Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series

The article introduces the concepts of pseudostarlikeness and pseudoconvexity in direction absolutely converges $\Pi_0=\{s\in\mathbb{C}^p\colon \text{Re}\,s<0\}$, $p\in\mathbb{N},$ multiple Dirichlet series form$$ F(s)=e^{(h,s)}+\sum\nolimits_{\|(n)\|=\|(n^0)\|}^{+\infty}f_{(n)}\exp\{(\lambda_{(n)},s)\}, \quad s=(s_1,...,s_p)\in {\mathbb C}^p,\quad p\geq 1,$$where $ \lambda_{(n^0)}>h$, $\text{Re}\,s<0\Longleftrightarrow (\text{Re}\,s_1<0,...,\text{Re}\,s_p<0)$,$h=(h_1,...,h_p)\in R}^p_+$, $(n)=(n_1,...,n_p)\in N}^p$, $(n^0)=(n^0_1,...,n^0_p)\in $\|(n)\|=n_1+...+n_p$ sequences$\lambda_{(n)}=(\lambda^{(1)}_{n_1},...,\lambda^{(p)}_{n_p})$ are such that $0<h_j<\lambda^{(j)}_1<\lambda^{(j)}_k<\lambda^{(j)}_{k+1}\uparrow+\infty$as $k\to+\infty$ for every $j\in\{1,...,p\}$, $(a,c)=a_1c_1+...+a_pc_p$ $a=(a_1,...,a_p)$ $c=(c_1,...,c_p)$. We say $a>c$ if $a_j\ge c_j$ all $1\le j\le p$ there exists at least one $j$ $a_j> c_j$. Let ${\bf b}=(b_1,...,b_p)$ $\partial_{{\bf b}}F( {s})=\sum\limits_{j=1}^p b_j\dfrac{\partial F( {s})}{\partial {s}_j}$ be derivative $F$ b}$. In this paper, particular, following assertions were obtained: 1) If b}>0$ and$\sum\limits_{\|(n)\|=k_0}^{+\infty}(\lambda_{(n)},{\bf b})|f_{(n)}|\le (h,{\bf b})$then {s})\not=0$ $\Pi_0:=\{s\colon i.e. is conformal $\Pi_0$ b}$ (Proposition 1).2) function pseudostarlike order $\alpha\in [0,\,(h,{\bf b}))$ type$\beta >0$ if$\Big|\frac{\partial_{{\bf {s})}{F(s)}-(h, {\bf b})\Big|<\beta\Big|\frac{\partial_{{\bf {s})}{F(s)}-(2\alpha-(h, b}))\Big|,\quad s\in \Pi_0.$Let $0\le \alpha<(h,{\bf b})$ $\beta>0$. ispseudostarlike $\alpha$ type $\beta$ b}> 0$, it sufficient case, when $f_{(n)}\le necessary that$\sum\limits_{\|(n)\|=k_0}^{+\infty}\{((1+\beta)\lambda_{(n)}-(1-\beta)h,{\bf b})-2\beta\alpha\}|f_{(n)}|\le 2\beta ((h,{\bf b})-\alpha)$ (Theorem 1).