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

Abstract

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 H2,2{H_{2,2}} or H3,1{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 𝒦ℝ\u2062(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.