In this paper, we investigate the properties of neighbourhoods of functions for the classes U CV (α) and Sp(α). First we established an inclusion relationship between them and proved a necessary and sufficient condition interms of convolutions for a function f to be in Sp(α). Next we show that the class Sp(α) is closed under convolution with functions f (z) which are convex univalent. The resul...

We prove that if Y is a closed subspace of a Banach space X such that Y and X/Y admit an equivalent asymptotically uniformly smooth norm, then X also admits an equivalent asymptotically uniformly smooth norm. The proof is based on the use of the Szlenk index and yields a few other applications to renorming theory.