On the Fekete--Szeg\"{o} type functionals for starlike and convex functions


Abstract: In the paper we discuss two functionals of the Fekete--Szeg\"{o} type: $\Phi_f(\mu) = a_2 a_4-\mu a_3{}^2$ and $\Theta_f(\mu) = a_4-\mu a_2a_3$ for an analytic function $f(z) = z+a_2z^2+a_3z^3+\ldots$, $z\in\Delta$, ($\Delta = \{z\in\mathbb{C}:|z|<1\}$) and a real number $\mu$. We focus our research on the estimation of $|\Phi_f(\mu)|$ and $|\Theta_f(\mu)|$, while $f$ is either in $\mathcal{S}^\ast$ (the class of starlike functions) or in $\mathcal{K}$ (the class of convex functions).

Keywords: Starlike functions, convex functions, Hankel determinant, functional of Fekete--Szeg\"{o} type

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