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

Abstract

In this paper we consider the Hankel determinant H2(3) = a3a5 − a4 2 defined for the coefficients of a function f which belongs to the class S of univalent functions or to its subclasses: S * of starlike functions, K of convex functions and R of functions whose derivative has a positive real part. Bounds of |H2(3)| for these classes are found; the bound for R is sharp. Moreover, the sharp results for starlike functions and convex functions for which a2 = 0 are obtained. It is also proved that max{|H2(3)| : f ∈ S} is greater than 1.

Some Further Coefficient Bounds on a New Subclass of Analytic Functions

“…we know H 2,3 f β = |H 2,3 ( f )| for every β ∈ R. This makes it possible to assume that when estimating |H 2,3 ( f )|, one selected coefficient of f is a non-negative real number, see [38]. Assume b 2 is real and p 1 := p ∈ [0, 2].…”

Two types of the second Hankel determinant for the class U and the general class S

“…(9) As it has been shown in [2, p. 57], if f is given by (1), then the coefficients a 2 , a 3 , a 4 and a 5 are expressed by the Grunsky's coefficients ω 2p−1,2q−1 of the function f2 given by (7) in the following way:…”

Sharp Coefficient Bounds for Starlike Functions Associated with Cosine Function

“…Several authors have determined the maximum value of |H 2,1 ( f )| and the upper bound for |H 2,2 ( f )| for various subclasses of A, see for example [13,14,22]. Results regarding the second Hankel determinant H 2,3 ( f ) could be found also in the recent papers like [23,28,32]. The determinant H 2,3 ( f ) is not much studied in the literature.…”