The Fekete Szeg&amp;amp;#246; Functional and Second Hankel Determinant for a Certain Sublass of Analytic Functions

Rasheed O Ayinla,Timothy O Opoola

doi:10.4236/am.2019.1012074

Abstract

Let S denote the class of functions that are analytic, normalized and univalent in the open unit disk E = {z: |z| S* and C respectively. A new subclass of analytic functions that generalize some known subclasses of analytic functions was defined and investigated. We obtained coefficient bounds, upper estimates for the Fekete-Szeg&#246 functional and the Hankel determinant.

Highlights

Let A denote the class of functions f ( z ) =z + a2 z2 + a3 z3 + a4 z4 + (1.1){ } which are analytic in the open unit dis= k U z : z < 1 and satisfy the condition f (0) = 0 and f ′(0) = 1.Let S denote the subclass of A consisting of univalent in U
Let S denote the class of functions that are analytic, normalized and univalent in the open unit dis= k E {z : z < 1}
Subclasses of S are the class of starlike and convex functions denoted by S* and C respectively

Summary

Introduction

Pommerenke [4] investigated the Hankel determinant of areally mean p-valent functions, univalent functions as well as starlike functions. Noor [5] investigated the Hankel determinant problem for the class of functions with bounded boundary rotation. Janteng et al [6] studied the sharp upper bound for second Hankel determinant H2= (2) a2a4 − a32 for univalent functions whose derivative has positive real parts. Lee et al [7] obtained bounds on second Hankel determinants belonging to the subclasses of Ma-Minda starlike and convex functions. We obtained the coefficient bound, Fekete-Szegö functional and second Hankel determinant for the functions belonging to the subclass Cn (β ,γ ).

Preliminary Lemmas

Main Results

Conclusion