Third Hankel determinants for two classes of analytic functions with real coefficients

Abstract

In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

The sharp bound of the third Hankel determinant of the kth-root transformation for bounded turning functions

“…> 0 in the unit disc D. As a consequence of this paper, many articles containing results associated with the third Hankel determinant (see [9,10,17,18,19,21,24,26]) for specific subsets of holomorphic functions were obtained. For our study in this paper, we choose H 3,1,k (f ), called as the third order Hankel determinant of the k throot tranformation for bounded turning functions.…”

The Sharp Bound of the Third Hankel Determinant of kth- Root Transformation for Functions Associated with Lemniscate of Bernoulli

“…> 0 in the unit disc D. As a consequence of this paper, many articles containing results associated with the third Hankel determinant (see [11,14,21,22,23,24,26]) for specific subsets of holomorphic functions were obtained. For our study in this paper, we choose H 3,1,k (f ), called as third Hankel determinant of k th root tranformation for the family of S * L .…”

The sharp bound of the third Hankel determinant for inverse of convex functions

“…In recent years, many authors are working on obtaining upper bounds (see [2,8,17,18,20]) and a few papers were devoted to the estimation of sharp upper bound to H 3,1 (f ), for certain subclasses of analytic functions (see [3,6,7,9,10,19]).…”