Some lower bound for holomorphic functions at the boundary

Some lower bound for holomorphic functions at the boundary

**Authors : **

Bülent NafiÖrnek^{1*}

**Author Address : **

^{1}Department of Computer Engineering, Amasya University, Merkez-Amasya 05100, Turkey.

*Corresponding author.

**Abstract : **

In this paper, a boundary version of the Schwarz lemma for classes $mathcal{%H(alpha )}$ is investigated. For the function $f(z)=1+c_{1}z+c_{2}z^{2}+...$defined in the unit disc such that $f(z)in mathcal{H(alpha )}$ $left(0<alpha leq 1 ight) $, we estimate a modulus of the angular derivative of $f(z)$ function at the boundary point $b$ with $f(b)=e^{frac{pi alpha }{2}%}$. The sharpness of these inequalities is also proved.

**Keywords : **

Holomorphic function, Jack’s lemma, Angular derivative.

**DOI : **

**Article Info : **

*Received : * September 17, 2017; *Accepted : * December 12, 2017.