Representation with majorant of the Schwarz lemma at the boundary

Abstract

Let f be a holomorphic function in the unit disc and |f(z)?1| < 1 for |z| < 1. We generalize the uniqueness portion of Schwarz?s lemma and provide sufficient conditions on the local behavior of f near a finite set of boundary points that needed for f to be a finite Blaschke product.

Uniqueness part of Schwarz lemma for driving point impedance functions

“…In recent years, a boundary version of Schwarz lemma was investigated in S.G. Krantz [10], D. Burns and S.G. Krantz [5], Dov Chelts [6], H. P. Boas [4], M. Mateljevi´c [12], Ornek-Akyel [19,20] and a few other authors' papers. They studied the uniqueness portion of the Schwarz lemma.…”

Note on the uniqueness holomorphic function on the unit disk

“…Recently similar problems were investigated in ( [6]) and ( [7]). For more detail literature and the other types of the results, we refer to ( [8]), ( [9]), ( [5]), ( [10]) and references therein.…”

Uniqueness part of the Schwarz lemma at the boundary

“…For historical background about the Schwarz lemma and its applications on the boundary of the unit disc, we refer to (see [16]). Also, the similar to considered problem is studied in ( [15]).…”

“…They studied the uniqueness(or, rigidity) part of the Schwarz lemma. Also, the similar to considered problem is studied in ( [15]).…”