Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series

Abstract

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