Subclasses of analytic functions associated with Pascal distribution series

Abstract

In the present paper we determine necessary and sufficient conditions for the Pascal distribution series to be in the subclasses S(k, λ) and C(k, λ) of analytic functions. Further, we consider an integral operator related to Pascal distribution series. Some interesting special cases of our main results are also considered.

An Application of Pascal Distribution Series on Certain Analytic Functions Associated with Stirling Numbers and Sălăgean Operator

“…Several authors have used this operator to define and consider the properties of certain known and new classes of analytic univalent functions (see, for example, [25,26]). In the present paper, and due to the earlier works (see, for example, [11,16,18]), we find a necessary and sufficient condition and inclusion relation for the Pascal distribution series to be in the class P k (λ, α) of analytic functions associated with the Stirling numbers and Sȃlȃgean differential operator. Furthermore, we consider an integral operator related to the Pascal distribution series.…”

“…Motivated by several earlier results on connections between various subclasses of analytic and univalent functions, using hypergeometric functions, generalized Bessel functions, Struve functions, Poisson distribution series, and Pascal distribution series (see, for example, [12], [13][14][15], [7][8][9][16][17][18][19][20][21][22][23], [24]), we determine a necessary and sufficient condition for Υ m σ (z) to be in our class P k (λ, α). Furthermore, we give sufficient conditions for…”

Classes of Harmonic Functions Related to Mittag-Leffler Function

“…∪ {0}. Inclusion relations between different subclasses of analytic and univalent functions by using hypergeometric functions [10,31], generalized Bessel function [32][33][34] and by the recent investigations related with distribution series [35][36][37][38][39][40][41], were studied in the literature. Very recently, several authors have investigated mapping properties and inclusion results for the families of harmonic univalent functions, including various linear and nonlinear operators (see [42][43][44][45][46][47][48]).…”

On Miller–Ross-Type Poisson Distribution Series

“…In recent years, several researchers used this distribution series [19,20] and other distribution series such as Poisson distribution series [21][22][23][24][25][26], Pascal distribution series [27][28][29][30], hypergeometric distribution series [31][32][33][34][35][36], and the Mittag-Leffler-type Poisson distribution [37] to obtain some necessary and sufficient conditions for these distributions to belong to certain classes of analytic functions defined in U. In the present paper we obtain some necessary and sufficient conditions for the Miller-Ross-type Poisson distribution series k ε α,c to be in our classes S * T (γ, β) and K T (γ, β).…”