Growth Theorems for Holomorphic Functions Under Geometric Conditions for the Image

Abstract

Let \(f\) be a holomorphic function in the unit disc with image lying in the angle \(D_{\theta }=\{z\in \mathbb C :|\text{ Arg }(z)| 0,\) where \(C_{r}\) is the circle of radius \(r\) centered at the origin and \(m\) is the length-measure on \(C_{r}.\) Suppose next that \( f(\mathbb D )\subset \mathbb D \) and \(f(0)=0.\) Littlewood’s generalization of Schwarz lemma asserts that for every \(w\in f(\mathbb D ),\) we have \(|w|\le \prod _{j}|z_{j}(w)|,\) where \(\{z_{j}(w)\}\) are the pre-images of \(w.\) We prove a similar bound for the modulus \(|w|,\) involving the logarithmic capacity of the image, without the assumption \(f(\mathbb D )\subset \mathbb D .\) The main tools in the proofs are Steiner and circular symmetrization, Green function, hyperbolic and modulus metric, condensers and polarization.