Growth and monotonicity properties for elliptically schlicht functions

Abstract

Let f f be a holomorphic function of the unit disc D , \mathbb {D}, with f ( D ) ⊂ D f(\mathbb {D})\subset \mathbb {D} and f ( 0 ) = 0 f(0)=0 . Littlewood’s generalization of Schwarz’s lemma asserts that for every w ∈ f ( D ) , w\in f(\mathbb {D}), we have | w | ≤ ∏ j | z j | , |w|\leq \prod _{j}|z_{j}|, where { z j } j \{z_{j}\}_j are the pre-images of w . w. We consider elliptically schlicht functions and we prove an analogous bound involving the elliptic capacity of the image. For these functions, we also study monotonicity theorems involving the elliptic radius and elliptic diameter.