Abstract

An analytic function f(z) in the unit disc D is called stable if sn(f,·)/f ? 1/f holds for all for n ? ?0. Here sn stands for the nth partial sum of the Taylor expansion about the origin of f, and ? denotes the subordination of analytic functions in D. We prove that (1 - z ?, ? ? [-1, 1], are stable. The stability of (1 + z)/(1 - z) turns out to be equivalent to a famous result of Vietoris on non-negative trigonometric sums. We discuss some generalizations of these results, and related conjectures, always with an eye on applications to positivity results for trigonometric and other polynomials. © 2003 Elsevier Inc. All rights reserved.