An Internal Polya Inequality for $\mathbb{C}$-Convex Domains in $\mathbb{C}^{n}$

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(z) =\sum_{k=0}^{\infty }a_{k}/z^{k+1} $ at $\infty $, and $H_{i}(f) :=\det (a_{k+l}) _{k,l=0}^{i}$ be the related Hankel determinants. The classical Polya theorem [11] says that
$$ \limsup_{i\to \infty }\vert H_{i}(f) \vert ^{1/i^{2}}\leq d(K) , $$
where $d(K) $ is the transfinite diameter of $K$. The main result of this paper is a multivariate analog of the Polya inequality for a weighted Hankel-type determinant constructed from the Taylor series of a function analytic on a $\mathbb{C}$-convex (= strictly linearly convex) domain in $\mathbb{C}^{n}$.