On holomorphic motions of n-symmetric functions

Abstract

We generalize a problem examined by Duren on the univalence of a family of n-symmetric functions generated by integrals of functions of the form exp(λζn). Our approach is based on the use of the inverse Faber transform, of the Martio-Sarvas univalence criterion, and of the λ-lemma of Ma né, Sad, and Sullivan. We also put forward a conjecture on the univalence of a family of n-symmetric functions, which is a weakened form of the Danikas-Ruscheweyh conjecture on the univalence of an integral transform of holomorphic functions. © 2010 Pleiades Publishing, Ltd.