Sufficient conditions for univalence obtained by using Briot-Bouquet differential subordination

Abstract

The concept of differential subordination was introduced in [3] by S.S. Miller and P.T. Mocanu and the concept of strong differential subordination was introduced in [1], [2] by J.A. Antonino and S. Romaguera.In [5] we have studied the strong differential subordinations in the general case and in [6] we have studied the first order linear strong differential subordinations. In this paper we study the second order linear strong differential subordinations. Our results may be applied to deduce sufficient conditions for univalence in the unit disc, such as starlikeness, convexity, alpha-convexity, close-to-convexity respectively.

Sufficient conditions for univalence obtained by using the Ruscheweyh-Bernardi differential-integral operator

“…Studying subordination properties by using differential and integral operators is a classic topic still of interest at this time, interesting results being currently obtained in forms of criteria for univalence of functions. A recent approach in using operators is to mix a differential and an integral operator as it the case in the very recent papers [1], [16] and [19]. This idea is also used in the present paper for introducing a new differential-integral operator mixing Ruscheweyh differential operator and Bernardi integral operator and by using it, a new class of univalent functions.…”

Starlikeness Condition for a New Differential-Integral Operator

“…This theory has remarkable applications allowing easier proofs of already known results and facilitating the emergence of new ones. The idea of combining integral and differential operators is illustrated in the very recent paper [4] where a differential-integral operator was defined and using the method of the subordination chains, differential subordinations in their special Briot-Bouquet form were studied obtaining their best dominant and, as a consequence, criteria containing sufficient conditions for univalence were formulated. Similar work containing subordination results related to a class of univalent functions obtained by the use of an operator introduced by using a differential operator and an integral one can be seen in [5].…”