Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions

Abstract

In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series $E_{\varphi }^{\ast }(\varsigma ,z)$ to belong to the inclusive subclass ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,Q_0)$. Necessary and sufficient conditions are also given for the generalized distribution series $E_{\varphi }^{\ast }(\varsigma ,z)\hslash$ and the integral operator $J_{\varsigma }^{\varphi }\left(z\right)$ to be in the inclusive subclass ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,0)$. Further, we provide a number of corollaries, which improve the existing ones that are available in some recent studies. The results presented here not only improve the earlier studies, but also give rise to a number of new results for particular choices of $Q_3,Q_2,Q_1$ and $Q_0$.

1. Preliminaries

The generalized distribution series (GDS) are utilized in mathematics, especially in distribution theory. In order to tackle items that might not be considered classical functions, such as Dirac delta functions and the like, distributions are utilized to expand traditional ideas of functions. Non-integrable functions, like Dirac functions, can be included in the distributions, which are mathematical objects that are interpreted as functions. A function or another mathematical model, like a series, can be used to depict a distribution. The generalized distribution series is widely used in particle physics, diffusion and motion mathematical models, and differential equation theory, particularly in equations with Dirac indices.

In Geometric Function Theory, several researchers have obtained new necessary and sufficient conditions associated with some special functions such as the Wright function [1], Fox–Wright function [2], generalized Bessel functions [3], hypergeometric functions [4], and generalized hypergeometric functions [5], among others.

Let D denote the sum of the convergent series of the form

$$D=\sum _{\tau =0}^{\infty }{\upsilon }_{\tau },$$

where ${\upsilon }_{\tau }\ge 0$ for all $\tau \in \mathbb{N}$.

Recently, some researchers such as Porwal [6], Güney and Yıldızhan [7], Kota [8], Kota and El-Ashwah [9], Yalçın et al. [10] and Porwal and Murugusundaramoorthy [11] have established a relationship between the subclasses of univalent functions and GDS given by (2).

In [12], Porwal suggested extending the discrete probability distribution using the probability mass function, which is provided by

$$𝘍\left(\tau \right)={\displaystyle \frac{{\upsilon }_{\tau }}{D}},\tau \in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}.$$

(1)

Obviously, $𝘍\left(\tau \right)\ge 0$ and $\sum _{\tau }𝘍\left(\tau \right)=1.$

Furthermore, he established the following series:

$$\varphi \left(x\right)=\sum _{\tau =0}^{\infty }{\upsilon }_{\tau }{x}^{\tau }.$$

(2)

This series clearly converges for $-1<x\le 1$. A number of well-known discrete probability distributions can be obtained by specializing the values of ${\upsilon }_{\tau }$ (see [12]). (I confirm)

Using the coefficients as the generalized distribution probabilities, and taking into account Equations (1) and (2) above, we obtain the following:

$$H_{\varphi }\left(z\right)=z+\sum _{\tau =2}^{\infty }{\displaystyle \frac{{\upsilon }_{\tau -1}}{D}}{z}^{\tau },$$

and

$$T{H}_{\varphi }\left(z\right)=z-\sum _{\tau =2}^{\infty }{\displaystyle \frac{{\upsilon }_{\tau -1}}{D}}{z}^{\tau }.$$

The functions $E_{\varphi }(\varsigma ,z)$ and $E_{\varphi }^{\ast }(\varsigma ,z)$ are defined as follows:

$$\begin{array}{ccc}\hfill E_{\varphi }(\varsigma ,z)& =& (1-\varsigma )H_{\varphi }\left(z\right)+\varsigma z{H}_{\varphi }^{\prime }\left(z\right)\hfill \\ & =& z+\sum _{\tau =2}^{\infty }(\varsigma (\tau -1)+1){\displaystyle \frac{{\upsilon }_{\tau -1}}{D}}{z}^{\tau },0\le \varsigma \le 1,z\in \mathsf{\Lambda }=\left\{z\in \mathbb{C}:\left|z\right|<1\right\},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill E_{\varphi }^{\ast }(\varsigma ,z)& =& 2z-E_{\varphi }(\varsigma ,z)\hfill \\ & =& z-\sum _{\tau =2}^{\infty }(\varsigma (\tau -1)+1){\displaystyle \frac{{\upsilon }_{\tau -1}}{D}}{z}^{\tau },0\le \varsigma \le 1,z\in \mathsf{\Lambda }.\hfill \end{array}$$

Let A be the class of analytic and univalent functions ℏ of the form

$$\hslash \left(z\right)=z+\sum _{\tau =2}^{\infty }{C}_{\tau }{z}^{\tau },z\in \mathsf{\Lambda },$$

(3)

such that $\hslash \left(0\right)={\hslash }^{\prime }\left(0\right)-1=0.$

Additionally, denote by $A^{-}$ the subclass of A of functions of the form

$$\hslash \left(z\right)=z-\sum _{\tau =2}^{\infty }\left|C_{\tau }\right|z^{\tau },z\in \mathsf{\Lambda }.$$

(4)

Now, by the convolution product $(\ast ),$ we define the linear operator $\mathrm{\Pi }(\epsilon ,\tau ,z):A\to A$ as follows:

$$E_{\varphi }^{\ast }(\varsigma ,z)\hslash =E_{\varphi }^{\ast }(\varsigma ,z)\ast \hslash \left(z\right)=z+\sum _{\tau =2}^{\infty }(\varsigma (\tau -1)+1){\displaystyle \frac{{\upsilon }_{\tau -1}}{D}}{C}_{\tau }{z}^{\tau }.$$

Definition 1

([13]). A function $\hslash \in A$ is said to be in the subclass ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,Q_0)$ if and only if

$$\sum _{\tau =2}^{\infty }\left(Q_3{\tau }^3+Q_2{\tau }^2+Q_1\tau +Q_0\right)\left|C_{\tau }\right|\le \eta ,$$

where $Q_0,Q_1,Q_2,Q_3\in \mathbb{R}$ and $\eta >0.$

Remark 1.

For some $p(0\le p\le 1)$, $\theta (0\le \theta <1)$, $\hslash \in A^{-}$ and specialization for the coefficients $Q_0,Q_1,Q_2,Q_3$ and η in Definition 1, we obtain the following subclasses studied by Kota and El-Ashwah [9]:

 1. 

$\hslash \in TF(p,\theta )\equiv {\mathrm{\Pi }}_{1-\theta }(0,$$p,$$1-p,0)$ if and only if

$$\sum _{\tau =2}^{\infty }\tau (1-p+p\tau )C_{\tau }\le 1-\theta .$$

 2. 

$\hslash \in TH(p,\theta )\equiv {\mathrm{\Pi }}_{1-\theta }(0,p,1-p,-\theta )$ if and only if

$$\sum _{\tau =2}^{\infty }(\left(\tau -1\left)\right(p\tau +1)+1-\theta \right)C_{\tau }\le 1-\theta .$$

 3. 

$\hslash \in TJ(p,\theta )\equiv {\mathrm{\Pi }}_{1-\theta }(0,0,p,1-p)$ if and only if

$$\sum _{\tau =2}^{\infty }\left(p(\tau -1)+1\right)C_{\tau }\le 1-\theta .$$

 4. 

$\hslash \in TX(p,\theta )\equiv {\mathrm{\Pi }}_{1-\theta }(0,p,-\theta ,$$1-p)$ if and only if

$$\sum _{\tau =2}^{\infty }\left(\left(\tau -1\left)\right(p(\tau +1)-1\right)+\tau (1-\theta )\right)C_{\tau }\le 1-\theta .$$

 5. 

$\hslash \in TX(0,\theta )\equiv {\mathrm{\Pi }}_{1-\theta }(0,0,-\theta ,$$1)$ if and only if

$$\sum _{\tau =2}^{\infty }\left(1-\tau )+\tau (1-\theta )\right)C_{\tau }\le 1-\theta .$$

Remark 2.

For some $\alpha \in \mathbb{C}-\left\{0\right\},\gamma (0\le \gamma \le 1)$, $\beta (0<\beta \le 1)$, $\hslash \in A^{-}$ and specialization for the coefficients $Q_0,Q_1,Q_2,Q_3$ and η in Definition 1, we obtain the following subclasses studied by Yalçın et al. [10]:

 1. 

$𝘍\in S(\alpha ,\gamma ,\beta )\equiv {\mathrm{\Pi }}_{\beta \left|\alpha \right|}(0,\gamma ,\gamma \beta \left|\alpha \right|-2\gamma +1,\left(1-\gamma \right)\left(\beta \left|\alpha \right|-1\right))$ if and only if

$$\sum _{\tau =2}^{\infty }\left(\gamma (\tau -1)+1\right)(\tau +\beta \left|\alpha \right|-1)C_{\tau }\le \beta \left|\alpha \right|.$$

 2. 

$\hslash \in R(\alpha ,\gamma ,\beta )\equiv {\mathrm{\Pi }}_{\beta \left|\alpha \right|}(0,\gamma ,1-\gamma ,0)$ if and only if

$$\sum _{\tau =2}^{\infty }\tau \left(\gamma (\tau -1)+1\right)C_{\tau }\le \beta \left|\alpha \right|.$$

Remark 3.

We observe that our class ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,Q_0)$ includes, as its special cases, various other classes that were introduced and studied in several works, such as the subclasses TG$(\lambda ,\alpha )$ and TK$(\lambda ,\alpha )$ [14], T$(\lambda ,\alpha )$ and $\mathcal{C}(\lambda ,\alpha )$ [15], S_p$(\alpha ,\beta )$ and $\mathcal{UCV}(\alpha ,\beta )$ [16], ${\mathcal{L}}^{\theta }(A,B,\gamma )$ [17], $\mathcal{S}(\gamma ,\alpha ,p)$ [18] and $\mathcal{K}(\gamma ,\alpha ,p)$ [19], among other subclasses.

The organization of this paper will be as follows: In Section 2, we give a sufficient and necessary condition for the generalized distribution series $E_{\varphi }^{\ast }(\varsigma ,z)$ to be in the inclusive subclass ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,Q_0)$. The inclusion relation ${\mathcal{Y}}^{\tau }({\Delta }_1,{\Delta }_2)\subset {\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,0)$ will be proven in Section 3. In Section 4, we give a sufficient and necessary condition for the integral operator $J_{\varsigma }^{\varphi }\left(z\right)=\underset{0}{\overset{z}{\int }}{\displaystyle \frac{T{H}_{\varphi }\left(u\right)}{u}}du$ to be in the subclass ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,0)$. In the last section of this paper, we provide a number of specific corollaries, which improve the existing ones in some recent works.

2. Necessary and Sufficient Condition

In this section, we give the following necessary and sufficient condition for the function $E_{\varphi }^{\ast }(\varsigma ,z)$ to be in the inclusive subclass ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,Q_0).$

Theorem 1.

If $0\le \varsigma \le 1$ and $\eta >0,$ then $E_{\varphi }^{\ast }(\varsigma ,z)\in {\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,Q_0)$ if and only if

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{D}}\left[Q_3{\varphi }^{\prime \prime \prime }\left(1\right)+(6Q_3+Q_2){\varphi }^{\prime \prime }\left(1\right)+(7Q_3+3Q_2+Q_1){\varphi }^{\prime }\left(1\right)+\right.\hfill \\ & & \left.\left(Q_3+Q_2+Q_1+Q_0\right)\left(\varphi \left(1\right)-\varphi \left(0\right)\right)\right]\hfill \\ & \le & \eta .\hfill \end{array}$$

(5)

Proof. 

To prove that $E_{\varphi }^{\ast }(\varsigma ,z)\in {\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,Q_0)$, from Definition 1, it suffices to show that

$$\sum _{\tau =2}^{\infty }\left(Q_3{\tau }^3+Q_2{\tau }^2+Q_1\tau +Q_0\right){\displaystyle \frac{{\upsilon }_{\tau -1}}{D}}\le \eta .$$

Now,

$$\begin{array}{ccc}& & \sum _{\tau =2}^{\infty }\left(Q_3{\tau }^3+Q_2{\tau }^2+Q_1\tau +Q_0\right){\displaystyle \frac{{\upsilon }_{\tau -1}}{D}}\hfill \\ & =& {\displaystyle \frac{1}{D}}\sum _{\tau =2}^{\infty }\left[Q_3(\tau -1)(\tau -2)(\tau -3)+(6Q_3+Q_2)(\tau -1)(\tau -2)\right.\hfill \\ & & \left.+(7Q_3+3Q_2+Q_1)(\tau -1)+\left(Q_3+Q_2+Q_1+Q_0\right)\right]{\upsilon }_{\tau -1}\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{D}}\left(Q_3\sum _{\tau =2}^{\infty }(\tau -1)(\tau -2)(\tau -3){\upsilon }_{\tau -1}+(6Q_3+Q_2)\sum _{\tau =2}^{\infty }(\tau -1)(\tau -2){\upsilon }_{\tau -1}\right.\hfill \\ & & +\left.(7Q_3+3Q_2+Q_1)\sum _{\tau =2}^{\infty }(\tau -1){\upsilon }_{\tau -1}+\left(Q_3+Q_2+Q_1+Q_0\right)\sum _{\tau =2}^{\infty }{\upsilon }_{\tau -1}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{D}}\left(Q_3\sum _{\tau =1}^{\infty }\tau (\tau -1)(\tau -2){\upsilon }_{\tau }+(6Q_3+Q_2)\sum _{\tau =1}^{\infty }\tau (\tau -1){\upsilon }_{\tau }\right.\hfill \\ & & +\left.(7Q_3+3Q_2+Q_1)\sum _{\tau =1}^{\infty }\tau {\upsilon }_{\tau }+\left(Q_3+Q_2+Q_1+Q_0\right)\sum _{\tau =1}^{\infty }{\upsilon }_{\tau }\right)\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{D}}\left[Q_3{\varphi }^{\prime \prime \prime }\left(1\right)+(6Q_3+Q_2){\varphi }^{\prime \prime }\left(1\right)+(7Q_3+3Q_2+Q_1){\varphi }^{\prime }\left(1\right)\right.\hfill \\ & & \left.+\left(Q_3+Q_2+Q_1+Q_0\right)\left(\varphi \left(1\right)-\varphi \left(0\right)\right)\right].\hfill \end{array}$$

But this expression is bounded above by $\eta$ if and only if (5) holds. Thus, the proof is complete. □

3. Inclusion Properties

A function $\hslash \in A$ is said to be in the subclass ${\mathcal{Y}}^{\mu }({\Delta }_1,{\Delta }_2),(\mu \in \mathbb{C}-\left\{0\right\},-1\le {\Delta }_2<{\Delta }_1\le 1)$ if it satisfies the inequality (Dixit and Pal [20]).

$$\left|{\displaystyle \frac{{\hslash }^{\prime }\left(z\right)-1}{({\Delta }_1-{\Delta }_2)\mu -{\Delta }_2[{\hslash }^{\prime }\left(z\right)-1]}}\right|<1(z\in \mathsf{\Lambda }).$$

Note that, if $\mu =1,$${\Delta }_1=\xi$ and ${\Delta }_2=-\xi (0<\xi \le 1)$, then we obtain the subclass of functions $\hslash \in A$ satisfying the following inequality (Caplinger and Causey [21]):

$$\left|{\displaystyle \frac{{\hslash }^{\prime }\left(z\right)-1}{{\hslash }^{\prime }\left(z\right)+1}}\right|<\xi ,(z\in \mathsf{\Lambda }).$$

Lemma 1

([20]). If $\hslash \in {\mathcal{Y}}^{\mu }({\Delta }_1,{\Delta }_2)$ is of form (3), then

$$\left|C_{\tau }\right|\le {\displaystyle \frac{\left|\mu \right|({\Delta }_1-{\Delta }_2)}{\tau }},\tau \in \{2,3,4,\dots \}.$$

The result is sharp.

Theorem 2.

If $0\le \varsigma \le 1$, $\eta >0$ and $\hslash \in {\mathcal{Y}}^{\mu }({\Delta }_1,{\Delta }_2),$ then $E_{\varphi }^{\ast }(\varsigma ,z)\hslash \in {\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,0)$ if

$$Q_3{\varphi }^{\prime \prime }\left(1\right)+(3Q_3+Q_2){\varphi }^{\prime }\left(1\right)+\left(Q_3+Q_2+Q_1\right)\left(\varphi \left(1\right)-\varphi \left(0\right)\right)\le {\displaystyle \frac{\eta D}{({\Delta }_1-{\Delta }_2)\left|\mu \right|}}.$$

(6)

Proof. 

Let ℏ be of the form (3), belonging to the subclass${\mathcal{Y}}^{\mu }({\Delta }_1,{\Delta }_2)$. By Definition 1, it suffices to show that

$$\sum _{\tau =2}^{\infty }\left(Q_3{\tau }^3+Q_2{\tau }^2+Q_1\tau \right){\displaystyle \frac{{\upsilon }_{\tau -1}}{D}}\left|{\hslash }_{\tau }\right|\le \eta .$$

Since ℏ$\in {\mathcal{Y}}^{\mu }({\Delta }_1,{\Delta }_2)$, then by Lemma 1, we obtain

$$\begin{array}{ccc}& & \sum _{\tau =2}^{\infty }\left(Q_3{\tau }^3+Q_2{\tau }^2+Q_1\tau \right){\displaystyle \frac{{\upsilon }_{\tau -1}}{D}}\left|{\hslash }_{\tau }\right|\hfill \\ & \le & {\displaystyle \frac{({\Delta }_1-{\Delta }_2)\left|\mu \right|}{D}}\sum _{\tau =2}^{\infty }\left(Q_3{\tau }^2+Q_2\tau +Q_1\right){\upsilon }_{\tau -1}\hfill \end{array}$$

$$={\displaystyle \frac{({\Delta }_1-{\Delta }_2)\left|\mu \right|}{D}}\sum _{\tau =2}^{\infty }\left[Q_3(\tau -1)(\tau -2)+(3Q_3+Q_2)(\tau -1)+\left(Q_3+Q_2+Q_1\right)\right]{\upsilon }_{\tau -1}$$

$$={\displaystyle \frac{({\Delta }_1-{\Delta }_2)\left|\mu \right|}{D}}\left[Q_3\sum _{\tau =1}^{\infty }\tau (\tau -1){\upsilon }_{\tau }+(2Q_3+Q_2)\sum _{\tau =1}^{\infty }\tau {\upsilon }_{\tau }+\left(Q_3+Q_2+Q_1\right)\sum _{\tau =1}^{\infty }{\upsilon }_{\tau }\right]$$

$$={\displaystyle \frac{({\Delta }_1-{\Delta }_2)\left|\mu \right|}{D}}\left[Q_3{\varphi }^{\prime \prime }\left(1\right)+(3Q_3+Q_2){\varphi }^{\prime }\left(1\right)+\left(Q_3+Q_2+Q_1\right)\left(\varphi \left(1\right)-\varphi \left(0\right)\right)\right].$$

Hence, the condition is obtained if the inequality (6) is satisfied. □

4. An Integral Operator ${\mathbf{J}}_{\mathbf{\varsigma }}^{\mathbf{\varphi }}\left(\mathbf{z}\right)$

Theorem 3.

If $0\le \varsigma \le 1$ and $\eta >0,$ then $J_{\varsigma }^{\varphi }\left(z\right)=\underset{0}{\overset{z}{\int }}{\displaystyle \frac{T{H}_{\varphi }\left(u\right)}{u}}du$$\in {\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,0)$ if and only if

$$Q_3{\varphi }^{\prime \prime }\left(1\right)+(3Q_3+Q_2){\varphi }^{\prime }\left(1\right)+\left(Q_3+Q_2+Q_1\right)\left(\varphi \left(1\right)-\varphi \left(0\right)\right)\le \eta D.$$

(7)

Proof. 

Since

$$J_{\varsigma }^{\varphi }\left(z\right)=\underset{0}{\overset{z}{\int }}{\displaystyle \frac{T{H}_{\varphi }\left(u\right)}{u}}du\equiv z-\sum _{\tau =2}^{\infty }{\displaystyle \frac{{\upsilon }_{\tau -1}}{\tau D}}{z}^{\tau }.$$

By Definition 1, it suffices to show that

$$\sum _{\tau =2}^{\infty }\left(Q_3{\tau }^3+Q_2{\tau }^2+Q_1\tau \right){\displaystyle \frac{{\upsilon }_{\tau -1}}{\tau D}}\equiv {\displaystyle \frac{1}{D}}\sum _{\tau =2}^{\infty }\left(Q_3{\tau }^2+Q_2\tau +Q_1\right){\upsilon }_{\tau -1}\le \eta .$$

Now,

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{D}}\sum _{\tau =2}^{\infty }\left(Q_3{\tau }^2+Q_2\tau +Q_1\right){\upsilon }_{\tau -1}\hfill \\ & =& {\displaystyle \frac{1}{D}}\sum _{\tau =2}^{\infty }\left[Q_3(\tau -1)(\tau -2)+(3Q_3+Q_2)(\tau -1)+\left(Q_3+Q_2+Q_1\right)\right]{\upsilon }_{\tau -1}\hfill \end{array}$$

$$={\displaystyle \frac{1}{D}}\left[Q_3\sum _{\tau =1}^{\infty }\tau (\tau -1){\upsilon }_{\tau }+(3Q_3+Q_2)\sum _{\tau =1}^{\infty }\tau {\upsilon }_{\tau }+\left(Q_3+Q_2+Q_1\right)\sum _{\tau =1}^{\infty }{\upsilon }_{\tau }\right]$$

$$={\displaystyle \frac{1}{D}}\left[Q_3{\varphi }^{\prime \prime }\left(1\right)+(3Q_3+Q_2){\varphi }^{\prime }\left(1\right)+\left(Q_3+Q_2+Q_1\right)\left(\varphi \left(1\right)-\varphi \left(0\right)\right)\right].$$

Hence, the condition is obtained if the inequality (7) is satisfied. □

5. Corollaries and Special Cases

The specialization of the parameters $Q_3,Q_2,Q_1,Q_0$ and $\eta$ in our theorems lead to many results, some of which have been examined by numerous authors. For instance, the following.

Let $Q_3=p\varsigma ,$$Q_2=\varsigma +p(1-2\varsigma ),$$Q_1=1-\varsigma +p(\varsigma -1),$$Q_0=0$ and $\eta =1-\theta$ in Theorem 1. Thus, we obtain the following corollary, which is due to Kota and El-Ashwah ([9], Theorem 2.1).

Corollary 1.

If $0\le \varsigma \le 1,$$0\le p\le 1,$$0\le \theta <1,$ then $E_{\varphi }^{\ast }(\varsigma ,z)\in TF(p,\theta )$ if and only if

$${\displaystyle \frac{1}{D}}\left[p\varsigma {\varphi }^{\prime \prime \prime }\left(1\right)+(p(4\varsigma +1)+\varsigma ){\varphi }^{\prime \prime }\left(1\right)+(2p(\varsigma +1)+2\varsigma +1){\varphi }^{\prime }\left(1\right)+\varphi \left(1\right)-\varphi \left(0\right)\right]\le 1-\theta .$$

Let $Q_3=p\varsigma ,$$Q_2=p(1-2\varsigma )+\varsigma ,$$Q_1=1+p(\varsigma -1)-\varsigma (\theta +1),$$Q_0=\theta (\varsigma -1)$ and $\eta =1-\theta$ in Theorem 1. Thus, we obtain the following corollary, which is due to Kota and El-Ashwah ([9], Theorem 2.2).

Corollary 2.

If $0\le \varsigma \le 1,$$0\le p\le 1,$$0\le \theta <1,$ then $E_{\varphi }^{\ast }(\varsigma ,z)\in TH(p,\theta )$ if and only if

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{D}}\left[p\varsigma {\varphi }^{\prime \prime \prime }\left(1\right)+(p(4\varsigma +1)+\varsigma ){\varphi }^{\prime \prime }\left(1\right)+((2p+1)(\varsigma +1)+\right.\hfill \\ & & \left.\varsigma (1-\theta )){\varphi }^{\prime }\left(1\right)+(1-\theta )\left(\varphi \left(1\right)-\varphi \left(0\right)\right)\right]\hfill \\ & \le & 1-\theta .\hfill \end{array}$$

Let $Q_3=0,$$Q_2=p\varsigma ,$$Q_1=p(1-2\varsigma )+\varsigma ,$$Q_0=1-\varsigma +p(\varsigma -1)$ and $\eta =1-\theta$ in Theorem 1. Thus, we obtain the following corollary, which is due to Kota and El-Ashwah ([9], Theorem 2.3).

Corollary 3.

If $0\le \varsigma \le 1,$$0\le p\le 1,$$0\le \theta <1,$ then $E_{\varphi }^{\ast }(\varsigma ,z)\in TJ(p,\theta )$ if and only if

$${\displaystyle \frac{1}{D}}\left[p\varsigma {\varphi }^{\prime \prime }\left(1\right)+(\varsigma (p+1)+p){\varphi }^{\prime }\left(1\right)+\varphi \left(1\right)-\varphi \left(0\right)\right]\le 1-\theta .$$

Let $Q_3=p\varsigma ,$$Q_2=p(1-\varsigma )-\theta \varsigma ,$$Q_1=\varsigma (1-p)-\theta (1-\varsigma ),$$Q_0=(1-\varsigma )(1-p)$ and $\eta =1-\theta$ in Theorem 1. Thus, we obtain the following corollary, which is due to Kota and El-Ashwah ([9], Theorem 2.5).

Corollary 4.

If $0\le \varsigma \le 1,$$0\le p\le 1,$$0\le \theta <1,$ then $E_{\varphi }^{\ast }(\varsigma ,z)\in TX(p,\theta )$ if and only if

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{D}}\left[p\varsigma {\varphi }^{\prime \prime \prime }\left(1\right)+(p(5\varsigma +1)-\theta \varsigma ){\varphi }^{\prime \prime }\left(1\right)+(3p(\varsigma +1)+\varsigma (1-2\theta )-\theta ){\varphi }^{\prime }\left(1\right)+\right.\hfill \\ & & \left.\left(1-\theta \right)\left(\varphi \left(1\right)-\varphi \left(0\right)\right)\right]\le 1-\theta .\hfill \end{array}$$

Let $\varsigma =0$ in Corollary 4; thus, we obtain the following corollary, which is due to Kota and El-Ashwah ([9], Theorem 2.4).

Corollary 5.

If $0\le p\le 1,$$0\le \theta <1,$ then $T{H}_{\varphi }\left(z\right)\equiv E_{\varphi }^{\ast }(0,z)\in TX(p,\theta )$ if and only if

$${\displaystyle \frac{1}{D}}\left[p{\varphi }^{\prime \prime }\left(1\right)+(3p-\theta ){\varphi }^{\prime }\left(1\right)+\left(1-\theta \right)\left(\varphi \left(1\right)-\varphi \left(0\right)\right)\right]\le 1-\theta .$$

Let $Q_3=0,$$Q_2=0,$$Q_1=\gamma \beta \left|\alpha \right|-2\gamma +1,$$Q_0=(1-\gamma )(\beta \left|\alpha \right|-1)$ and $\eta =\beta \left|\alpha \right|$ in Theorem 1. Thus, we obtain the following corollary, which is due to Yalçın et al. [10] (Theorem 2.1).

Corollary 6.

If $\alpha \in \mathbb{C}-\left\{0\right\},0\le \gamma \le 1,0<\beta \le 1,$ then $T{H}_{\varphi }\left(z\right)\in S(\alpha ,\gamma ,\beta )$ if

$$\gamma {\varphi }^{\prime \prime }\left(1\right)+(1+\gamma \left(\beta \left|\alpha \right|+1\right)){\varphi }^{\prime }\left(1\right)\le \beta \left|\alpha \right|\varphi \left(0\right).$$

(8)

Let $Q_3=Q_0=0,$$Q_2=\gamma ,$$Q_1=1-\gamma$ and $\eta =\beta \left|\alpha \right|$ in Theorem 1. Thus, we obtain the following corollary, which is due to Yalçın et al. [10] (Theorem 2.2).

Corollary 7.

If $\alpha \in \mathbb{C}-\left\{0\right\},0\le \gamma \le 1,0<\beta \le 1,$ then $T{H}_{\varphi }\left(z\right)\in R(\alpha ,\gamma ,\beta )$ if

$$\gamma {\varphi }^{\prime \prime }\left(1\right)+(2\gamma +1){\varphi }^{\prime }\left(1\right)\le (\beta \left|\alpha \right|-1)D+\varphi \left(0\right).$$

Let $Q_3=0,$$Q_2=\gamma ,$$Q_1=1-\gamma$ and $\eta =\beta \left|\alpha \right|$ in Theorem 3. Thus, we obtain the following two corollaries, which are due to Yalçın et al. [10] (Theorem 3.1).

Corollary 8.

If $\alpha \in \mathbb{C}-\left\{0\right\},0\le \gamma \le 1,0<\beta \le 1,$ then $J_{\varsigma }^{\varphi }\left(z\right)\in R(\alpha ,\gamma ,\beta )$ if

$$\gamma {\varphi }^{\prime \prime }\left(1\right)+\varphi \left(1\right)-\varphi \left(0\right)\le \beta \left|\alpha \right|D.$$

Remark 4.

Note that

 1. 

By taking ${\upsilon }_{\tau }={\displaystyle \frac{k^{\tau }}{\tau !}},$$k>0,$ the results of the aforementioned theorems reduce to the results for te Poisson distribution series;

 2. 

By taking ${\upsilon }_{\tau }={\displaystyle \frac{{\left(a\right)}_{\tau }{k}^{\tau }}{{\left(b\right)}_{\tau }\tau !}},$$k>0,$ then the results of the aforementioned theorems reduce to the results for the confluent hypergeometric distribution series;

 3. 

By taking ${\upsilon }_{\tau }={\displaystyle \frac{{\left(a\right)}_{\tau }{\left(c\right)}_{\tau }{k}^{\tau }}{{\left(b\right)}_{\tau }\tau !}},$$k>0,$ then the results of the aforementioned theorems reduce to the results for the hypergeometric distribution-type series;

 4. 

By taking ${\upsilon }_{\tau }={\displaystyle \frac{k^{\tau }}{\mathsf{\Gamma }(q\tau +d)}},$$k>0,$ then the results of the aforementioned theorems reduce to the results for the Mittag–Leffler-type Poisson distribution series;

 5. 

By taking ${\upsilon }_{\tau }={\displaystyle \frac{k^{\tau }}{\tau !}},$$k>0,$ in Corollary 2, then we have the results obtained by Lashin et al. [22] (Theorem 7);

 6. 

By taking ${\upsilon }_{\tau }={\displaystyle \frac{k^{\tau }}{\tau !}},$$k>0$ and $\varsigma =0$ in Corollary 2, then we have the results of Lashin et al. [22] (Corollary 8);

 7. 

By taking ${\upsilon }_{\tau }={\displaystyle \frac{k^{\tau }}{\tau !}},$$k>0,$ in Corollary 3, then we have the results of Lashin et al. [22] (Theorem 9);

 8. 

By taking ${\upsilon }_{\tau }={\displaystyle \frac{k^{\tau }}{\tau !}},$$k>0$ and $\varsigma =0$ in Corollary 3, then we have the results of Frasin [23] (Corollary 5.1);

 9. 

By taking ${\upsilon }_{\tau }={\displaystyle \frac{k^{\tau }}{\tau !}},$$k>0,$ in Corollary 5, then we have the results of Lashin et al. [22] (Theorem 9);

 10. 

If we put $Q_3=0,$$Q_2=\lambda$, $Q_1=1-\lambda ,$$Q_0=-\alpha ,$$\eta =1-\alpha$ and $Q_3=\lambda$, $Q_2=1-\lambda ,$ $Q_1=-\alpha ,$$Q_0=0,$$\eta =1-\alpha$, respectively, then we obtain the necessary and sufficient conditions for functions in the classes TG$(\lambda ,\alpha )$ and TK$(\lambda ,\alpha ),$ obtained by Porwal [6];

 11. 

If we put $Q_3=Q_2=0$, $Q_1=1-\alpha \lambda ,$$Q_0=\alpha (\lambda -1),$$\eta =1-\alpha$ and $Q_3=Q_0=0$, $Q_2=1-\alpha \lambda ,$$Q_1=\alpha (\lambda -1),$$\eta =1-\alpha$, respectively, then we obtain the necessary and sufficient conditions for functions in the classes T$(\lambda ,\alpha )$ and C$(\lambda ,\alpha ),$ obtained by Porwal [12];

 12. 

If we put $Q_3=Q_2=0$, $Q_1=\beta +1,$$Q_0=-(\alpha +\beta ),$ $\eta =1-\alpha$ and $Q_3=Q_0=0$, $Q_2=\beta +1,$$Q_1=-(\alpha +\beta ),$$\eta =1-\alpha$, respectively, then we obtain the necessary and sufficient conditions for functions in the classes S_p$(\alpha ,\beta )$ and UCV$(\alpha ,\beta ),$ obtained by Porwal and Murugusundaramoorthy [11];

 13. 

If we put $Q_3=Q_2=Q_0=0,$$Q_1=\left|\beta \right|+1,$$\eta =(1-\gamma )(B-A)cos\theta$, then we obtain the necessary and sufficient conditions for functions in the class L^θ$(A,B,\gamma ),$ obtained by Kota [8];

 14. 

If we put $Q_3=Q_2=0$, $Q_1=(1-p)sec\gamma -p(\gamma +1),$$Q_0=(p-1)(sec\gamma +\gamma -1),$$\eta =1-\alpha$ and $Q_3=Q_0=0$, $Q_2=(1-p)sec\gamma -p(\gamma +1),$$Q_1=(p-1)(sec\gamma +\gamma -1),$$\eta =1-\alpha ,$ respectively, then we obtain the necessary and sufficient conditions for functions in the classes S$(\gamma ,\alpha ,p)$ and K$(\gamma ,\alpha ,p),$ obtained by Güney and Yıldızhan [7].

6. Conclusions

In this paper, we have derived some necessary and sufficient conditions for the generalized distribution series $E_{\varphi }^{\ast }(\varsigma ,z)$ to be in the inclusive subclass ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,Q_0),$ and the generalized distribution series $E_{\varphi }^{\ast }(\varsigma ,z)\hslash$ and the integral operator $J_{\varsigma }^{\varphi }\left(z\right)$ to be in this subclass ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,0)$. Additionally, we have provided a number of specific corollaries, some of which improve some results available in the literature. This work encourages researchers to find new necessary and sufficient conditions for functions defined by the generalized distribution series $E_{\varphi }^{\ast }(\varsigma ,z)$ or other special functions to be in the subclass ${\mathrm{\Pi }}_{\eta }(Q_3,Q_2,Q_1,Q_0)$.