An Internal Polya Inequality for ℂ-Convex Domains in ℂn

Abstract

Let K ⊂ ℂ be a polynomially convex compact set, f be a function analytic in a domain \(\overline{\mathbb{C}} \backslash K\) with Taylor expansion \(f(z) = \sum\nolimits_{k = 0}^\infty {{a_k}/{z^{k + 1}}} \) at ∞, and \({H_i}(f): = {\rm{det}}({a_{k + l}})_{k,l = 0}^i\) be the related Hankel determinants. The classical Polya theorem [11] says that \(\mathop {{\rm{lim\; sup}}}\limits_{i \to \infty } \;{\rm{|}}{H_i}(f){{\rm{|}}^{1/{i^2}}} \le d(K),\) 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 ℂ-convex (= strictly linearly convex) domain in ℂⁿ.