Area Estimates of Images of Disks under Analytic Functions

Abstract

Let Dr := {z = x + iy ϵ C : |z| < r}, r ≤ 1. For a normalized analytic function f in the unit disk D := D1, estimating the Dirichlet integral (Formula presented.)., is an important classical problem in complex analysis. Geometrically, Δ(r, f) represents the area of the image of Dr under f counting multiplicities. In this paper, our main objective is to estimate areas of images of Dr under non-vanishing analytic functions of the form (z/f)μ, μ > 0, in principal powers, when f ranges over certain classes of analytic and univalent functions in Dr. © 2021, Springer-Verlag GmbH Germany & The Editorial Office of AMS.