Pseudostarlikeness and Pseudoconvexity of Multiple Dirichlet Series

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 $0c$ 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