Third Hankel Determinants for Subclasses of Univalent Functions

Abstract

Abstract. The main aim of this paper is to discuss the third Hankel determinants for three classes: S * of starlike functions, K of convex functions and R of functions whose derivative has a positive real part. Moreover, the sharp results for twofold and threefold symmetric functions from these classes are obtained.Mathematics Subject Classification. 30C50.

The Bound of the Hankel Determinant of the Third Kind for Starlike Functions

“…Although the constant 8/9 improves essentially the estimates found in [1] and [19], it is not the best possible. To find the sharp estimate of the Hankel determinant H 3,1 ( f ) for starlike functions is still an open problem.…”

“…Estimating each term of the right hand of (1.4) Babalola [1] showed that |H 3,1 ( f )| ≤ 16. In [19] Zaprawa by a suitable grouping and using Lemma 1 due to Livingston [11] proved that |H 3,1 ( f )| ≤ 1.…”

The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions with Real Coefficients

“…For the estimates on the Hankel determinant H 3,1 ( f ) over the class S * , Babalola [17] obtained the inequality |H 3,1 ( f )| ≤ 16. And Zaprawa [18] improved the result by proving |H 3,1 ( f )| ≤ 1. Next, Kwon et.…”

On Hankel Determinant $${{\varvec{H}}}_\mathbf{2}{} \mathbf{(3)}$$ H 2 ( 3 ) for Univalent Functions

“…In recent years many mathematicians have investigated Hankel determinants for various classes of functions contained in A. These studies focus on the main subclasses of class S consisting of univalent functions (see, [1,3,[8][9][10][14][15][16]21,22,24,25]). A few papers are devoted to some subclasses of S σ of bi-univalent functions (see, [4,17]).…”