Some generalization properties of $p$-valent functions

Abstract

Let $\mathcal{A}_{p}$ be the class of functions $f(z)$ of the form $$f(z)=z^{p}+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+\cdots, (p\in \mathbb{N}=\{1,2,3,\ldots\})$$ which are analytic in the open unit disc $\mathbb{U}$. In this article, we consider some generalization properties of the functions in $\mathcal{A}_{p}$ and generalize results by applying  fractional derivatives.