A continuous extension of the de la Vallée Poussin means

Abstract

Among the many interesting results of their 1958 paper, G. Polya and I. J. Schoenberg studied the de la Vallee Poussin means of analytic functions. These are polynomial approximations of a given analytic function on the unit disk obtained by taking Hadamard products of the functionf with certain polynomialsV n (z), wheren is the degree of the polynomial. The polynomial approximationsV n *f converge locally uniformly tof asn→∞. In this paper, we define a subordination chainV λ (z),γ>0, |z|<1, of convex mappings of the disk that for integer values is the same as the previously definedV n (z). Iff is a conformal mapping of the diskD onto a convex domain, thenV λ *f→f locally uniformly as λ→∞, and in fact $$V_{\lambda _1 } * f(\mathbb{D}) \subset V_{\lambda _2 } * f(\mathbb{D}) \subset f(\mathbb{D})$$ when λ2 > λ1. We also consider Hadamard products of theV λ with complex-valued harmonic mappings of the disk.