On the Schwarz lemma at the upper half plane

Abstract

In this paper, we give a simple proof for the boundary Schwarz lemma at the upper half plane. Considering that f (z) is a holomorphic function defined on the upper half plane, we derive inequalities for the modulus of derivative of f (z), |f'(0)|; by assuming that the f (z) function is also holomorphic at the
boundary point z = 0 on the real axis with f(0) = Imf(i).