta$.}The corresponding results for Hadamard compositions of such series are also established.