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 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 U denoted by D^m,b_{λ1λ2 p,α}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 D^m,b_{λ1λ2 p,α}f(z) we introduce and investigate two new subclasses S^m,b_λ1,λ2,p,α(μ,ν) and TS^m,b_λ1,λ2,p,α(μ,ν), 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 ∈ Σ_p,α to be in the class S^m,b_λ1,λ2,p,α(μ,ν) and a necessary and sufficient condition for functions f ∈ T_p,α will be obtained. Some corollaries are also pointed out. Moreover, we determine the extreme points of functions belong to the class TS^m,b_λ1,λ2,p,α(μ,ν).

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