Area Problem for Univalent Functions in the Unit Disk with Quasiconformal Extension to the Plane

Abstract

Let Δ (r, f) denote the area of the image of the subdisk |z|<r,0<r≤1, under an analytic function f in the unit disk | z| < 1. Without loss of generality, in this context, we consider only the analytic functions f in the unit disk with the normalization f(0) = 0 = f′(0) - 1. We set Ff(z) = z/ f(z). Our objective in this paper is to obtain a sharp upper bound of Δ (r, Ff) , when f varies over the class of normalized analytic univalent functions in the unit disk with quasiconformal extension to the entire complex plane. © 2018, Iranian Mathematical Society.