On the rigidity part of Schwarz Lemma

Abstract

We consider the rigidity part of Schwarz Lemma. Let f be a holomorphic function in the unit disc D and |ℜf(z)| <1 for |z| < 1. We generalize the rigidity of holomorphic function and provide sufficient conditions on the local behaviour of f near a finite set of boundary points that needs f to be a finite Blaschke product. For a different version of the rigidity theorems of D. Burns-S. Krantz and D. Chelst, we present some more general results in which the bilogaritmic concave majorants are used. The strategy of these results relies on a special version of Phragmen-Lindelof princible and Harnack inequality.