First order differential subordination associated with Cassini curve

Abstract

Let p be an analytic function defined on the open unit disc with 01p. In this paper, we determine the conditions for  such that certain subordination properties hold for pz is subjected to certain geometric conditions involving the expressions 1',zpz 1'/zpzpz and 21'/zpzpz of which each is subordinated to 1cz and the condition for  is determined.