General coefficient estimates for bi-univalent functions: a new approach

Oqlah Al-Refai,Mohammed Ali

doi:10.3906/mat-1910-100

Abstract

We prove for univalent functions f(z)=z+∑∞k=nakzk;(n≥2) in the unit disk U={z:|z|<1}) with f−1(w)=w+∑∞k=nbkwk;(|w|<r0(f),r0(f)≥14) that b2n−1=na2n−a2n−1andbk=−akfor(n≤k≤2n−2). As applications, we find estimates for |an| whenever f is bi-univalent, bi-close-to-convex, bi-starlike, bi-convex, or for bi-univalent functions having positive real part derivatives in U. Moreover, we estimate |na2n−a2n−1| whenever f is univalent in U or belongs to certain subclasses of univalent functions. The estimation method can be applied for various subclasses of bi-univalent functions in U and it helps to improve well-known estimates and to generalize some known results as shown in the last section. % You shouldn't use formulas and citations in the abstract.

Highlights

Introduction and preliminaries LetA denote the class of functions f (z) of the form ∞f (z) = z + akzk, k=2 (1.1)which are analytic in the open unit disk U = {z ∈ C : |z| < 1} and normalized by f (0) = f ′(0) − 1 = 0 .let S be the subclass of A consisting of univalent functions in U
We prove for univalent functions f (z) =
We find estimates for |an| whenever f is bi-univalent, bi-close-to-convex, bi-starlike, bi-convex, or for bi-univalent functions having positive real part derivatives in U

Summary

Introduction

We estimate |na2n − a2n−1| whenever f is univalent in U or belongs to certain subclasses of univalent functions. Let S be the subclass of A consisting of univalent functions in U . The class P consists of analytic functions p satisfying p(0) = 1 and Re{p(z)} > 0 , (z ∈ U) . For every f ∈ S defined by (1.1), that the inverse function f −1 exists and has the form f −1(w) = w − a2w2 + (2a22 − a3)w3 − (5a32 − 5a2a3 + a4)w4 + · · · ,

Results

Conclusion