Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series

Abstract

Let $p\in {\Bbb N}$, $s=(s_1,\ldots,s_p)\in {\Bbb C}^p$, $h=(h_1,\ldots,h_p)\in {\Bbb R}^p_+$, $(n)=(n_1,\ldots,n_p)\in {\Bbb N}^p$ and the sequences $\lambda_{(n)}=(\lambda^{(1)}_{n_1},\ldots,\lambda^{(p)}_{n_p})$ are such that $0<\lambda^{(j)}_1<\lambda^{(j)}_k<\lambda^{(j)}_{k+1}\uparrow+\infty$ as $k\to\infty$ for every $j=1,\ldots,p$. For $a=(a_1,\ldots,a_p)$ and $c=(c_1,\ldots,c_p)$ let $(a,c)=a_1c_1+\ldots+a_pc_p$, and we say that $a>c$ if $a_j> c_j$ for all $1\le j\le p$. For a multiple Dirichlet series \begin{align*}F(s)=e^{(s,h)}+\sum\limits_{\lambda_{(n)}>h}f_{(n)}\exp\{(\lambda_{(n)},s)\}\end{align*} absolutely converges in $\Pi^p_0=\{s:\text{Re}\,s<0\}$, concepts of pseudostarlikeness and pseudoconvexity are introduced and criteria for pseudostarlikeness and the pseudoconvexity are proved. Using the obtained results, we investigated neighborhoods of multiple Dirichlet series, Hadamard compositions, and properties of solutions of some differential equations.