On Hankel Determinant {{varvec{H}}}_mathbf{2}{} mathbf{(3)} for Univalent Functions

Paweł Zaprawa

doi:10.1007/s00025-018-0854-1

Abstract

In this paper we consider the Hankel determinant H_2(3) = a_3a_5 - a_4{}^2 defined for the coefficients of a function f which belongs to the class mathcal {S} of univalent functions or to its subclasses: S^* of starlike functions, mathcal {K} of convex functions and mathcal {R} of functions whose derivative has a positive real part. Bounds of |H_2(3)| for these classes are found; the bound for mathcal {R} is sharp. Moreover, the sharp results for starlike functions and convex functions for which a_2=0 are obtained. It is also proved that max {|H_2(3)|: fin mathcal {S}} is greater than 1.

Highlights

Let Δ be the unit disk {z ∈ C : |z| < 1} and A be the family of all functions f analytic in Δ, normalized by the condition f (0) = f (0) − 1 = 0
Math the main subclasses of class S consisting of univalent functions
A few papers are devoted to some subclasses of Sσ of bi-univalent functions

Summary

Introduction

Let Δ be the unit disk {z ∈ C : |z| < 1} and A be the family of all functions f analytic in Δ, normalized by the condition f (0) = f (0) − 1 = 0. The functions in A are of the form f (z) = z + a2z2 + a3z3 + · · ·. Pommerenke (see, [19,20]) defined the k-th Hankel determinant for a function f as an an+1 . In recent years many mathematicians have investigated Hankel determinants for various classes of functions contained in A.

Results

Auxiliary Lemmas

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