Differential subordination related with exponential functions

Abstract

In this article, we study some first-order differential subordinations. We determine the conditions on β  such that 1 + β zp′(z) ≺ √1+cz) and 1 + β zp′(z) ≺ 1 + √(2z+ ¹/2 z²  implies p(z) ≺ e^z. Similarly, some results are also obtained for the expressions 1 + β zp′ (z)/p(z) and 1 + β zp′(z) / p²(z). We also give applications of these results to obtain some sufficient conditions for a function to be starlike related to exponential functions.