Some classes of \(p\)-valent analytic functions defined by certain integral operator (Q1888258)

Let \({\mathcal A}(p)\), \(p\in \mathbb{N}\) be the class of functions \(f(z)=z^{p}+\sum_{k=1}^{\infty}a_{p+k}z^{p+k}\) analytic in the open unit disk \(E= \{z:| z| <1\}\). The author denotes by \(I_{n+p-1}:{\mathcal A}(p)\rightarrow {\mathcal A}(p)\) the integral operator defined by \[ I_{n+p-1}f(z)=(f_{n+p-1}^{(-1)}\ast f)(z)=\left[\frac{z^{p}}{(1-z)^{n+p}}\right]^{(-1)}\ast f(z),\quad n>-p \] Using this integral operator a new subclass \(S_{n}(p,\alpha)\) of \({\mathcal A}(p)\), \(0\leq\alpha<p\) is introduced as follows: \[ S_{n}(p,\alpha)=\left\{f\in {\mathcal A}(p):I_{n+p-1}f\in S(p,\alpha)\right\} \] where \(S(p,\alpha)\) denotes the class of \(p\)-valently starlike functions of order \(\alpha\), \(0\leq\alpha<p\). The author obtains some interesting properties of the class \(S_{n}(p,\alpha)\).