Note on some classes of holomorphic functions related to Jack’s and Schwarz’s lemma

Abstract

In this paper we discuss holomorphic mappings f of the unit disc U and corresponding index defined as If (z) = zf?(z) f(z). We are interested in finding bounds on the growth of functions f and related issues, if there are known some properties of If on U. Our main tool in accomplishing this connection is Jack?s lemma. As a special case, we got estimates on the growth of some classes of ?-starlike functions as well as some interesting generalisations.