Univalence criteria related with Ruscheweyh and Salagean derivatives and Loewner Chains

TL;DR

This paper uses Loewner chains to establish new sufficient conditions for the univalence of functions involving Ruscheweyh and Salagean derivatives, simplifying existing criteria and extending known results in complex analysis.

Contribution

It introduces new, simpler univalence criteria for integral operators involving Ruscheweyh and Salagean derivatives using Loewner chains, extending classical results.

Findings

Derived sufficient conditions for univalence involving Ruscheweyh and Salagean derivatives.

Reproduced classical univalence criteria as special cases.

Proposed new simpler conditions for univalence of integral operators.

Abstract

In this paper we obtain, by the method of Loewner chains, some sufficient conditions for the analyticity and the univalence of the functions defined by an integral operator. These conditions in- volves Ruscheweyh and Salagean derivative operator in the open unit disk. In particular cases, we find the well-known conditions for univalency established by Becker [3], Ahlfors [2], Kanas and Srivastava [8] and others for analytic mappings f : U ! C: Also, we obtain the corresponding new, useful and simpler conditions for this integral operator.