## List of issues > Series «Mathematics». 2015. Vol. 11

##
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 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

*T*

*S*

^{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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