New conditions for Pascal distribution series to be in a certain class of analytic functions

Abstract

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. New necessary and sufficient conditions involving Stirling numbers for the series ϒθs(ς) written by the Pascal distribution are discussed for the subclass Tκ(ϵ,α). Also, we provide sufficient condition for the inclusion relation Iθs(Rϖ(E,D))⊂Tκ(ϵ,α). Further, we consider the properties of integral operator related to Pascal distribution series. New special cases as consequences of the main results are also obtained.

Keywords

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