Neighborhoods of Dirichlet series absolutely convergent in a half-plane

Анотація

For an absolutely convergent in the half-plane ${\Pi }_0=\left\{s: \mathit{R}\mathit{e} s<0\right\}$ Dirichlet series $F\left(s\right)=e^s+\sum _{k=1}^{\infty }{f}_{k}exp\left\{s{\lambda }_k\right\}$ the set $O_{j,\delta }\left(F\right)$ of Dirichlet series $G\left(s\right)=e^s+\sum _{k=1}^{\infty }{g}_k exp\left\{s{\lambda }_k\right\}$ such that $\sum _{k=1}^{\infty }{\lambda }_k^j\left|g_k-f_k\right|\le \delta$ is called a neighborhood of the function $F$. It is proved that each function $G$ from the neighborhood $O_{1,1}\left(E\right)$ of the function  $E\left(s\right)=e^s$ is a pseudostarlike function in ${\Pi }_0$, and if $G$ is pseudostarlike in ${\Pi }_0$ and $g_k\le 0$ for all $k\ge 1$ then $G\in O_{1,1}\left(E\right)$. A similar connection exists between $O_{2,1}\left(E\right)$ and pseudoconvex functions in ${\Pi }_0$. The neighborhoods $O_{1,\delta }\left(F\right)$ and $O_{2,\delta }\left(F\right)$ are investigated also in the cases when $F$ is either pseudostarlike or pseudoconvex of the order $\alpha \in [0,1)$ and the type $\beta \in (0,1)$. Assuming that $0\le \alpha <1$, $0<\beta <{\beta }_1\le 1$, the coefficients $f_k$ and $g_k$ are negative and the function $F$ is pseudostarlike of the order $\alpha$ and the type $\beta$ it is proved, in particular, that if $G\in O_{1,\delta }\left(F\right)$ with $\delta =\frac{2(1-\alpha )({\beta }_1-A\beta )}{1+{\beta }_1}$, where $A=\frac{(1+{\beta }_1){\lambda }_1-2\alpha {\beta }_1-(1-{\beta }_1)}{(1+\beta ){\lambda }_1-2\alpha \beta -(1-\beta )}$, then $G$ is pseudostarlike of the order $\alpha$ and the type ${\beta }_1$. On the contrary, if $G$ is pseudostarlike of the order $\alpha$ and the type ${\beta }_1$ then $G\in O_{1,\delta }\left(F\right)$ with

$\delta =2{\lambda }_1(1-\alpha )\left(\frac{{\beta }_1}{{\lambda }_1(1+{\beta }_1)-2\alpha {\beta }_1-(1-{\beta }_1)}+\frac{\beta }{{\lambda }_1(1+\beta )-2\alpha \beta -(1-\beta )}\right)$.