On the Fekete–Szegö type functionals for starlike and convex functions

Abstract

In the paper we discuss two functionals of the Fekete–Szegö type: $Phi f$ (μ) = a2a4 − $μa3^2$ and $Theta f$ (μ) = a4 − μa2a3 for an analytic function f (z) = z + $a2z^2$ + $a3z^3$ + . . . , z ∈ ∆ , ( ∆ = {z ∈ C : |z| < 1} ) and a real number μ .We focus our research on the estimation of |$Phi f$ (μ)| and |$Theta f$(μ)| , while f is either in S∗ (the class of starlike functions) or in K (the class of convex functions).