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|