Some Remarks Associated With the Schwarz Lemma on the Boundary

Abstract

The aim of this paper is to introduce the class of the analytic functions called $\mathcal{N}\left( \alpha ,c\right) $ and to investigate the various properties of the functions belonging to this class. For the functions in this class, some inequalities related to the angular derivative have been obtained. That is, the modulus of the angular derivative of the function $%
\frac{zf^{\prime }(z)}{f(z)}$ under hypothesis $
\sum\limits_{p=2}^{\infty }\{ ( p-1) +\vert c(
1+\alpha ) +\alpha ( p-1) \vert\} \vert
c_{p}\vert \leq \vert c( 1+\alpha ) \vert $ is estimated from below.