Fuzzy differential subordination associated with generalized Mittag-Leffler type Poisson distribution

Abstract

This article introduces a new operator defined by the convolution of the Poisson distribution series and the generalized Mittag-Leffler function. Applying this operator, we study a novel subclass of analytic functions using the technique of fuzzy differential subordination. Also, we prove some significant results related to the fuzzy differential relation for this novel class. These results are applied to particular functions to showcase the significance of our findings.

Keywords:

• Analytic functions
• fuzzy differential subordination
• Mittag-Leffler function
• open unit disk
• poisson distribution

1. Introduction

In 1965, Lofti A. Zadeh gave the concept of Fuzzy set theory in his article’ Fuzzy sets’ Zadeh (1965). This paper is a master piece to the study of Fuzzy set theory and piqued the attention of researchers from various domains such as mathematics, sciences, social science and engineering. It began from the idea that components of human thinking are linguistic terms not the numbers. Over time, the study of Fuzzy theory has indeed expanded and found applications in various branches of science. Fuzzy theory deals with uncertainty and provides a framework for reasoning and decision-making where exact binary logic is insufficient. For example, Fuzzy set theory is applicable in Fuzzy geometric transformations, Image processing and Computer vision, Optimization and Decision-making, etc. The flexibility and ability to handle uncertainty makes Fuzzy theory a valuable tool in addressing real-world problems in various fields (see (Coffin & Taylor, 1996; Ganesh & Arivazhagan, 2017; Teodorescu, 2011; Tzes & Borowiec, 1996)).

The notion of differential subordination has been developed and studied within the field of Complex analysis. Diverse branches of the area, such as Geometric function theory and Fractional calculus have extensively used this concept to introduce and investigate various classes of analytic functions (seeJabeen & Saliu, 2023; Mihsin et al., 2022; Raza et al., 2021; Saliu et al., 2022)). Although the concept of differential subordination has started some decades ago but the work of Mocanu and Miller in 1981 and their monograph’ Differential Subordination; Theory and Application’ Miller, & Mocanu, (2000). is the gateway to the in-depth study of theory of differential subordination. In 2011, (Oros, and Oros. (2012) , extended the idea of differential subordination and systematically developed the concept of Fuzzy differential subordination. This notion integrates ideas from Fuzzy set theory and Complex analysis by handling uncertainty in the context of analytical functions. As a result, the novel area has witness significant developments in Geometric function theory. For more information in this direction, one may see (El-Deeb & Lupas, 2020; Kanwal et al., 2023; Lupaş & Oros, 2021; Shah et al., 2023; Shah et al., 2023).

Inspired by its vast applications, we used the technique of Fuzzy differential subordination to introduce a subclass of analytic functions involving Poisson distribution and generalized Mittag-Leffler functions. Furthermore, we investigate some interesting results associated with the Fuzzy differential subordination for this novel class. The motivation behind the present article is the current developments and principally the recent works of Lupas & Oros, 2021; Lupaş & Oros, 2015 We start by defining some subclasses of analytic functions necessary to introduce our novel class.

Let $\Omega =\left\{\varsigma \in \mathbb{C}:\left|\varsigma \right|<1\right\}$ be an open unit disc on a complex plane and $\mathbb{H}(\Omega )$ the class of holomorphic functions in $\Omega .$ We denote $${\mathbb{A}}_n=\left\{f\in \mathbb{H}\left(\Omega \right):f\left(\varsigma \right)=\varsigma +a_{n+1}{\varsigma }^{n+1}+\dots ,\hspace{1em}\varsigma \in \Omega \right\}$$ to be the class of normalized analytic functions with ${\mathbb{A}}_1=\mathbb{A}.$ For $f\in \mathbb{A},$ we have the series representation $$f(\varsigma )=\varsigma +\sum _{n=2}^{\infty }{a}_n{\varsigma }^n,\hspace{1em}\forall \varsigma \in \Omega .$$

Also, we let $$\mathbb{H}[a,n]=\left\{f\in \mathbb{H}\left(\Omega \right):f\left(\varsigma \right)=a+a_{n+1}{\varsigma }^{n+1}+\dots ,\hspace{1em}\varsigma \in \Omega ,a\in \mathbb{C}\right\}$$

Let S denote the class of normalized univalent functions and $\mathbf{C}\subset S$ the class of all convex functions. Analytically, $f\in \mathbf{C}$ if and only if $$\mathfrak{R}\left(\frac{{\left(\varsigma f^{\prime }(\varsigma )\right)}^{\prime }}{f^{\prime }(\varsigma )}\right)>0,\hspace{1em}\varsigma \in \Omega .$$

Let $f$ and $g$ be in ${\mathbb{A}}_n.$ Then $f$ is subordinate to $g$ denoted by $f\prec g$ if there exists analytic function w with $|w(\varsigma )|<1$ and $w(0)=0$ such that $f(\varsigma )=g(w(\varsigma )),$ for all $\varsigma \in \Omega .$

Fuzzy differential subordination extends the concept of differential subordination to fuzzy logic. The notion of fuzzy differential subordination involves knowing the properties of a differential expression on a fuzzy set for a function, allow one to determin the properties of that function on a given fuzzy set. We introduce the following notions of the Fuzzy set.

Definition 1.1

Gal, & Ban, (1996). Let X be a non-empty set. A Fuzzy subset A of X is the pair $(A,F_A)$ such that F_A: $X\to \left[0,1\right]$ is defined to be the membership function of A as (1) $$F_A(x)=\{\begin{array}{cc}1\hfill & \text{if\hspace{0.5em}}x\in A,\hfill \\ 0\hfill & \text{if\hspace{0.5em}}x\notin A,\hfill \end{array}$$(1) and $A=\left\{x\in X:0<F_A(x)\le 1\right\},$ is the support of the Fuzzy set consisting of all the elements with non-zero membership value of set A. So, we can also write as $A=\mathit{\text{supp}}(A,F_A).$

It is worth mentioning that for a Fuzzy subset, 0 is the smallest and 1 is the highest membership value. We denote the membership of empty set $\Phi \subseteq X$ as $F_{\Phi }(x)=0,x\in X,$ while the set X itself has the membership value as $F_X(x)=1,x\in X.$

Definition 1.2

(Lupaş, and Oros (2021) . Let ${\varsigma }_0$ be a fixed point in $\Omega .$ Let $g,h\in \mathbb{H}(\Omega )$ be two analytic functions. The function g is said to be the fuzzy subordinate to the function h and written as $g(\varsigma ){\prec }_{F}h(\varsigma )$ for $\varsigma \in \Omega ,$ or we can also write $g{\prec }_{F}h$ if the following conditions are true:

1. $g({\varsigma }_0)=h({\varsigma }_0),$

2. $F_{g(\Omega )}g(\varsigma )\le F_{h(\Omega )}h(\varsigma ),\hspace{1em}\varsigma \in \Omega .$

The relationship between notions of classical subordination and fuzzy subordination was discussed by Oros et al. in (Oros & Oros, 2011). In this paper, they proved that every classical subordination is not equivalent to fuzzy subordination. However, if we assume $h(\varsigma )$ is univalent, then $g(\varsigma ){\hspace{0.17em}\prec \hspace{0.17em}}_{F}h(\varsigma )$ if and only if $g(\varsigma )\prec h(\varsigma ).$ It means that the classical subordination and fuzzy subordination coincide if and only if $h$ is univalent. For a clear understanding of this fact, (Oros, and Oros (2011) present the following examples.

Example 1.3.

Let $f(\varsigma )=\varsigma +{\varsigma }^2/4$ and $g(\varsigma )={\varsigma }^2/(\varsigma +2)$ be defined in $\Omega .$ Clearly, these functions are not univalent in $\Omega ,$ but $f(0)=g(0).$ If $F_{f(\Omega )}f(\varsigma )\le F_{g(\Omega )}g(\varsigma ),\varsigma \in \Omega ,$ then $f{ \prec }_F\hspace{0.17em}g.$ The condition $F_{f(\Omega )}f(\varsigma )\le F_{g(\Omega )}g(\varsigma )$ implies $f(\Omega )\subset g(\Omega ).$ On the other hand, using the definition of subordination, $f\cancel{\prec }g$ since $g$ is not univalent in $\Omega .$

The subsequent example shows when the two subordinations coincide.

Example 1.4.

Let $f(\varsigma )=\sqrt{1+\varsigma }$ and $g(\varsigma )=\frac{1+\varsigma }{1-\varsigma }.$ It is clear that $f(0)=g(0).$ Furthermore, if we consider $F_{f(\Omega )}f(\varsigma )\le F_{g(\Omega )}g(\varsigma ),\varsigma \in \Omega ,$ then $f{\prec }_{F}g,$ which implies that $f(\Omega )\subset g(\Omega ).$ Since $g$ is univalent, then $f\prec g.$

Conversely, suppose $f\prec g.$ It is easy to see that $f(\varsigma )$ maps the open unit disc onto the right half of Lemniscate of Bernoulli i.e. $f(\Omega )=\{w\in \mathbb{C}:|w^2-1|\le 1\}$ and $g(\varsigma )$ maps the open unit disc onto the right half of the complex plane. Therefore, $f(\Omega )\subset g(\Omega ),$ which means $F_{f(\Omega )}f(\varsigma )\le F_{g(\Omega )}g(\varsigma ).$ Also, $f(0)=g(0),$ and hence $f{\prec }_{F}g.$

Definition 1.5

(Oros, and Oros (2012). Consider $\Psi :{\mathbb{C}}^3\times \Omega \to \mathbb{C}$ and let h be a univalent function in $\Omega$ with, (2) $$\Psi \left(a,0,0;0\right)=h\left(0\right)=a.$$(2)

For an analytic function ρ in $\Omega$ with $\rho (0)=a,$ if the following fuzzy differential subordination of second order is satisfied (3) $$F_{\Psi \left({\mathbb{C}}^3\times \Omega \right)}\Psi \left[\rho \left(\varsigma \right),\varsigma \rho \prime \left(\varsigma \right),{\varsigma }^2{\rho }^{″}\left(\varsigma \right);\varsigma \right]\le F_{h\left(\Omega \right)}h\left(\varsigma \right)\hspace{1em}\varsigma \in \Omega .$$(3)

That is, (4) $$\Psi \left[\rho \left(\varsigma \right),\varsigma \rho \prime \left(\varsigma \right),{\varsigma }^2{\rho }^{″}\left(\varsigma \right);\varsigma \right]{\prec }_{F}h\left(\varsigma \right),\hspace{1em}\varsigma \in \Omega .$$(4)

Then, ρ will be the fuzzy solution of fuzzy differential subordination. A univalent function $q$ will be the fuzzy dominant to the fuzzy differential solution if $$F_{\rho (\Omega )}\rho (\varsigma )\le F_{q(\Omega )}q(\varsigma ),\hspace{1em}\varsigma \in \Omega$$ that is, (5) $$\rho \left(\varsigma \right){\prec }_{F}q\left(\varsigma \right),$$(5)

and it satisfies (Coffin & Taylor, 1996) for all function ρ. A fuzzy dominant $\tilde{q}$ that satisfies the following condition (6) $$\tilde{q}\left(\varsigma \right){\prec }_{F}q\left(\varsigma \right),$$(6) for all fuzzy dominants $q$ of (Coffin & Taylor, 1996) is called fuzzy best dominant.

Swedish Mathematician Gosta Mittag-Leffler introduced Mittag-Leffler function in 1903 while studying summations of some divergent series (Mittag-Leffler (1903). The Mittag Leffler function has the form $$E_{\alpha }\left(\varsigma \right)=\sum _{n=0}^{\infty }\frac{{\varsigma }^n}{\Gamma \left(\alpha n+1\right)}\hspace{1em}\varsigma \in \mathbb{C},\hspace{1em}\mathfrak{R}\left(\alpha \right)>0,$$ which was extended by Wiman (1905). and defined as: $$E_{\alpha ,\beta }\left(\varsigma \right)=\sum _{n=0}^{\infty }\frac{{\varsigma }^n}{\Gamma \left(\alpha n+\beta \right)}\hspace{1em}\varsigma ,\alpha ,\beta \in \mathbb{C},\hspace{1em}\mathfrak{R}\left(\alpha ,\beta \right)>0.$$

This function is normalized as: $${\mathbb{E}}_{\alpha ,\beta }\left(\varsigma \right)=\varsigma +\sum _{n=2}^{\infty }\frac{{\varsigma }^n}{\Gamma \left(\alpha \left(n-1\right)+\beta \right)},$$ where $\varsigma ,\alpha ,\beta \in \mathbb{C},\hspace{0.5em}\mathfrak{R}(\alpha ,\beta )>0$ and $\beta \ne 0,-1,-2,-3,\dots .$ The concept of generalized Mittag-Lefller functions presented in Srivastava, and El-Deeb (2021). plays a key role in the theory of Fractional calculus and has applications in modeling complex physical systems with memory effects. Due to the significance of Mittag-Leffler functions, many researchers used it in Geometric function theory (see Kanwal, 2022; Noreen et al., 2019; Srivastava et al., 2022). In a more general sense, Attiya (2021) used generalized Mittag-Leffler functions to introduce the following function: $$Q_{\alpha ,\beta }^{\eta ,k}(\varsigma )=\varsigma +\sum _{n=2}^{\infty }\frac{\Gamma \left(\alpha +\beta \right)\Gamma \left(\eta +k\right){\varsigma }^n}{\Gamma \left(\alpha \eta +\beta \right)\Gamma \left(\alpha \eta +k\right)n!}\hspace{1em}\varsigma \in \Omega ,$$ where $\alpha ,\beta ,\eta \in \mathbb{C};$ $\mathfrak{R}(\alpha )>\mathrm{\text{max}}\left\{0,\mathfrak{R}(k)-1\right\}$ and k is the complex number such that $\mathfrak{R}(k)>0.$

Distribution series for random variables are studied in Geometric function theory over the span of last decade. Posisson Distribution series was developed by Poisson (2013). The series provides a mathematical framework for understanding and analyzing random processes with a discrete event structure. For a random variable X, the probability mass function of Poisson distribution with parameter m > 0 is define as follows: $$f\left(x\right)=P(X=x)=\frac{e^{-m}}{x!}{m}^x,\hspace{1em}x=0,1,2,\dots .$$

In 2014, Porwal (2014) introduced a series having the coefficients as probability of Poisson Distribution. In the article, he applied this series on univalent functions. Moreover, Porwal and Dixit presented Mittag-Leffler type poisson distribution series in Porwal, and Dixit (2017) . The probability mass function of this distribution with parameter m > 0 is $P(x=r)=\frac{m^r}{\Gamma (\alpha r+\beta ){\mathbb{E}}_{\alpha ,\beta }(m)},\hspace{1em}r=0,1,2,\dots .$ The probability mass function for $Q_{\alpha ,\beta }^{\eta ,k}(z)$ with parameter m > 0 can also be defined in a similar way as $P(x=r)=\frac{m^r}{\Gamma (\alpha r+\beta )Q_{\alpha ,\beta }^{\eta ,k}(m)},\hspace{1em}r=0,1,2,\dots .$ Moreover, Frasin et al. (2022) defined power series whose coefficients involve probabilities of Poisson distribution series of Mittag-Leffler type. Frasin et al. (2021) introduced the following series with generalized Mittag-Leffler function: (7) $${\psi }_{\alpha ,\beta }^{m,\eta ,k}(\varsigma )=\varsigma +\sum _{n=2}^{\infty }\frac{m^{n-1}}{\Gamma (\alpha (n-1)+\beta )Q_{\alpha ,\beta }^{\eta ,k}\left(m\right)}{\varsigma }^n\hspace{1em}m\in \left[0,1\right],\varsigma \in \Omega .$$(7)

By using the technique of convolution, we define the following new operator.

Definition 1.6

(Frasin et al., 2021). Let $f(\varsigma )=\varsigma +{\sum }_{n=2}^{\infty }{a}_n{\varsigma }^n,\varsigma \in \Omega .$ Define $$P_{\alpha ,\beta }^{m,\eta ,k}(\varsigma )={\psi }_{\alpha ,\beta }^{m,\eta ,k}(\varsigma )\ast f(\varsigma ).$$

Using (Frasin et al., 2021) in the above expression, we get the following function (8) $$P_{\alpha ,\beta }^{m,\eta ,k}(\varsigma )=\varsigma +\sum _{n=2}^{\infty }\frac{m^{n-1}}{\Gamma (\alpha (n-1)+\beta )Q_{\alpha ,\beta }^{\eta ,k}\left(m\right)}{a}_n{\varsigma }^n,$$(8) where $\alpha ,\eta ,\beta ,k\in \mathbb{C};$ $\mathfrak{R}(\alpha )>\mathrm{\text{max}}\left\{0,\mathfrak{R}(k)-1\right\};$ $\mathfrak{R}(k)>0,\hspace{0.5em}m\in \left[0,1\right].$

For convenience, we will consider our parameters to be real-valued for $\varsigma \in \Omega .$ We introduce the following class of analytic functions using the operator $P_{\alpha ,\beta }^{m,\eta ,k}.$

Definition 1.7

(Frasin et al., 2021). Consider $0\le \gamma <1.$ A function $f\in \mathbb{A}$ belongs to class ${\overline{\overline{\mathit{F}}}}_{\alpha ,\beta }^{m,\eta ,k}(\gamma )$ if the following inequality holds true: $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)\prime (\Omega )}(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime >\gamma ,\hspace{1em}\varsigma \in \Omega .$$

2. Preliminary results

In this section, we present some important results which are significant to our study.

Lemma 2.1

(Miller, and Mocanu (2000) . Let $h\in \mathbb{A}$ and $$\phi \left(\varsigma \right)=\frac{1}{\varsigma }{\int }_0^{\varsigma }h\left(t\right)\mathit{\text{dt}},\hspace{1em}\varsigma \in \Omega$$ if $$\mathfrak{R}\left(1+\frac{\varsigma h\prime \left(\varsigma \right)}{h^{″}\left(\varsigma \right)}\right)>-\frac{1}{2},\hspace{1em}\varsigma \in \Omega ,$$ then $\phi \in \mathit{C}.$

Lemma 2.2

(Oros, and Oros (2012)). Let h be a convex function and let $\lambda \in {\mathbb{C}}^{*}=\mathbb{C}-\left\{0\right\}$ with $\mathfrak{R}(\lambda )>0.$ If ρ is analytic in $\Omega$ and $\rho (0)=h(0),$ define $\Phi :{\mathbb{C}}^2\times \Omega \to \mathbb{C}$ by $$\Phi \left[\rho \left(\varsigma \right),\varsigma \rho \prime \left(\varsigma \right);\varsigma \right]=\rho \left(\varsigma \right)+\frac{1}{\lambda }\varsigma \rho \prime \left(\varsigma \right)$$ such that $\Phi$ is analytic in $\Omega$ and $$F_{\Phi \left({\mathbb{C}}^2\times \Omega \right)}\left(\rho \left(\varsigma \right)+\frac{1}{\lambda }\varsigma \rho \prime \left(\varsigma \right)\right)\le F_{h\left(\Omega \right)}h\left(\varsigma \right)\Rightarrow \rho \left(\varsigma \right)+\frac{1}{\lambda }\varsigma \rho \prime \left(\varsigma \right){\prec }_{F}h\left(\varsigma \right).$$

Then $$F_{\rho \left(\Omega \right)}\rho \left(\varsigma \right)\le F_{q\left(\Omega \right)}q\left(\varsigma \right)\le F_{h\left(\Omega \right)}h\left(\varsigma \right)\Rightarrow \rho \left(\varsigma \right){\prec }_{F}q\left(\varsigma \right){\prec }_{F}h\left(\varsigma \right)\hspace{1em}\varsigma \in \Omega ,$$ where $q(\varsigma )=\frac{\lambda }{n{\varsigma }^{\frac{\lambda }{n}}}{\int }_0^{\varsigma }h(t)t^{\lambda /n-1}\mathit{\text{dt}},\hspace{1em}\varsigma \in \Omega$ is convex in open unit disc $\Omega$ and fuzzy best dominant.

Lemma 2.3.

[21] Let $g\in \mathbf{C}$ such that $$h(\varsigma )=g(\varsigma )+\mathit{\text{nλςg}}(\varsigma ),\varsigma \in \Omega ,n\in \mathbb{N},\lambda >0.$$

Define a fucntion ρ as $$\rho (\varsigma )=g(0)+{\rho }_n{\varsigma }^n+{\rho }_{n+1}{\varsigma }^{n+1}+\cdots ,\hspace{1em}\varsigma \in \Omega$$ such that $$F_{\rho \left(\Omega \right)}\left(\rho \left(\varsigma \right)+\varsigma \lambda \rho \left(\varsigma \right)\right)\le F_{h\left(\Omega \right)}h\left(\varsigma \right)\Rightarrow \rho \left(\varsigma \right)+\varsigma \lambda \rho \left(\varsigma \right){\prec }_{F}h\left(\varsigma \right).$$

Then $$F_{\rho \left(\Omega \right)}\rho \left(\varsigma \right)\le F_{g\left(\Omega \right)}g\left(\varsigma \right)\Rightarrow \rho \left(\varsigma \right){\prec }_{F}g\left(\varsigma \right).$$

This result is sharp.

3. Main results

In this section, we prove our main results.

Theorem 3.1.

Let $l\in \mathbf{C}$ and suppose that $h(\varsigma )=l(\varsigma )+\frac{1}{s+2}\varsigma l\prime (\varsigma ).$ If $f\in {\overline{\overline{\mathit{F}}}}_{\alpha ,\beta }^{m,\eta ,k}(\gamma )$ and (9) $$G(\varsigma )=I^{s}f(\varsigma )=\frac{s+2}{{\varsigma }^{s+1}}{\int }_0^{\varsigma }{t}^{s}f(t)\mathit{\text{dt}},$$(9) then the following differential subordination (10) $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)\prime (\Omega )}(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime \le F_{h(\Omega )}h(\varsigma )\Rightarrow (P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime {\prec }_{F}h(\varsigma )$$(10) implies (11) $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}G)\prime (\Omega )}(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime \le F_{l(\Omega )}l(\varsigma )\Rightarrow (P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime {\prec }_{F}l(\varsigma ).$$(11)

This result is sharp.

Proof.

Since ${\varsigma }^{s+1}G(\varsigma )=(s+2){\int }_0^{\varsigma }{t}^{s}f(t)\mathit{\text{dt}},$ then a simple computation gives $(s+1)G(\varsigma )+\varsigma G\prime (\varsigma )=(s+2)f(\varsigma ).$ Since $P_{\alpha ,\beta }^{m,\eta ,k}$ is a linear operator, we can write the above equation in the following form (12) $$(s+1)P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma )+\varsigma (P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime =(s+2)(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ).$$(12)

On differentiating (Kanwal, 2022) w.r.t Ϛ, we get (13) $$(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime +\frac{1}{s+2}\varsigma {(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))}^{″}=(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime .$$(13)

From fuzzy differential subordination (Jabeen & Saliu, 2023), we can write (Lupaş & Oros, 2021) as (14) $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}F)(\Omega )}\left(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma )\prime +\frac{1}{s+2}\varsigma {(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))}^{″}\right)\le F_{h(\Omega )}\left(l(\varsigma )+\frac{1}{s+2}\varsigma l\prime (\varsigma )\right).$$(14)

Set (15) $$q(\varsigma )=((P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime$$(15) such that $q\in \mathbb{H}[1,n].$ Then substituting (Miller and Mocanu, 2000) into (Lupaş & Oros, 2015), we obtain $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)(\Omega )}\left(q(\varsigma )+\frac{1}{s+2}\varsigma q\prime (\varsigma )\right)\le F_{h(\Omega )}\left(l(\varsigma )+\frac{1}{s+2}\varsigma l\prime (\varsigma )\right).$$

Applying Lemma (2.3), we have $F_{q(\Omega )}q(\varsigma )\le F_{l(\Omega )}l(\varsigma ).$ That is, $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime (\Omega )}(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime {\prec }_F{F}_{l(\Omega )}l(\varsigma ),$$ which implies $$(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime {\prec }_{F}l(\varsigma ).$$

Hence, we get the result. □

Theorem 3.2.

Assume that $$h(\varsigma )=\frac{1+(2\gamma -1)\varsigma }{1+\varsigma },\hspace{1em}\gamma \in [0,1).$$

Let I^s denote the operator given in (Ganesh & Arivazhagan, 2017). Then $$I^s{\stackrel{\hspace{0.17em}}{\stackrel{¯¯}{\mathit{F}}}}_{\alpha ,\beta }^{m,\eta ,k}\left(\gamma \right)\subset {\stackrel{\hspace{0.17em}}{\stackrel{¯¯}{\mathit{F}}}}_{\alpha ,\beta }^{m,\eta ,k}\left({\gamma }^{*}\right),$$ where (16) $${\gamma }^{*}=2\gamma -1+(\gamma +2)(2-2\gamma ){\int }_0^1\frac{t^{s+2}}{t+1}\mathit{\text{dt}}.$$(16)

Proof.

Following the initial procedures of the proof of Theorem 3.1, we obtain (17) $$F_{q(\Omega )}\left(q(\varsigma )+\frac{1}{m+2}\varsigma q\prime (\varsigma )\right){\prec }_F{F}_{h(\Omega )}h(\varsigma ),$$(17) where $$q(\varsigma )=(P_{\alpha ,\beta }^{m,\eta ,k}G\prime (\varsigma )).$$

Using Lemma (2.2), we get $$F_{q(\Omega )}q(\varsigma )\le F_{l(\Omega )}l(\varsigma )\le F_{h(\Omega )}h(\varsigma ),$$ which implies $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}G)\prime (\Omega )}(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime \le F_{l(\Omega )}l(\varsigma )\le F_{h(\Omega )}h(\varsigma ).$$

Thus, (18) $$\begin{array}{c}l(\varsigma )=\frac{s+2}{{\varsigma }^{s+2}}{\int }_0^{\varsigma }{t}^{s+1}\frac{1+(2\gamma -1)t}{1+t}\mathit{\text{dt}}\\ =(2\gamma -1)+\frac{(s+2)(2-2\gamma )}{{\varsigma }^{s+2}}{\int }_0^{\varsigma }\frac{t^{s+1}}{1+t}\mathit{\text{dt}}.\end{array}$$(18)

By Lemma (2.2), it can easily be seen that $l(\varsigma )\in C$ and is the best fuzzy dominant. Therefore, we have $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}G)\prime (\Omega )}(P_{\alpha ,\beta }^{m,\eta ,k}G(\varsigma ))\prime \ge \underset{{}_{|z=1|}}{\mathrm{\text{min}}}{F}_{l(\Omega )}l(\varsigma )=F_{l(\Omega )}l(1)$$ and $${\gamma }^{*}=l(1).$$

Hence, the proof is complete. □

Theorem 3.3.

Let function $l\in \mathbf{C}$ with $l(0)=1$ and $$h(\varsigma )=l(\varsigma )+l\prime (\varsigma ),\hspace{1em}\varsigma \in \Omega .$$

Let f $\in \mathbb{A}$ satisfy the following fuzzy differential subordination (19) $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)\prime (\Omega )}(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime \le F_{h(\Omega )}h(\varsigma )\Rightarrow (P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime {\prec }_{F}h(\varsigma ).$$(19)

Then (20) $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)(\Omega )}\frac{(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))}{\varsigma }\le F_{l(\Omega )}l(\varsigma )\Rightarrow \frac{(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))}{\varsigma }{\prec }_{F}l(\varsigma ).$$(20)

The result is sharp.

Proof.

Let $$q(\varsigma )=\frac{P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )}{\varsigma }=\frac{\varsigma +{\sum }_{n=2}^{\infty }\frac{m^{n-1}}{\Gamma (\alpha (n-1)+\beta )Q_{\alpha ,\beta }^{\eta ,k}(m)}{a}_n{\varsigma }^n}{\varsigma }$$ $$=1+{\sum }_{n=2}^{\infty }\frac{m^{n-1}}{\Gamma (\alpha (n-1)+\beta )Q_{\alpha ,\beta }^{\eta ,k}(m)}{a}_n{\varsigma }^{n-1}.$$

Then $$q(\varsigma )+\varsigma q\prime (\varsigma )=(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime .$$

Therefore, from (Oros & Oros, 2011), we arrive at $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)\prime (\Omega )}(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime \le F_{h(\Omega )}h(\varsigma ),$$ which implies that $$F_{q(\Omega )}(q(\varsigma )+\varsigma q\prime (\varsigma ))\le F_{h(\Omega )}h(\varsigma )=F_{l(\Omega )}(l(\varsigma )+\varsigma l\prime (\varsigma )).$$

Using Lemma (2.3), we obtain $$F_{q(\Omega )q(\varsigma )}\le F_{l(\Omega )l(\varsigma )}\Rightarrow \frac{P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )}{\varsigma }\le F_{l(\varsigma )}l(\varsigma ),$$ which means $$\frac{P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )}{\varsigma }\prec l(\varsigma ).$$

We demonstrate Theorem 3.3 with following example. □

Example 3.4.

Let $l(\varsigma )=\frac{1}{1+\varsigma }$ with $l(0)=1,l\prime (\varsigma )=\frac{1}{{(1+\varsigma )}^2}$ and $l^{″}(\varsigma )=\frac{-2}{{(1+\varsigma )}^3}.$ Then (21) $$h(\varsigma )=l(\varsigma )+\varsigma l\prime (\varsigma )=\frac{1}{1+\varsigma }+\varsigma \frac{1}{{\left(1+\varsigma \right)}^2},h(\varsigma )=\frac{1+2\varsigma }{{(1+\varsigma )}^2}.$$(21)

Take $f(\varsigma )=\varsigma +3{\varsigma }^2+4{\varsigma }^3.$ For $\alpha =2,\beta =2,m=1,\eta =1,k=1,$ the operator $P_{\alpha ,\beta }^{m,\eta ,k}$ on $f(\varsigma )$ gives $$P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )=P_{2,2}^{1,1,1}f(\varsigma )=\varsigma +0.48{\varsigma }^2+0.64{\varsigma }^3.$$

Also, $$\frac{P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )}{\varsigma }=\frac{P_{2,2}^{1,1,1}f(\varsigma )}{\varsigma }=1+0.48\varsigma +0.64{\varsigma }^2$$ and $$\left(P_{2,2}^{1,1,1}f(\varsigma )\right)\prime =1+0.96\varsigma +1.92{\varsigma }^2.$$

By Theorem 3.3, we have $$1+0.96\varsigma +1.92{\varsigma }^2{\prec }_F\frac{1}{1+\varsigma }+\frac{\varsigma }{{\left(1+\varsigma \right)}^2},$$ which means $$1+0.48\varsigma +0.64{\varsigma }^2{\prec }_F\frac{1}{1+\varsigma }.$$

Theorem 3.5.

Consider $h\in H(\Omega )$ such that $h(0)=1$ with $$\mathfrak{R}(1+\frac{\varsigma h^{″}(\varsigma )}{h\prime (\varsigma )})>-\frac{1}{2}.$$

If $f\in \mathbb{A}$ satisfies the following fuzzy differential subordination (22) $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)\prime (\Omega )}\left(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )\right)\prime \le F_{h(\Omega )}h(\varsigma )\Rightarrow \left(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )\right)\prime {\prec }_{F}h(\varsigma ),$$(22) then (23) $$F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)(\Omega )}\frac{(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))}{\varsigma }\le F_{l(\Omega )}l(\varsigma )\Rightarrow \frac{P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )}{\varsigma }{\prec }_{F}l(\varsigma ),$$(23) where $l(\varsigma )=\frac{1}{\varsigma }{\int }_0^{\varsigma }h(t)\mathit{\text{dt}}$ is convex and best fuzzy dominant.

Proof.

Let (24) $$\begin{array}{c}q(\varsigma )=\frac{P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )}{\varsigma }=1+\sum _{n=2}^{\infty }\frac{m^{n-1}}{\Gamma (\alpha (n-1)+\beta )Q_{\alpha ,\beta }^{\eta ,k}}{a}_n{\varsigma }^{n-1},\end{array}$$(24)

where $q\in H[1,n].$ From Lemma (2.1), we know that $l(\varsigma )=\frac{1}{\varsigma }{\int }_0^{\varsigma }h(t)\mathit{\text{dt}}$ belongs to $\mathbf{C}$ and satisfies fuzzy differential subordination (Oros & Oros, 2011). Since $l(\varsigma )+\varsigma l\prime (\varsigma )=h(\varsigma ),$ l is fuzzy best dominant. Now, we observe that $q(\varsigma )+\varsigma q\prime (\varsigma )=(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime ,\hspace{1em}\varsigma \in \Omega .$ So (Poisson, 2013), becomes $F_{q(\Omega )}(q(\varsigma )+\varsigma q\prime (\varsigma ))\le F_{h(\Omega )}h(\varsigma ).$ By using Lemma 3, we obtain $F_{q(\Omega )}q(\varsigma )\le F_{l(\Omega )}l(\varsigma )\Rightarrow F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)(\Omega )}\frac{(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))}{\varsigma }\le F_{l(\Omega )}l(\varsigma ).$ Thus, (25) $$\frac{P_{\alpha ,\beta }^{m,\eta ,k}f\left(\varsigma \right)}{\varsigma }{\prec }_{F}l(\varsigma ),$$(25) which completes the proof. □

Corollary 3.6. Let $h(\varsigma )=(1+(2\mu -1)\varsigma )/(1+\varsigma ),0\le \mu <1,\varsigma \in \Omega ,$ and $f\in \mathbb{A}$ satisfies the following relation $F_{(P_{\alpha ,\beta }^{m,\eta ,k}f)\prime (\Omega )}(P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma ))\prime \le F_{h(\Omega )}h(\varsigma )$ $\Rightarrow P_{\alpha ,\beta }^{m,\eta ,k}f(\varsigma )\prime {\prec }_{F}h(\varsigma ).$ Then the function $l(\varsigma$) given by $l(\varsigma )=2\mu -1+\frac{2(1-\mu )}{\varsigma }\mathrm{\text{log}}\hspace{0.17em}(1+\varsigma )$ is convex and fuzzy best dominant.

Example 3.7.

Let $h(\varsigma )=\frac{1+A\varsigma }{1+B\varsigma }$ with $h(0)=1$ such that $-1\le B<A\le 1,$ we get $$\begin{array}{c}\mathfrak{R}\left(\frac{\varsigma h^{″}}{h\prime }+1\right)=\mathfrak{R}\left(\varsigma .\frac{-2B(A-B)}{{\left(1+B\varsigma \right)}^3}.\frac{{\left(1+B\varsigma \right)}^2}{A-B}+1\right)\\ =\mathfrak{R}\left(\frac{1-B\varsigma }{1+B\varsigma }\right)\\ =\mathfrak{R}\left(\frac{1-B^2{p}^2}{1+B^2{p}^2+2\mathit{\text{Bp}}\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}\theta }\right)>0>-\frac{1}{2}.\end{array}$$

Therefore, from Lemma (2.1), it implies that the function f is convex. Let us consider $f(\varsigma )=\varsigma +{\varsigma }^2.$ For $\alpha =2,\beta =1,m=0.5,\eta =1,k=4,$ the operator

$P_{2,1}^{0.5,1,4}f(\varsigma )=\varsigma +0.0156{\varsigma }^2.$ Also, $\frac{P_{2,1}^{0.5,1,4}f(\varsigma )}{\varsigma }=1+0.0156\varsigma$ and $(P_{2,1}^{0.5,1,4}f(\varsigma ))\prime =1+0.0312\varsigma .$ Since $$\begin{array}{c}l(\varsigma )=\frac{1}{\varsigma }{\int }_0^{\varsigma }h(t)\mathit{\text{dt}}=\frac{1}{\varsigma }{\int }_0^{\varsigma }\left(\frac{1+\mathit{\text{At}}}{1+\mathit{\text{Bt}}}\right)\mathit{\text{dt}}\\ =\frac{A}{B}-\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}\left(1+B\varsigma \right)\left(A-B\right)}{B^2\varsigma },\end{array}$$ then by Theorem 3.5, $$1+0.0312(\varsigma ){\prec }_F\frac{1+A\varsigma }{1+B\varsigma }.$$

Thus, we get $$1+0.0156(\varsigma ){\prec }_F\frac{A}{B}-\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}\left(1+B\varsigma \right)\left(A-B\right)}{B^2\varsigma },\hspace{1em}\varsigma \in \Omega ,$$ which implies that $$l(\varsigma )=\frac{1}{\varsigma }{\int }_0^{\varsigma }h(t)\mathit{\text{dt}}=\frac{1}{\varsigma }{\int }_0^{\varsigma }\left(\frac{1+\mathit{\text{At}}}{1+\mathit{\text{Bt}}}\right)\mathit{\text{dt}}$$ is a fuzzy best dominant.

4. Conclusion

In the present article, we used the operator associated with generalized Mittag-Leffler function and Poisson distribution to introduce a novel subclass of analytic functions defined by a means of fuzzy differential subordination. Some significant results related to second order fuzzy differential subordination are obtained. Moreover, some examples are given as an application of our results.

The work presented in this article can be generalized in a variety of ways. Geometric properties, like coefficient estimates, inclusion relations, and distortion results for this class can be established. Furthermore, the idea presented in this article can also be expanded in the realm of fractional calculus. Results related to higher order fuzzy differential subordination can also be established.

4.1. Subject classification codes

[2010]Primary 30C45, 30C50

Notes on contributor(s)

The authors worked jointly and equally on this manuscript. All the authors approved the final version of the manuscript.

Acknowledgement(s)

The authors are thankful to the Vice Chancellors of Fatima Jinnah Women University and University of the Gambia for providing excellent research and academic environment.

Disclosure statement

The authors declare no conflicts of interest.

Additional information

Funding

There is no external funding available for this article.