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 f < 1 for |z| < 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 f (b) according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and z 0 0. Moreover, two examples for our results are considered.

Bound estimates for the derivative of driving point impedance functions

“…V. N. Dubinin [5] strengthened the inequality f (b) ≥ 1 by involving zeros of the function f (z). In the last 15 years, there have been tremendous studies on Schwarz lemma at the boundary (see, [1], [5], [10], [11], [12], [18], [19] and references therein). In addition, M. Mateljević has given a review about some properties of hyperbolic metrics and various versions of Schwarz lemma in [13].…”