On Miller–Ross-Type Poisson Distribution Series

Abstract

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 ${\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta )$ and ${\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta )$ of analytic functions with negative coefficients. Furthermore, we investigate several inclusion properties of the class ${\mathcal{Y}}^{\sigma }(V,W)$ associated of the operator ${\mathbb{I}}_{\alpha ,c}^{\epsilon }$ 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.

1. Definitions and Preliminaries

Special functions are very important in the study of geometric function theory, applied mathematics, physics, statistics and many other subjects. In [1], Kenneth S. Miller and Bertram Ross introduced the special function, which is called the Miller–Ross function defined by

$${\mathbb{E}}_{\alpha ,c}\left(\xi \right)={\xi }^{\nu }\sum _{s=0}^{\infty }\frac{{\left(c\xi \right)}^s}{\Gamma (s+\alpha +1)},(\alpha ,c,\xi \in \mathbb{C}).$$

(1)

Observe that the function ${\mathbb{E}}_{\alpha ,c}$ contains many well-known functions as special cases, for example, ${\mathbb{E}}_{0,1}\left(\xi \right)=\xi e^{\xi },$ ${\mathbb{E}}_{1,1}\left(\xi \right)=e^{\xi }-1,$ ${\mathbb{E}}_{2,1}\left(\xi \right)=\frac{2}{\xi }{e}^{\xi }-\frac{2}{\xi }-2,$ ${\mathbb{E}}_{3,1}\left(\xi \right)=\frac{3(e^{\xi }-{\xi }^2-2\xi -2)}{{\xi }^2},$ ${\mathbb{E}}_{\frac{1}{2},\frac{1}{2}}\left(\xi \right)=e^{{}^{\frac{\xi }{2}}}\sqrt{\frac{\pi }{2}}\sqrt{\xi }erf\sqrt{\frac{\xi }{2}},$ where $erf$ is the error function defined by $erf\left(\xi \right)=\frac{2}{\sqrt{\pi }}{\displaystyle {\int }_0^{\xi }}{e}^{-t^2}dt$.

Let $E_{\varsigma ,\mu }\left(\xi \right)$ be the two parameters Mittag–Leffler function [2] defined by

$$E_{\varsigma ,\mu }\left(\xi \right)=\sum _{s=0}^{\infty }\frac{{\xi }^s}{\Gamma (\varsigma s+\mu )},(\xi ,\varsigma ,\mu \in \mathbb{C},\mathrm{Re}\left(\varsigma \right)>0,\mathrm{Re}\left(\mu \right)>0).$$

(2)

Several properties of Mittag–Leffler function and generalized Mittag–Leffler function can be found e.g., in ([3,4,5,6,7,8,9,10,11]).

If $\mu =1$, from (2) we get the Mittag–Leffler function of one parameter [12]

$$E_{\varsigma }\left(\xi \right)=\sum _{s=0}^{\infty }\frac{{\xi }^s}{\Gamma (\varsigma s+1)},(\xi ,\varsigma \in \mathbb{C},\mathrm{Re}\left(\varsigma \right)>0).$$

From (1) and (2), the Miller–Ross function can be expressed as

$${\mathbb{E}}_{\alpha ,c}\left(\xi \right)={\xi }^{\alpha }{E}_{1,1+\alpha }\left(c\xi \right).$$

Let $\mathfrak{U}=\{\xi \in \mathbb{C}:\left|\xi \right|<1\}$ and $\mathcal{A}$ denote for the class of analytic functions given by the expansion

$$f\left(\xi \right)=\xi +\sum _{s=2}^{\infty }{a}_s{\xi }^s,\xi \in \mathfrak{U}.$$

(3)

Further, let $\mathcal{T}$ be the subclass of $\mathcal{A}$ consisting of functions of the form

$$f\left(\xi \right)=\xi -\sum _{s=2}^{\infty }\left|a_s\right|{\xi }^s,\xi \in \mathfrak{U}.$$

(4)

Given two functions $f,g\in \mathcal{A}$, where $f\left(z\right)=z+{\displaystyle \sum _{n=2}^{\infty }}{a}_n{z}^n$ and $g\left(z\right)=z+{\displaystyle \sum _{n=2}^{\infty }}{b}_n{z}^n$, their Hadamard product or convolution $f\left(z\right)\ast g\left(z\right)$ is defined by (see, [13,14])

$$f\left(z\right)\ast g\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{b}_n{z}^n,(z\in \mathfrak{U}).$$

(5)

Let

$${\mathbb{S}}^{*}\left(\gamma \right)=\left\{f\in \mathcal{A}:\Re \left(\frac{\xi f^{\prime }\left(\xi \right)}{f\left(\xi \right)}\right)>\gamma ,\xi \in \mathfrak{U},\right\}$$

and

$$\mathbb{K}\left(\gamma \right)=\left\{f\in \mathcal{A}:\Re \left(1+\frac{\xi f^{″}\left(\xi \right)}{f^{\prime }\left(\xi \right)}\right)>\gamma ,\xi \in \mathfrak{U},\right\}$$

denote the subclasses of $\mathcal{A}$ which are starlike and convex of order $\gamma (0\le \gamma <1)$, respectively. Let ${\mathbb{S}}_{\mathcal{T}}^{*}\left(\gamma \right)$ and ${\mathbb{K}}_{\mathcal{T}}\left(\gamma \right)$ be the subfamilies of ${\mathbb{S}}^{*}\left(\gamma \right)$ and $\mathbb{K}\left(\gamma \right)$, respectively, whose functions are of the form (4).

The generalization of the classes ${\mathbb{S}}^{*}\left(\gamma \right)$ and $\mathbb{K}\left(\gamma \right)$ of functions $f\in \mathcal{A}$ given by the classes ${\mathbb{S}}^{*}(\gamma ,\beta )$ and $\mathbb{K}(\gamma ,\beta ),$ which are satisfies the conditions:

$${\mathbb{S}}^{*}(\gamma ,\beta )=\left\{f\in \mathcal{A}:\Re \left(\frac{\xi f^{\prime }\left(\xi \right)+\beta {\xi }^2{f}^{″}\left(\xi \right)}{f\left(\xi \right)}\right)>\gamma ,\xi \in \mathfrak{U},0\le \gamma <1,0\le \beta <1\right\}$$

and

$$\mathbb{K}(\gamma ,\beta )=\left\{f\in \mathcal{A}:\Re \left(\frac{{\left(\xi f^{\prime }\left(\xi \right)+\beta {\xi }^2{f}^{″}\left(\xi \right)\right)}^{\prime }}{f^{\prime }\left(\xi \right)}\right)>\gamma ,\xi \in \mathfrak{U},0\le \gamma <1,0\le \beta <1\right\},$$

respectively. Let

$${\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta )={\mathbb{S}}^{*}(\gamma ,\beta )\cap \mathcal{T}\mathrm{and}{\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta )=\mathbb{K}(\gamma ,\beta )\cap \mathcal{T}.$$

Clearly, we have ${\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,0)=$ ${\mathbb{S}}_{\mathcal{T}}^{*}\left(\gamma \right)$ and ${\mathbb{K}}_{\mathcal{T}}(\gamma ,0)={\mathbb{K}}_{\mathcal{T}}\left(\gamma \right).$

For $\sigma \in \mathbb{C}\backslash \left\{0\right\}$ and $-1\le W<V\le 1,$ Dixit and Pal [15] introduced the class ${\mathcal{Y}}^{\sigma }(V,W)$ of all analytic functions in $\mathfrak{U}$, defined as:

$${\mathcal{Y}}^{\sigma }(V,W)=\left\{f:f\in \mathcal{A}\mathrm{and}\left|\frac{f^{\prime }\left(\xi \right)-1}{(V-W)\sigma -W[f^{\prime }\left(\xi \right)-1]}\right|<1,\xi \in \mathfrak{U}\right\}.$$

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 $\xi$ 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:

$$P(X=k)=\frac{e^{-\epsilon }}{k!}{\epsilon }^k,k=0,1,2,\dots ,$$

(6)

and $\epsilon >0$ is the parameter of the distribution, then a random variable X follows a Poisson distribution.

Recently, with coefficients are Miller–Ross-type Poisson distribution Şeker et al. [16] (see also, [17]) defined the following power series

$${\mathbb{F}}_{\alpha ,c}^{\epsilon }\left(\xi \right):=\xi +\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}{\xi }^s,\xi \in \mathfrak{U},$$

(7)

where $\alpha >-1$, $c>0$.

We note that if we put $\alpha =0$ and $c=1$ in (7), we get the Poisson distribution series introduced by Porwal [18].

Furthermore, Şeker et al. [16] defined the series

$${\mathbb{k}}_{\alpha ,c}^{\epsilon }\left(\xi \right):=2\xi -{\mathbb{F}}_{\alpha ,c}^{\epsilon }\left(\xi \right)=\xi -\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}{\xi }^s,\xi \in \mathfrak{U}.$$

(8)

Now by the convolution, we construct the linear operator ${\mathbb{I}}_{\alpha ,c}^{\epsilon }:\mathcal{A}\to \mathcal{A}$ to be

$${\mathbb{I}}_{\alpha ,c}^{\epsilon }f\left(\xi \right):={\mathbb{F}}_{\alpha ,c}^{\epsilon }\left(\xi \right)\ast f\left(\xi \right)=\xi +\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}{a}_s{\xi }^s,(\xi \in \mathfrak{U},\alpha >-1,c>0).$$

(9)

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 $\mathfrak{U}.$ In the present paper we obtain some necessary and sufficient conditions for the Miller–Ross-type Poisson distribution series ${\mathbb{k}}_{\alpha ,c}^{\epsilon }$ to be in our classes ${\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta )$ and ${\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta )$. Furthermore, we associate these subclasses with the class ${\mathcal{Y}}^{\sigma }(V,W)$, and finally, we give necessary and sufficient conditions for the function f such that the operator ${\mathbb{G}}_{\alpha ,c}^{\epsilon }f\left(\xi \right)={\int }_0^{\xi }\frac{{\mathbb{k}}_{\alpha ,c}^{\epsilon }\left(t\right)}{t}dt$ belongs to class ${\mathcal{Y}}^{\sigma }(V,W)$.

2. Preliminary Lemmas

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

Lemma 1

([25]). A function $f\in \mathcal{T}$ in the class ${\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta )$ if and only if

$$\sum _{s=2}^{\infty }\left(s+\beta s(s-1)-\gamma \right)\left|a_s\right|\le 1-\gamma ,$$

(10)

where $0\le \gamma <1,0\le \beta <1.$

Lemma 2

([25]). A function $f\in \mathcal{T}$ in the class ${\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta )$ if and only if

$$\sum _{s=2}^{\infty }s\left(s+\beta s(s-1)-\gamma \right)\left|a_s\right|\le 1-\gamma ,$$

where $0\le \gamma <1,0\le \beta <1.$

Lemma 3

([15]). If f ∈ ${\mathcal{Y}}^{\sigma }(V,W)$ is of the form (4), then

$$\left|a_s\right|\le (V-W)\frac{\left|\sigma \right|}{s},s\in \mathbb{N}-\left\{1\right\}.$$

(11)

In this paper, we will assume that $0\le \gamma <1,0\le \beta <1,\sigma \in \mathbb{C}\backslash \left\{0\right\}$, and $-1\le W<V\le 1$ unless otherwise stated.

3. Necessary and Sufficient Conditions

The necessary and sufficient condition for ${\mathbb{k}}_{\alpha ,c}^{\epsilon }$ to be in the class ${\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta )$ is given by the following

Theorem 1.

Let $\alpha >-1$ and $c>0$, then ${\mathbb{k}}_{\alpha ,c}^{\epsilon }\in {\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta )$ if and only if

$$\frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta {\epsilon }^2{\mathbb{E}}_{\alpha -1,c}\left(\epsilon \right)+\left(2\beta (1-\alpha )+1\right)\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)+\left((1-\alpha )(1-\beta \alpha )-\gamma \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\le 1-\gamma .$$

(12)

Proof. 

Since ${\mathbb{k}}_{\alpha ,c}^{\epsilon }$ is defined by (8), in view of Lemma 1 it suffices to verify that

$$\sum _{s=2}^{\infty }\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\le 1-\gamma .$$

(13)

Writing

$$s^2=(\alpha +s-1)(\alpha +s-2)+(3-2\alpha )(\alpha +s-1)+{(1-\alpha )}^2$$

and

$$s=\left(\alpha +s-1\right)+\left(1-\alpha \right)$$

in (13), we have

$$\begin{array}{cc}\hfill & \sum _{s=2}^{\infty }\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\hfill \\ \hfill & =\sum _{s=2}^{\infty }\left(\beta s^2+s(1-\beta )-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\hfill \\ \hfill & =\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \sum _{s=2}^{\infty }(\alpha +s-1)(\alpha +s-2)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\right.\hfill \\ \hfill & +\left(2\beta (1-\alpha )+1\right)\sum _{s=2}^{\infty }(\alpha +s-1)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\hfill \\ \hfill & \left.+\left((1-\alpha )(1-\beta \alpha )-\gamma \right)\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\right]\hfill \\ \hfill & =\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha -2)}+\left(2\beta (1-\alpha )+1\right)\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha -1)}\right.\hfill \\ \hfill & \left.+\left((1-\alpha )(1-\beta \alpha )-\gamma \right)\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\right]\hfill \\ \hfill & =\frac{c}{{\mathbb{E}}_{\nu ,c}\left(\epsilon \right)}\left[\beta \epsilon \sum _{s=0}^{\infty }\frac{{\epsilon }^{\nu }{\left(c\epsilon \right)}^s}{\Gamma (s+\nu )}+\left(2\beta (1-\nu )+1\right)\epsilon \sum _{s=0}^{\infty }\frac{{\epsilon }^{\nu }{\left(c\epsilon \right)}^s}{\Gamma (s+\nu +1)}\right.\hfill \\ \hfill & \left.+\left((1-\alpha )(1-\beta \alpha )-\gamma \right)\epsilon \sum _{s=0}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^s}{\Gamma (s+\alpha +2)}\right]\hfill \\ \hfill & =\frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta {\epsilon }^2{\mathbb{E}}_{\alpha -1,c}\left(\epsilon \right)+\left(2\beta (1-\alpha )+1\right)\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)+\left((1-\alpha )(1-\beta \alpha )-\gamma \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right],\hfill \end{array}$$

which is bounded above by $1-\gamma$ if and only if

$$\frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta {\epsilon }^2{\mathbb{E}}_{\alpha -1,c}\left(\epsilon \right)+\left(2\beta (1-\alpha )+1\right)\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)+\left((1-\alpha )(1-\beta \alpha )-\gamma \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\le 1-\gamma .$$

□

Now, we obtain a necessary and sufficient condition for ${\mathbb{k}}_{\alpha ,c}^{\epsilon }$ to be in the class ${\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta ).$

Theorem 2.

Let $\alpha >-1$ and $c>0$, then ${\mathbb{k}}_{\alpha ,c}^{\epsilon }\in {\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta )$ if and only if

$$\begin{array}{cc}\hfill & \frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta {\epsilon }^3{\mathbb{E}}_{\alpha -2,c}\left(\epsilon \right)+\left[\beta (6-3\alpha )+(1-\beta )\right]{\epsilon }^2{\mathbb{E}}_{\alpha -1,c}\left(\epsilon \right)\right.\hfill \\ \hfill & +\left[\beta (3{\alpha }^2-9\alpha +7)+\left(1-\beta \right)(3-2\alpha )-\gamma \right]\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)\hfill \\ \hfill & \left.+\left[\beta {(1-\alpha )}^3+\left(1-\beta \right){(1-\alpha )}^2-\gamma \left(1-\alpha \right)\right]{\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\hfill \\ \hfill & \le 1-\gamma .\hfill \end{array}$$

(14)

Proof. 

By Lemma 1 we show that

$$\sum _{s=2}^{\infty }s\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\le 1-\gamma .$$

(15)

Writing

$$\begin{array}{cc}\hfill s^3& =(\alpha +s-1)(\alpha +s-2)(\alpha +s-3)+(6-3\alpha )(\alpha +s-1)(\alpha +s-2)\hfill \\ \hfill & (3{\alpha }^2-9\alpha +7)(\alpha +s-1)+{(1-\alpha )}^3,\hfill \end{array}$$

$$s^2=(\alpha +s-1)(\alpha +s-2)+(3-2\alpha )(\alpha +s-1)+{(1-\alpha )}^2$$

and

$$s=\left(\alpha +s-1\right)+\left(1-\alpha \right)$$

in (15), we have

$$\begin{array}{cc}\hfill & \sum _{s=2}^{\infty }s\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\hfill \\ \hfill & =\sum _{s=2}^{\infty }\left(\beta s^3+s^2(1-\beta )-\gamma s\right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\hfill \\ \hfill & =\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \sum _{s=2}^{\infty }(\alpha +s-1)(\alpha +s-2)(\alpha +s-3)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\right.\hfill \\ \hfill & +\left[\beta (6-3\alpha )+(1-\beta )\right]\sum _{s=2}^{\infty }(\alpha +s-1)(\alpha +s-2)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\hfill \\ \hfill & +\left[\beta (3{\alpha }^2-9\alpha +7)+\left(1-\beta \right)(3-2\alpha )-\gamma \right]\sum _{s=2}^{\infty }(\alpha +s-1)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\hfill \\ \hfill & \left.+\left[\beta {(1-\alpha )}^3+\left(1-\beta \right){(1-\alpha )}^2-\gamma \left(1-\alpha \right)\right]\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\right]\hfill \\ \hfill & =\frac{1}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha -3)}\right.\hfill \\ \hfill & +\left[\beta (6-3\alpha )+(1-\beta )\right]\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha -2)}\hfill \\ \hfill & +\left[\beta (3{\alpha }^2-9\alpha +7)+\left(1-\beta \right)(3-2\alpha )-\gamma \right]\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha -1)}\hfill \\ \hfill & \left.+\left[\beta {(1-\alpha )}^3+\left(1-\beta \right){(1-\alpha )}^2-\gamma \left(1-\alpha \right)\right]\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\right]\hfill \\ \hfill & =\frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \epsilon \sum _{s=0}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^s}{\Gamma (s+\alpha -1)}\right.\hfill \\ \hfill & +\left[\beta (6-3\alpha )+(1-\beta )\right]\epsilon \sum _{s=0}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^s}{\Gamma (s+\alpha )}\hfill \\ \hfill & +\left[\beta (3{\alpha }^2-9\alpha +7)+\left(1-\beta \right)(3-2\alpha )-\gamma \right]\epsilon \sum _{s=0}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^s}{\Gamma (s+\alpha +1)}\hfill \\ \hfill & \left.+\left[\beta {(1-\alpha )}^3+\left(1-\beta \right){(1-\alpha )}^2-\gamma \left(1-\alpha \right)\right]\epsilon \sum _{s=0}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^s}{\Gamma (s+\alpha +2)}\right]\hfill \\ \hfill & =\frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta {\epsilon }^3{\mathbb{E}}_{\alpha -2,c}\left(\epsilon \right)+\left[\beta (6-3\alpha )+(1-\beta )\right]{\epsilon }^2{\mathbb{E}}_{\alpha -1,c}\left(\epsilon \right)\right.\hfill \\ \hfill & +\left[\beta (3{\alpha }^2-9\alpha +7)+\left(1-\beta \right)(3-2\alpha )-\gamma \right]\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)\hfill \\ \hfill & \left.+\left[\beta {(1-\alpha )}^3+\left(1-\beta \right){(1-\alpha )}^2-\gamma \left(1-\alpha \right)\right]{\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right],\hfill \end{array}$$

but the final phrasing bounded above by $1-\gamma$ if and only if (14) holds. □

4. Inclusion Relations

The inclusion relations of the class ${\mathcal{Y}}^{\sigma }(V,W)$ associated of the operator ${\mathbb{I}}_{\alpha ,c}^{\epsilon }$ defined by (9) proved in this section.

Theorem 3.

Let $\alpha >-1$ and $c>0.$ If $f\in {\mathcal{Y}}^{\sigma }(V,W)$ and

$$\begin{array}{cc}\hfill & \frac{(V-W)c\left|\sigma \right|}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta {\epsilon }^2{\mathbb{E}}_{\alpha -1,c}\left(\epsilon \right)+\left(2\beta (1-\alpha )+1\right)\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)\right.\hfill \\ \hfill & \left.+\left((1-\alpha )(1-\beta \alpha )-\gamma \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\hfill \\ \hfill & \le 1-\gamma \hfill \end{array}$$

(16)

is satisfied then ${\mathbb{I}}_{\alpha ,c}^{\epsilon }f\in {\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta ).$

Proof. 

By Lemma 2 it is sufficient to show that

$$\sum _{s=2}^{\infty }s\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left|a_s\right|\le 1-\gamma .$$

(17)

Since $f\in {\mathcal{Y}}^{\sigma }(V,W)$, then by Lemma 3, we have

$$\left|a_s\right|\le \frac{(V-W)\left|\sigma \right|}{s}.$$

Therefore, it is enough to show that

$$\begin{array}{cc}\hfill & \sum _{s=2}^{\infty }s\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left|a_s\right|\hfill \\ \hfill & \le (V-W)\left|\sigma \right|\left[\sum _{s=2}^{\infty }\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\right]\le 1-\gamma .\hfill \end{array}$$

(18)

Using the similar computations like in the proof of in Theorem 1 it follows that the inequality (18) is satisfied whenever (16) holds. □

Theorem 4.

Let $\alpha >-1$ and $c>0.$ If $f\in {\mathcal{Y}}^{\sigma }(V,W)$ and

$$\frac{(V-W)\left|\sigma \right|c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)+\left(1-\beta \alpha \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\le 1-\gamma$$

(19)

is satisfied then ${\mathbb{I}}_{\alpha ,c}^{\epsilon }f\in {\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta ).$

Proof. 

By Lemma 1 it is sufficient to show that

$$\sum _{s=2}^{\infty }\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left|a_s\right|\le 1-\gamma .$$

Since $f\in {\mathcal{Y}}^{\sigma }(V,W)$, using the inequality (11) of Lemma 3, we have

$$\begin{array}{cc}\hfill & \sum _{s=2}^{\infty }\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left|a_s\right|\hfill \\ \hfill & \le (V-W)\left|\sigma \right|\left[\sum _{s=2}^{\infty }\left[\left(s\beta -\beta \right)+\left(1-\frac{\gamma }{s}\right)\right]\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\right]\hfill \\ \hfill & \le (V-W)\left|\sigma \right|\left[\sum _{s=2}^{\infty }\left[\left(s\beta -\beta \right)+1\right]\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\right]\hfill \\ \hfill & =(V-W)\left|\sigma \right|\left[\sum _{s=2}^{\infty }\left[\beta \left(\alpha +s-1\right)+\left(1-\beta \alpha \right)\right]\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\right]\hfill \\ \hfill & =\frac{(V-W)\left|\sigma \right|}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \sum _{s=2}^{\infty }\left(\alpha +s-1\right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}+\left(1-\beta \alpha \right)\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\right]\hfill \\ \hfill & =\frac{(V-W)\left|\sigma \right|}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha -1)}+\left(1-\beta \alpha \right)\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha )}\right]\hfill \\ \hfill & =\frac{(V-W)\left|\sigma \right|c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \epsilon \sum _{s=0}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^s}{\Gamma (s+\alpha +1)}+\left(1-\beta \alpha \right)\epsilon \sum _{s=0}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^s}{\Gamma (s+\alpha +2)}\right]\hfill \\ \hfill & =\frac{(V-W)\left|\sigma \right|c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta \epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)+\left(1-\beta \alpha \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right],\hfill \end{array}$$

this final phrasing is bounded above by $\left(1-\gamma \right)$ if and only if (19) holds. □

5. The Operator ${\mathbb{G}}_{\alpha ,c}^{\epsilon }\left(\xi \right)$

Theorem 5.

Let $\alpha >-1$ and $c>0$. If the integral operator ${\mathbb{G}}_{\alpha ,c}^{\epsilon }$ is given by

$${\mathbb{G}}_{\alpha ,c}^{\epsilon }\left(\xi \right):={\int }_0^{\xi }\frac{{\mathbb{k}}_{\alpha ,c}^{\epsilon }\left(t\right)}{t}dt,\xi \in \mathfrak{U},$$

(20)

then ${\mathbb{G}}_{\alpha ,c}^{\epsilon }\in {\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta )$, if and only if

$$\begin{array}{cc}\hfill & \frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta {\epsilon }^2{\mathbb{E}}_{\alpha -1,c}\left(\epsilon \right)+\left(2\beta (1-\alpha )+1\right)\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)\right.\hfill \\ \hfill & \left.+\left((1-\alpha )(1-\beta \alpha )-\gamma \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\le 1-\gamma .\hfill \end{array}$$

Proof. 

By (8) it follows that

$${\mathbb{G}}_{\alpha ,c}^{\epsilon }\left(\xi \right)=\xi -\sum _{s=2}^{\infty }\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\frac{{\xi }^s}{s},\xi \in \mathfrak{U}.$$

Using Lemma 2, the integral operator ${\mathbb{G}}_{\alpha ,c}^{\epsilon }\left(\xi \right)$ belongs to ${\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta )$ if and only if

$$\sum _{s=2}^{\infty }\left(s+\beta s(s-1)-\gamma \right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\le 1-\gamma .$$

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 $\alpha >-1$ and $c>0$. Then the integral operator ${\mathbb{G}}_{\alpha ,c}^{\epsilon }$ given by (20) is in the class ${\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta )$, if and only if

$$\frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\beta {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)+\left(1-\beta \alpha \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\le 1-\gamma .$$

Proof. 

Using Lemma 1, the integral operator ${\mathbb{G}}_{\alpha ,c}^{\epsilon }\left(\xi \right)$ belongs to ${\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta )$ if and only if

$$\sum _{s=2}^{\infty }\left(s\beta -\beta \right)+\left(1-\frac{\gamma }{s}\right)\frac{{\epsilon }^{\alpha }{\left(c\epsilon \right)}^{s-1}}{\Gamma (s+\alpha ){\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\le 1-\gamma .$$

The complement is similar to proof of Theorem 4. □

6. Corollaries and Consequences

Putting $\beta =0$ in the previous theorems, we get the following special cases.

Corollary 1.

Let $\alpha >-1$ and $c>0$, then ${\mathbb{k}}_{\alpha ,c}^{\epsilon }\in {\mathbb{S}}_{\mathcal{T}}^{*}\left(\gamma \right)$ if and only if

$$\frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)+\left(1-\alpha -\gamma \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\le 1-\gamma .$$

Corollary 2.

Let $\alpha >-1$ and $c>0$, then ${\mathbb{k}}_{\alpha ,c}^{\epsilon }\in {\mathbb{K}}_{\mathcal{T}}\left(\gamma \right)$ if and only if

$$\begin{array}{cc}\hfill & \frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[{\epsilon }^2{\mathbb{E}}_{\alpha -1,c}\left(\epsilon \right)+\left(3-2\alpha -\gamma \right)\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)\right.\hfill \\ \hfill & \left.+\left(1-\alpha \right)(1-\alpha -\gamma ){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\le 1-\gamma .\hfill \end{array}$$

Corollary 3.

Let $\alpha >-1$ and $c>0.$ If $f\in {\mathcal{Y}}^{\sigma }(V,W)$ and

$$\frac{(V-W)c\left|\sigma \right|}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)+\left(1-\alpha -\gamma \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\le 1-\gamma$$

then ${\mathbb{I}}_{\alpha ,c}^{\epsilon }f\in {\mathbb{K}}_{\mathcal{T}}\left(\gamma \right).$

Corollary 4.

Let $\alpha >-1$ and $c>0.$ If $f\in {\mathcal{Y}}^{\sigma }(V,W)$ and

$$(V-W)\left|\sigma \right|c\frac{{\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\le 1-\gamma$$

then ${\mathbb{I}}_{\alpha ,c}^{\epsilon }f\in {\mathbb{S}}_{\mathcal{T}}^{*}\left(\gamma \right).$

Corollary 5.

Let $\alpha >-1$ and $c>0$. Then the integral operator ${\mathbb{G}}_{\alpha ,c}^{\epsilon }$ given by (20) is in the class ${\mathbb{K}}_{\mathcal{T}}\left(\gamma \right)$, if and only if

$$\frac{c}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\left[\epsilon {\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)+\left(1-\alpha -\gamma \right){\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)\right]\le 1-\gamma .$$

Corollary 6.

Let $\alpha >-1$ and $c>0$. Then the integral operator ${\mathbb{G}}_{\alpha ,c}^{\epsilon }$ given by (20) is in the class ${\mathbb{S}}_{\mathcal{T}}^{*}\left(\gamma \right)$, if and only if

$$\frac{c{\mathbb{E}}_{\alpha +1,c}\left(\epsilon \right)}{{\mathbb{E}}_{\alpha ,c}\left(\epsilon \right)}\le 1-\gamma .$$

7. Conclusions

Several researchers have used certain distribution series such as Poisson distribution series, Pascal distribution series, hypergeometric distribution series, and the Mittag–Leffler-type 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 $\mathfrak{U}$. In our study, necessary and sufficient conditions for Miller–Ross-type Poisson distribution series to be in the classes ${\mathbb{S}}_{\mathcal{T}}^{*}(\gamma ,\beta )$ and ${\mathbb{K}}_{\mathcal{T}}(\gamma ,\beta )$ of analytic functions with negative coefficients is obtained. We also investigate several inclusion properties of the class ${\mathcal{Y}}^{\sigma }(V,W)$ associated of the operator ${\mathbb{I}}_{\alpha ,c}^{\epsilon }$ 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 $\mathfrak{U}.$