Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions

Abstract

In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series Eϕ∗(ς,z) to belong to the inclusive subclass Πη(Q3,Q2,Q1,Q0). Necessary and sufficient conditions are also given for the generalized distribution series Eϕ∗(ς,z)ℏ and the integral operator Jςϕ(z) to be in the inclusive subclass Πη(Q3,Q2,Q1,0). Further, we provide a number of corollaries, which improve the existing ones that are available in some recent studies. The results presented here not only improve the earlier studies, but also give rise to a number of new results for particular choices of Q3,Q2,Q1 and Q0.

Keywords

• analytic
• univalent
• generalized distribution
• integral operator