On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis

Abstract

In this paper we consider two functionals of the Fekete–Szegö type: $\Phi _f(\mu ) = a_2 a_4-\mu a_3{}^2$ and $\Theta _f(\mu ) = a_4-\mu a_2a_3$ for analytic functions $f(z) = z+a_2z^2+a_3z^3+\ldots $, $z\in \Delta $, ($\Delta = \lbrace z\in \mathbb{C}:|z|<1\rbrace $) and for real numbers $\mu $. For $f$ which is univalent and convex in the direction of the imaginary axis, we find sharp bounds of the functionals $\Phi _f(\mu )$ and $\Theta _f(\mu )$. It is possible to transfer the results onto the class $\mathcal{K}_{\mathbb{R}}(i)$ of functions convex in the direction of the imaginary axis with real coefficients as well as onto the class $\mathcal{T}$ of typically real functions. As corollaries, we obtain bounds of the second Hankel determinant in $\mathcal{K}_{\mathbb{R}}(i)$ and $\mathcal{T}$.