Sălăgean Differential Operator in Connection with Stirling Numbers

Abstract

Sălăgean differential operator Dκ plays an important role in the geometric function theory, where many studies are using this operator to introduce new subclasses of analytic functions defined in the open unit disk. Studies of Sălăgean differential operator Dκ in connection with Stirling numbers are relatively new. In this paper, the differential operator Dκ involving Stirling numbers is considered. A new sufficient condition involving Stirling numbers for the series Υθs(ϰ) written with the Pascal distribution are discussed for the subclass Tκ(ϵ,♭). Also, we provide a sufficient condition for the inclusion relation IθsRϖ(E,D)⊂Tκ(ϵ,♭). Further, we consider the properties of an integral operator related to Pascal distribution series. New special cases as a consequences of the main results are also obtained.

Keywords

• analytic functions
• Hadamard product
• Stirling numbers
• Pascal distribution series