On certain classes of fractional $p$-valent analytic functions

The theory of analytic functions and more specific $p$-valent functions, is one of the most fascinating topics in one complex variable. There are many remarkable theorems dealing with extremal problems for a class of $p$-valent functions on the unit disk $\mathbb{U}$. Recently, many researchers have shown great interests in the study of differential operator. The objective of this paper is to define a new generalized derivative operator of $p$-valent analytic functions of fractional power in the open unit disk $\mathbb{U}$ denoted by $\mathcal{D}_ {\lambda_{1},\lambda_{2},p,\alpha}^{m,b}f(z)$. This operator generalized some well-known operators studied earlier, we mention some of them in the present paper. Motivated by the generalized derivative operator $\mathcal{D}_ {\lambda_{1},\lambda_{2},p,\alpha}^{m,b}f(z)$, we introduce and investigate two new subclasses $S^{m,b}_{\lambda_1,\lambda_2,p,\alpha}(\mu,\nu)$ and $T S^{m,b}_{\lambda_1,\lambda_2,p,\alpha}(\mu,\nu)$, which are subclasses of starlike $p$-valent analytic functions of fractional power with positive coefficients and starlike $p$-valent analytic functions of fractional power with negative coefficients, respectively. In addition, a sufficient condition for functions $f \in \Sigma_{p,\alpha}$ to be in the class $S^{m,b}_{\lambda_1,\lambda_2,p,\alpha}(\mu,\nu)$ and a necessary and sufficient condition for functions $f \in T_{p,\alpha}$ will be obtained. Some corollaries are also pointed out. Moreover, we determine the extreme points of functions belong to the class $T S^{m,b}_{\lambda_1,\lambda_2,p,\alpha}(\mu,\nu)$.