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

Let Delta (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 (r, Ff), when f varies over the class of normalized analytic univalent functions in the unit diskwith quasiconformal extension to the entire complex plane.