Applications of the Jack’s lemma for the meromorphic functions

Abstract

In this paper, we give some results on $\frac{zf^{\prime }(z)}{f(z)}$ for the certain classes of f(z) meromorphic functions. For the function $f(z)= \frac{1}{z}+c_{0}+c_{1}z+c_{2}z^{2}+\cdots$ defined in the punctured disc $U=\left\{ z:0<\left| z\right| <1\right\}$ such that $f(z)\in \mathcal {M}_{\alpha }$, we estimate a modulus of the angular derivative $\frac{zf^{\prime }(z)}{f(z)}$ function at the boundary point b with $\frac{ bf^{\prime }(b)}{f(b)}=-\alpha$. Moreover, the module of the angular derivative of the function $\frac{zf^{\prime }(z)}{f(z)}$ at the boundary point b will be strengthened from the below, taking into account the zeros of the function $\frac{zf^{\prime }(z)}{f(z)}+1$ different from $z=0$.