Univalence and Starlikeness of Certain Classes of Analytic Functions

Abstract

For the analytic functions ϕ(ζ)=ζ+∑k=n∞ϕkζk in the unit disk O, we calculate the values of n and α, where the condition ℜ1+ζϕ″(ζ)/ϕ′(ζ)>−α or ℜ1+ζϕ″(ζ)/ϕ′(ζ)1+α/2 yields univalence and starlikeness. Conditions imply ϕ in the class where all normalized analytic functions U, with ζ/ϕ(ζ)2ϕ′(ζ)−11 are obtained. Recent findings are gained, and unique cases are demonstrated. The generalization of the Jack lemma serves the proof of the main result and that our methodology is based on the idea of subordination.

Keywords

• analytic function
• subordination
• starlike function
• univalent function
• Jack lemma
• open unit disk