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 $\left\vert \Re f\right\vert <1$ for $\left\vert
z\right\vert <1$, 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 $\left\vert f^{\prime }(b)\right\vert$ according to the first
nonzero Taylor coefficient of about two zeros, namely $z=0$ and $z_{0}\neq 0$%
. Moreover, two example for our results is considered.