Boundary Schwarz Lemma for Holomorphic Functions

Abstract

In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function f (z) holomorphic in the unit disc and f (0) = 0 such that vertical bar R-f vertical bar < 1 for vertical bar z vertical bar < 1, we estimate a modulus of angular derivative of f (z) function at the boundary point b with f (b) = 1, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below vertical bar f'(b)vertical bar according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and z(0) not equal 0. Moreover, two examples for our results are considered.