On Miller–Ross-Type Poisson Distribution Series

: The objective of the current paper is to ﬁnd the necessary and sufﬁcient conditions for Miller–Ross-type Poisson distribution series to be in the classes S ∗T ( γ , β ) and K T ( γ , β ) of analytic functions with negative coefﬁcients. Furthermore, we investigate several inclusion properties of the class Y σ ( V , W ) associated of the operator I εα , c deﬁned by this distribution. We also take into consideration an integral operator connected to series of Miller–Ross-type Poisson distributions. Special cases of the main results are also considered

Let U = {ξ ∈ C : |ξ| < 1} and A denote for the class of analytic functions given by the expansion Further, let T be the subclass of A consisting of functions of the form Given two functions f , g ∈ A, where Hadamard product or convolution f (z) * g(z) is defined by (see, [13,14]) Let denote the subclasses of A which are starlike and convex of order γ(0 ≤ γ < 1), respectively.Let S * T (γ) and K T (γ) be the subfamilies of S * (γ) and K(γ), respectively, whose functions are of the form (4).
The generalization of the classes S * (γ) and K(γ) of functions f ∈ A given by the classes S * (γ, β) and K(γ, β), which are satisfies the conditions: Dixit and Pal [15] introduced the class Y σ (V, W) of all analytic functions in U, defined as: In the recent years, there has been a tremendous lot of interest in the distributions of the random variables.In statistics and probability theory, their probability density functions in the real variable x and the complex variable ξ have been crucial.Distributions have so been the subject of much study.Numerous distribution types, including the Binomial distribution, negative binomial distribution, Poisson distribution and geometric distribution, emerged from real-world circumstances.
If the probability density function is given by: and ε > 0 is the parameter of the distribution, then a random variable X follows a Poisson distribution.
We note that if we put α = 0 and c = 1 in ( 7), we get the Poisson distribution series introduced by Porwal [18].
Furthermore, Şeker et al. [16] defined the series Now by the convolution, we construct the linear operator I ε α,c : A → A to be 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).Furthermore, we associate these subclasses with the class Y σ (V, W), and finally, we give necessary and sufficient conditions for the function f t dt belongs to class Y σ (V, W).

Preliminary Lemmas
We require the following Lemmas in order to establish our main results.

Necessary and Sufficient Conditions
The necessary and sufficient condition for k ε α,c to be in the class S * T (γ, β) is given by the following Proof.Since k ε α,c is defined by (8), in view of Lemma 1 it suffices to verify that Writing in ( 13), we have Now, we obtain a necessary and sufficient condition for k ε α,c to be in the class K T (γ, β).
Proof.By Lemma 1 we show that Writing 3 , but the final phrasing bounded above by 1 − γ if and only if ( 14) holds.

Inclusion Relations
The inclusion relations of the class Y σ (V, W) associated of the operator I ε α,c defined by (9) proved in this section.
Proof.By Lemma 2 it is sufficient to show that Since f ∈ Y σ (V, W), then by Lemma 3, we have Therefore, it is enough to show that Using the similar computations like in the proof of in Theorem 1 it follows that the inequality ( 18) is satisfied whenever (16) holds.
Since f ∈ Y σ (V, W), using the inequality (11) of Lemma 3 , we have this final phrasing is bounded above by (1 − γ) if and only if (19) holds.

The
Proof.By (8) it follows that Using Lemma 2, the integral operator G ε α,c (ξ) belongs to K T (γ, β) if and only if We omit the remaining part of the proof because the remaining proof of Theorem 5 is similar to that of Theorem 1. Theorem 6.Let α > −1 and c > 0. Then the integral operator G ε α,c given by (20) is in the class S * T (γ, β), if and only if c The complement is similar to proof of Theorem 4.

Corollaries and Consequences
Putting β = 0 in the previous theorems, we get the following special cases.

Conclusions
Several researchers have used certain distribution series such as Poisson distribution series, Pascal distribution series, hypergeometric distribution series, and the Mittag-Lefflertype Poisson distribution to obtain some necessary and sufficient conditions for these distributions to belong to certain classes of analytic functions defined in the open disk U.In our study, necessary and sufficient conditions for Miller-Ross-type Poisson distribution series to be in the classes S * T (γ, β) and K T (γ, β) of analytic functions with negative coefficients is obtained.We also investigate several inclusion properties of the class Y σ (V, W) associated of the operator I ε α,c defined by this distribution.This study could inspire researchers to introduce new sufficient conditions for Miller-Ross-type Poisson distribution series to be in different classes of analytic functions with negative coefficients defined in U.