The Internal Polya Inequality for $\mathbb{C}$-convex Domains in ${\mathbb{C}}^n$

Abstract

Let $K\subset \mathbb{C}$ be a polynomially convex compact set, $f$ be a function analytic in a domain $\overline{\mathbb{C}}\setminus K$ with Taylor expansion $f\left(z\right)={\sum }_{k=0}^{\infty }\frac{a_k}{z^{k+1}}$ at $\infty$, and $H_s\left(f\right):=\det {\left(a_{k+l}\right)}_{k,l=0}^s$ related Hankel determinants. The classical Polya theorem \cite% {P} says that \[ \limsup_{s\rightarrow \infty }\left\vert H_{s}\left( f\right) \right\vert ^{1/s^{2}}\leq d\left( K\right) , \]% where $d\left(K\right)$ is the transfinite diameter of $K$. The main result of this paper is multivariate internal analogs of Polya's inequality for $\mathbb{C}$-convex (=strictly linearly convex) domains $D\subset {\mathbb{C}}^n$ and weighted Hankel-type determinants, constructed from the Taylor coefficients of a function $f\in A\left(D\right)$ at a given point ; therewith the weights are generated by $s$-indicatrices of the sequence of analytic functionals biorthogonal to the system of monomials in . It is proved by the reduction to the outer multivariate analog of Polya's inequality (Zakharyuta, Math. USSR Sbornik, \textbf{25 }% (1975)) and is based on the characterization of the strict linear convexity in terms of $s$-indicatrices (S. Znamenskii, Siberian Math. J. \textbf{26 } (1985)).