On certain classes of fractional p-valent analytic functions

Abstract

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 Dm,bλ1λ2 p,αf(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 Dm,bλ1λ2 p,αf(z) we introduce and investigate two new subclasses Sm,bλ1,λ2,p,α(μ,ν) and TSm,bλ1,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 ∈ Σp,α to be in the class Sm,bλ1,λ2,p,α(μ,ν) and a necessary and sufficient condition for functions f ∈ Tp,α will be obtained. Some corollaries are also pointed out. Moreover, we determine the extreme points of functions belong to the class TSm,bλ1,λ2,p,α(μ,ν).

Keywords

• analytic functions
• p-valent functions
• starlike functions
• derivative operator