Fuzzy Subordination Results for Meromorphic Functions Associated with Hurwitz–Lerch Zeta Function

Abstract

The notion of the fuzzy set was incorporated into geometric function theory in recent years, leading to the emergence of fuzzy differential subordination theory, which is a generalization of the classical differential subordination notion. This article employs a new integral operator introduced using the class of meromorphic functions, the notion of convolution, and the Hurwitz–Lerch Zeta function for obtaining new fuzzy differential subordination results. Furthermore, the best fuzzy dominants are provided for each of the fuzzy differential subordinations investigated. The results presented enhance the approach to fuzzy differential subordination theory by giving new results involving operators in the study, for which starlikeness and convexity properties are revealed using the fuzzy differential subordination theory.

1. Introduction

Since the publication of the first paper outlining the concept of subordination in fuzzy set theory in 2011 [1], the fields of complex analysis and fuzzy set theory have contributed to the study of geometric function theory (GFT). The idea was inspired by the general trend present in new research directions in many scientific and practical fields to integrate aspects of the fuzzy set theory initiated by Lotfi A. Zadeh [2]. The well-known notions and methods of the differential subordination theory established by Miller and Mocanu [3,4] were considered to be adapted by incorporating fuzzy set notions and to offer a new approach to the differential subordination theory. The research techniques provided by Miller and Mocanu were adopted by later publications examining fuzzy differential subordination and included elements from the previously established theory of differential subordination concerning key notions [5,6]. Scholars studying GFT quickly welcomed the concept, and all the traditional lines of research in this field were modified to take the new fuzzy features into consideration. Operator-related research is an important line of study in GFT. Such studies on developing fuzzy subordination results were published in 2013 [7], not long after the concept was introduced. The continued research interest [8,9,10] was followed by the addition of the dual notion of fuzzy differential superordination [11]. After that, the two dual notions were considered for obtaining new fuzzy differential subordination and superordination results [12,13,14,15]. To illustrate that the interest in this topic is continually expanding, we merely mention a handful of the many publications that have only been released during the current year [16,17,18,19]. Fuzzy differential subordination theory embedding fractional and quantum calculus has advanced significantly in recent years as a result of increasing influence in GFT. A fractional integral operator is employed in [20], the q-analogue of multiplier transformation is applied in [21], and certain quantum calculus operators are involved in the study published in [22].

The general context of the investigation is provided by the following elements.

Consider $\mathfrak{D}=\{\tau :\tau \in \mathbb{C}$ and $\left|\tau \right|<1\}$ representing the unit disc of the complex plane, $\mathcal{H}\left(\mathfrak{D}\right)$ representing the class of analytic functions in $\mathfrak{D},$ and the widely used classes:

$${\mathcal{A}}_{\kappa }=\left\{\mathfrak{f}\in \mathcal{H}\left(\mathfrak{D}\right):\mathfrak{f}\left(\mathit{\tau }\right)=\tau +a_{\kappa +1}{\tau }^{\kappa +1}+\cdots \tau \in \mathfrak{D}\right\},$$

and

$$\mathcal{H}[a,\kappa ]=\left\{\mathfrak{f}\in \mathcal{H}\left(\mathfrak{D}\right):\mathfrak{f}\left(\mathit{\tau }\right)=a+a_{\kappa }{\tau }^{\kappa }+a_{\kappa +1}{\tau }^{\kappa +1}+\cdots \tau \in \mathfrak{D}\right\},$$

for $a\in \mathbb{C}$, $\kappa \in \mathbb{N}=\{1,2,3,\cdots \}$, where $\mathcal{H}[1,1]=\mathcal{H}$.

Normalized convex functions in $\mathfrak{D}$ are represented by

$$\mathfrak{K}=\left\{\mathfrak{f}\in {\mathcal{A}}_{\kappa }:Re\left(1+{\displaystyle \frac{\tau {\mathfrak{f}}^{″}\left(\tau \right)}{{\mathfrak{f}}^{\prime }\left(\tau \right)}}\right)>0,\tau \in \mathfrak{D}\right\}.$$

The class of meromorphic functions represented by $\Sigma$ comprises functions written as

$$\mathfrak{f}\left(\mathit{\tau }\right)={\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{a}_{\kappa }{\tau }^{\kappa }(\tau \in {\mathfrak{D}}^{\ast }=\mathfrak{D}\setminus \left\{0\right\}),$$

(1)

where ${\mathfrak{D}}^{\ast }$ is the punctured unit disc defined by ${\mathfrak{D}}^{\ast }=\{\tau :\tau \in \mathbb{C}$ and $0<\left|\tau \right|<1\}.$

If $\mathfrak{f}\in \Sigma$ as in (1) and $\mathfrak{g}$ are given by

$$\mathfrak{g}\left(\tau \right)={\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{b}_{\kappa }{\tau }^{\kappa },$$

then the Hadamard product of $\mathfrak{f}$ and $\mathfrak{g}$ is represented by

$$(\mathfrak{f}\ast \mathfrak{g})\left(\tau \right)={\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{a}_{\kappa }{b}_{\kappa }{\tau }^{\kappa }.$$

Let ${\mathcal{S}}_{\Sigma }^{\ast }$ and ${\mathcal{C}}_{\Sigma }$ be the subclasses of $\Sigma$, which are meromorphic starlike and meromorphic convex in ${\mathfrak{D}}^{\ast }$, respectively, and stated by

$${\mathcal{S}}_{\Sigma }^{\ast }=\left\{\mathfrak{f}:\mathfrak{f}\in \Sigma and-Re\left({\displaystyle \frac{\tau {\mathfrak{f}}^{\prime }\left(\tau \right)}{\mathfrak{f}\left(\tau \right)}}\right)>0(\tau \in {\mathfrak{D}}^{\ast })\right\},$$

and

$${\mathcal{C}}_{\Sigma }=\left\{\mathfrak{f}:\mathfrak{f}\in \Sigma and-Re\left(1+{\displaystyle \frac{\tau {\mathfrak{f}}^{″}\left(\tau \right)}{{\mathfrak{f}}^{\prime }\left(\tau \right)}}\right)>0(\tau \in {\mathfrak{D}}^{\ast })\right\}.$$

Meromorphic functions are associated in recent studies regarding fuzzy differential subordination theory, such as [23,24]; hence, they are useful tools in developing this theory and will be used in this investigation. Meromorphic functions have been successfully associated with the Hurwitz–Lerch Zeta function in research involving new operators, such as [25,26,27].

The general Hurwitz–Lerch Zeta function $\phi (\tau ,\varrho ,\varkappa )$ is defined by (see [28]):

$$\phi (\tau ,\varrho ,\kappa )=\sum _{\kappa =0}^{\infty }{\displaystyle \frac{{\tau }^{\kappa }}{{(\kappa +\varkappa )}^{\varrho }}}$$

$$(\varkappa \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}=\left\{0,-1,-2,\dots \right\};\varrho \in \mathbb{C}\mathrm{where}\left|\tau \right|<1:Re\varrho >1\mathrm{when}\left|\tau \right|=1).$$

Numerous authors’ investigations have revealed a number of intriguing aspects and traits of the Hurwitz–Lerch Zeta function $\phi (\tau ,\varrho ,\kappa )$ ([29,30,31,32]). Inspired by the interesting results obtained using this function for defining new operators and conducting studies regarding differential subordination theory, in this research, the following function, denoted ${\mathfrak{g}}_{\varkappa }^{\varrho }\left(\tau \right)(\varkappa \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};\varrho \in \mathbb{C})$, is defined by

$${\mathfrak{g}}_{\varkappa }^{\varrho }\left(\tau \right)={\displaystyle \frac{{\varkappa }^{\varrho }}{\tau }}\phi (\tau ,\varrho ,\kappa )(\tau \in {\mathfrak{D}}^{\ast }).$$

(2)

Additionally, denote by ${\mathcal{L}}_{\varkappa }^{\varrho }\mathfrak{f}\left(\tau \right):\Sigma \to \Sigma$ the linear operator defined by:

$${\mathcal{L}}_{\varkappa }^{\varrho }\mathfrak{f}\left(\tau \right)={\mathfrak{g}}_{\varkappa }^{\varrho }\left(\tau \right)\ast \mathfrak{f}\left(\tau \right)(\varkappa \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};\varrho \in \mathbb{C};\tau \in {\mathfrak{D}}^{\ast }).$$

Then,

$${\mathcal{L}}_{\varkappa }^{\varrho }\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +1}}\right)}^{\varrho }{a}_{\kappa }{\tau }^{\kappa }.$$

It must be noted that:

(i)

${\mathcal{L}}_1^{\varrho }\mathfrak{f}\left(\tau \right)=P^{\varrho }\mathfrak{f}\left(\tau \right)(\varrho >0)$ (Aqlan et al. [33], with $p=1$);

(ii)

${\mathcal{L}}_{\varkappa }^{\varrho }\mathfrak{f}\left(\tau \right)=P_{\varkappa }^{\varrho }\mathfrak{f}\left(\tau \right)(\varrho ,\varkappa >0)$(Lashin [34]);

(iii)

${\mathcal{L}}_{\varkappa }^1\mathfrak{f}\left(\tau \right){=}_{\varkappa }\mathfrak{f}\left(\tau \right)(\varkappa >0)$ ([35], p. 11 and 389).

Finally, for $\mathfrak{f}\left(\tau \right)\in \Sigma ,\tau ,t_i\in {\mathfrak{D}}^{\ast }(i=1,2,3,\dots ,\kappa ),\kappa \in \mathbb{N}$ and $\varkappa \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-},$ we have:

$${\mathcal{L}}_1^{\mathfrak{0}}\mathfrak{f}\left(\tau \right)=\mathfrak{f}\left(\tau \right)\mathrm{and}{\mathcal{L}}_{\varkappa }^{\mathfrak{0}}\mathfrak{f}\left(\tau \right)=\mathfrak{f}\left(\tau \right);$$

$${\mathcal{L}}_{\varkappa }^{\kappa }\mathfrak{f}\left(\tau \right)={\displaystyle \frac{{\varkappa }^{\kappa }}{{\tau }^{\varkappa +1}}}{\int }_0^{\tau }{\displaystyle \frac{1}{t_1}}{\int }_0^{t_1}{\displaystyle \frac{1}{t_2}}{\int }_0^{t_2}\dots {\displaystyle \frac{1}{t_{\kappa -1}}}{\int }_0^{t_{\kappa -1}}{t}_{\kappa }^{\varkappa }\mathfrak{f}\left(t_{\kappa }\right)d{t}_{\kappa }d{t}_{\kappa -1}\dots d{t}_{2}d{t}_1(\mathfrak{f}\in \Sigma ;\tau \in {\mathfrak{D}}^{\ast }).$$

As a characteristic, it can be proved that:

$${\mathcal{L}}_{\varkappa }^{\varrho +1}\mathfrak{f}\left(\tau \right)={\displaystyle \frac{\varkappa }{{\tau }^{\varkappa +1}}}{\int }_0^{\tau }{t}^{\varkappa }{\mathcal{L}}_{\varkappa }^{\varrho }\mathfrak{f}\left(t\right)dt(\mathfrak{f}\in \Sigma ;\tau \in {\mathfrak{D}}^{\ast }).$$

Next, the following function is introduced:

$$\Psi (n,m;\tau )={\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{\displaystyle \frac{{\left(n\right)}_{\kappa +1}}{{\left(m\right)}_{\kappa +1}}}{\tau }^{\kappa }(n\in {\mathbb{C}}^{\ast }=\mathbb{C}\setminus \left\{0\right\};m\in {\mathbb{C}}^{\ast }=\mathbb{C}\setminus {\mathbb{Z}}_0^{-};\tau \in {\mathfrak{D}}^{\ast }),$$

(3)

where ${\left(\zeta \right)}_{\kappa }$ is the Pochhammer symbol defined, in terms of the Gamma function $\Gamma ,$ as

$${\left(\zeta \right)}_{\kappa }={\displaystyle \frac{\Gamma (\zeta +\kappa )}{\Gamma \left(\zeta \right)}}=\left\{\begin{array}{cc}1& (\kappa =0)\\ \zeta (\zeta +1)\dots (\zeta +\kappa +1)& (\kappa \in \mathbb{N}).\end{array}\right.$$

We note that

$$\Psi (n,m;\tau )={\displaystyle \frac{1}{\tau }}{}_2\Gamma _1(n,1;m;\tau ),$$

where

$${}_2\Gamma _1(n,v;m;\tau )=\sum _{\kappa =0}^{\infty }{\displaystyle \frac{{\left(n\right)}_{\kappa }{\left(v\right)}_{\kappa }}{{\left(m\right)}_{\kappa }{\left(1\right)}_{\kappa }}}{\tau }^{\kappa }(n,v,m\in \mathbb{C}andm\notin {\mathbb{Z}}_0^{-};\tau \in \mathfrak{D}),$$

is the Gaussian hypergeometric function.

Setting

$${\mathcal{L}}_{\varkappa }^{\varrho }\ast {\mathfrak{k}}_{\varkappa }^{\varrho }\left(\tau \right)={\displaystyle \frac{1}{\tau (1-\tau )}},$$

we have

$${\mathfrak{k}}_{\varkappa }^{\varrho }\left(\tau \right)={\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{\left({\displaystyle \frac{\kappa +\varkappa +1}{\varkappa }}\right)}^{\varrho }{\tau }^{\kappa }.$$

By using the operator ${\mathfrak{k}}_{\varkappa }^{\varrho }\left(\tau \right)$, a new operator ${\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )$ is introduced as follows:

$${\mathfrak{k}}_{\varkappa }^{\varrho }\left(\tau \right)\ast {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )=\Psi (n,m;\tau )(\tau \in {\mathfrak{D}}^{\ast }).$$

(4)

The linear operator, ${\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau ):\Sigma \to \Sigma ,$ is described by:

$${\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)={\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\ast \mathfrak{f}\left(\tau \right)(\varkappa ,m\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};n\in {\mathbb{C}}^{\ast };\varrho \in \mathbb{C},\tau \in {\mathfrak{D}}^{\ast }),$$

whose series expansion for $\varkappa ,m\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};n\in {\mathbb{C}}^{\ast };\varrho \in \mathbb{C},\tau \in {\mathfrak{D}}^{\ast }$ and for $\mathfrak{f}$ as in (1) is given by

$${\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)={\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +1}}\right)}^{\varrho }{\displaystyle \frac{{\left(n\right)}_{\kappa +1}}{{\left(m\right)}_{\kappa +1}}}{a}_{\kappa }{\tau }^{\kappa }.$$

(5)

We note that this operator ${\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )$ was introduced by El-Ashawh [36].

It is easily verified from the definition of the operator ${\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau ),$ that:

$$\tau {\left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }=\varkappa {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)-(\varkappa +1){\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)$$

(6)

and

$$\tau {\left({\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }=n{\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)-(n+1){\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)(n\in \mathbb{C}\setminus \{-1\}).$$

(7)

We note that

${\mathfrak{k}}_{\varkappa }^{\varrho }(\mu ,1;\tau )\mathfrak{f}\left(\tau \right)=I_{\varkappa ,\mu }^{\varrho }\mathfrak{f}\left(\tau \right)(\varkappa ,\mu \in {\mathbb{R}}^{+},\varrho \in {\mathbb{N}}_0)$ (Cho et al. [37]).

2. Definitions and Preliminaries

The following definitions and lemmas will be used as tools for proving the new results included in the following section.

Definition 1 

([1]). Let X be a non-empty set. An application $F:X\to \left[0,1\right]$ is called a fuzzy subset.

Definition 2 

([1]). A pair $\left(A,F_A\right)$, where $F_A:X\to \left[0,1\right]$ and

$$A=\left\{x\in X:0<F_A\left(x\right)\le 1\right\},$$

is called a fuzzy subset of X.

The set A is called the support of the fuzzy set $\left(A,F_A\right)$, and $F_A$ is called the membership function of the fuzzy set $\left(A,F_A\right)$.

One can also denote $A=$ supp $\left(A,F_A\right).$

Proposition 1 

([1]). $\left(i\right)$$\left(\mathfrak{b},F_{\mathfrak{b}}\right)=\left(\mathfrak{D},F_{\mathfrak{D}}\right)$ then $\mathfrak{b}=\mathfrak{D}$ where $\mathfrak{b}=\mathit{supp}(\mathfrak{b},F_{\mathfrak{b}})$ and $\mathfrak{D}=$$\mathit{supp}\left(\mathfrak{D},F_{\mathfrak{D}}\right).$

$\left(ii\right)$ If $\left(\mathfrak{b},F_{\mathfrak{b}}\right)\subseteq \left(\mathfrak{D},F_{\mathfrak{D}}\right),$ then $I\subseteq \mathfrak{D}$, where $I=\mathit{supp}(I,F_{\mathfrak{b}})$ and $\mathfrak{D}=\mathit{supp}\left(\mathfrak{D},F_{\mathfrak{D}}\right).$

Let $\mathfrak{f}$, $\mathfrak{g}\in \mathcal{H}\left(\mathfrak{D}\right).$ We say

$$\mathfrak{f}\left(\mathfrak{D}\right)=\mathit{supp}(\mathfrak{f}\left(\mathfrak{D}\right),F_{\mathfrak{f}\left(\mathfrak{D}\right)})=\{\mathfrak{f}\left(\tau \right):0<\left|F_{\mathfrak{f}\left(\mathfrak{D}\right)}\left(\mathfrak{f}\left(\tau \right)\right)\right|\le 1,\tau \in \mathfrak{D}\}$$

(8)

and

$$\mathfrak{g}\left(\mathfrak{D}\right)=\mathit{supp}(\mathfrak{g}\left(\mathfrak{D}\right),F_{\mathfrak{g}\left(\mathfrak{D}\right)})=\{\mathfrak{g}\left(\tau \right):0<\left|F_{\mathfrak{g}\left(\mathfrak{D}\right)}\left(\mathfrak{g}\left(\tau \right)\right)\right|\le 1,\tau \in \mathfrak{D}\}.$$

(9)

Definition 3 

([1]). Let ${\tau }_0\in \mathfrak{D}$ and $\mathfrak{f}$, $\mathfrak{g}\in \mathcal{H}\left(\mathfrak{D}\right).$ $\mathfrak{f}$ be said to be fuzzy subordinate to $\mathfrak{g}$ and written as $\mathfrak{f}{\prec }_F\mathfrak{g}$ or $\mathfrak{f}\left(\tau \right){\prec }_F\mathfrak{g}\left(\tau \right)$ if each of the subsequent requirements is satisfied:

$$\mathfrak{f}\left({\tau }_0\right)=\mathfrak{g}\left({\tau }_0\right)\mathit{and}\left|F_{\mathfrak{f}\left(\mathfrak{D}\right)}\left(\mathfrak{f}\left(\tau \right)\right)\right|\le \left|F_{\mathfrak{g}\left(\mathfrak{D}\right)}\left(\mathfrak{g}\left(\tau \right)\right)\right|,\tau \in \mathfrak{D},$$

Proposition 2 

([1]). Let ${\tau }_0\in \mathfrak{D}$ and $\mathfrak{f}$, $\mathfrak{g}\in \mathcal{H}\left(\mathfrak{D}\right).$ if $\mathfrak{f}\left(\tau \right){\prec }_F\mathfrak{g}\left(\tau \right),$$\tau \in \mathfrak{D},$ then

$$\begin{array}{ccc}\hfill \left(\mathfrak{i}\right)\mathfrak{f}\left({\tau }_0\right)& =& \mathfrak{g}\left({\tau }_0\right),\hfill \\ \hfill \left(\mathfrak{ii}\right)\mathfrak{f}\left(\mathfrak{D}\right)& \subseteq & \mathfrak{g}\left(\mathfrak{D}\right)\mathit{and}\left|F_{\mathfrak{f}\left(\mathfrak{D}\right)}\left(\mathfrak{f}\left(\tau \right)\right)\right|\le \left|F_{\mathfrak{g}\left(\mathfrak{D}\right)}\left(\mathfrak{g}\left(\tau \right)\right)\right|,\tau \in \mathfrak{D},\hfill \end{array}$$

where $\mathfrak{f}\left(\mathfrak{D}\right)$ and $\mathfrak{g}\left(\mathfrak{D}\right)$ are given by (8) and (9), respectively.

Definition 4 

([5]). Let $\Omega :{\mathbb{C}}^3\times \mathfrak{D}\to \mathbb{C}$ and let $\mathfrak{h}$ be an analytic function with Ω$(\mathrm{a},0,0,0)=\mathfrak{h}\left(0\right)=\mathrm{a}$. If ω is analytic in $\mathfrak{D}$ with $\omega \left(0\right)=\mathrm{a}$ and satisfies the (second-order) fuzzy differential subordination:

$$F_{\Omega ({\mathbb{C}}^3\times \mathfrak{D})}\left(\Omega (\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{″}\left(\tau \right);\tau )\right)\le \left|F_{\mathfrak{h}\left(\mathfrak{D}\right)}\left(\mathfrak{h}\left(\mathfrak{D}\right)\right)\right|,$$

(10)

i.e.,

$$\Omega \left(\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right),{\tau }^2{\omega }^{″}\left(\tau \right);\tau \right){\prec }_F\mathfrak{h}\left(\tau \right),\tau \in \mathfrak{D},$$

then ω is a fuzzy solution and ω is a fuzzy dominant if

$$\left|F_{\omega \left(\mathfrak{D}\right)}\omega \left(\tau \right)\right|\le \left|F_{\chi \left(\mathfrak{D}\right)}\chi \left(\tau \right)\right|,i.e.,\omega \left(\tau \right){\prec }_F\chi \left(\tau \right)\tau \in \mathfrak{D},$$

for all ω satisfying (8). A fuzzy dominant $\tilde{\chi }$ that satisfies

$$\left|F_{\tilde{\chi }\left(\mathfrak{D}\right)}\tilde{\chi }\left(\tau \right)\right|\le \left|F_{\chi \left(\mathfrak{D}\right)}\chi \left(\tau \right)\right|i.e.,\tilde{\chi }\left(\tau \right){\prec }_F\chi \left(\tau \right)\tau \in \mathfrak{D},$$

for all fuzzy dominants χ of (10) is the fuzzy best dominant of (10).

Lemma 1 

([35] Corollary 2.6g.2, p. 66). Assume that $\mathrm{Y}\in \mathcal{H}\left(\mathfrak{D}\right)$ and

$$\mu \left(\tau \right)={\displaystyle \frac{1}{n{\tau }^{\frac{1}{n}}}}{\int }_0^{\tau }{t}^{{\displaystyle \frac{1}{n}}-1}\mathrm{Y}\left(t\right)dt,(\tau \in \mathfrak{D}).$$

If

$$Re\left({\displaystyle \frac{\tau {\mathrm{Y}}^{″}\left(\tau \right)}{{\mathrm{Y}}^{\prime }\left(\tau \right)}}+1\right)>-{\displaystyle \frac{1}{2}}(\tau \in \mathfrak{D}),$$

then $\mu \in \mathfrak{K}.$

Lemma 2 

([38]). Assume $\mathrm{Y}$ is a convex function that satisfies $\mathrm{Y}\left(0\right)=a;$ let $\epsilon \in {\mathbb{C}}^{\ast }=\mathbb{C}\setminus \left\{0\right\}$ such that $Re\left(\epsilon \right)\ge 0.$ If $\omega \in \mathcal{H}[a,n]$ with $\omega \left(0\right)=\mathfrak{a}$ and $\Omega :{\mathbb{C}}^2\times \mathfrak{D}\to \mathbb{C}$, $\Omega (\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right))=\omega \left(\tau \right)+{\displaystyle \frac{1}{\epsilon }}\tau {\omega }^{\prime }\left(\tau \right)$ is analytic in $\mathfrak{D}$, then

$$\left|F_{\Omega ({\mathbb{C}}^3\times \mathfrak{D})}\left(\Omega (\omega \left(\tau \right)+{\displaystyle \frac{1}{\epsilon }}\tau {\omega }^{\prime }\left(\tau \right))\right)\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\iff \omega \left(\tau \right)+{\displaystyle \frac{1}{\epsilon }}\tau {\omega }^{\prime }\left(\tau \right){\prec }_F\mathrm{Y}\left(\tau \right)(\tau \in \mathfrak{D}),$$

implies

$$F_{\omega \left(\mathfrak{D}\right)}\left(\omega \left(\tau \right)\right)\le F_{\chi \left(\mathfrak{D}\right)}\left(\chi \left(\tau \right)\right)\le F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)i.e.,\omega \left(\tau \right){\prec }_F\chi \left(\tau \right),$$

where

$$\chi \left(\tau \right)={\displaystyle \frac{\epsilon }{n{\tau }^{\frac{\epsilon }{n}}}}\underset{0}{\overset{\tau }{\int }}{t}^{{\displaystyle \frac{\epsilon }{n}}-1}\mathrm{Y}\left(t\right)dt,$$

is best dominant and convex.

Lemma 3 

([38]). Assume that χ is a convex function in $\mathfrak{D}$, and let

$$\mathrm{Y}\left(\tau \right)=\chi \left(\tau \right)+n\upsilon \tau {\chi }^{\prime }\left(\tau \right),$$

$\upsilon >0$ and $n\in \mathbb{N}.$ If $\omega \in \mathcal{H}\left[\chi \right(0),n]$ and $\Omega :{\mathbb{C}}^2\times \mathfrak{D}\to \mathbb{C}$,

$$\Omega (\omega \left(\tau \right),\tau {\omega }^{\prime }\left(\tau \right))=\omega \left(\tau \right)+\upsilon n\tau {\omega }^{\prime }\left(\tau \right)$$

is analytic in $\mathfrak{D}$, then

$$\left|F_{\Omega ({\mathbb{C}}^2\times \mathfrak{D})}\left(\omega \left(\tau \right)+n\upsilon \tau {\omega }^{\prime }\left(\tau \right)\right)\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\iff \omega \left(\tau \right)+n\upsilon \tau {\omega }^{\prime }\left(\tau \right){\prec }_F\mathrm{Y}\left(\tau \right)(\tau \in \mathfrak{D}),$$

implies that

$$\left|F_{\omega \left(\mathfrak{D}\right)}\left(\omega \left(\tau \right)\right)\right|\le \left|F_{\chi \left(\mathfrak{D}\right)}\left(\chi \left(\tau \right)\right)\right|i.e.,\omega \left(\tau \right){\prec }_F\chi \left(\tau \right),$$

is the fuzzy best dominant.

The purpose of this investigation is to obtain fuzzy differential subordinations associated with the Hurwitz–Lerch Zeta function through the operator ${\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )$. For each fuzzy differential subordination obtained, the fuzzy best dominant will also be provided.

3. Main Results

Theorem 1. 

Let μ be a convex function in $\mathfrak{D}$, $\mu \left(0\right)=1,$ and

$$\mathrm{Y}\left(\tau \right)=\mu \left(\tau \right)+\tau {\mu }^{\prime }\left(\tau \right),\tau \in \mathfrak{D}.$$

If, for $\mathfrak{f}\in \Sigma$, satisfying

$$\left|F_{\Omega ({\mathbb{C}}^2\times {\mathfrak{D}}^{\ast })}\left[\varkappa \left[\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)-\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right]+\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right]\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|,$$

implies

$$\left[\varkappa \left[\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)-\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right]+\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right]{\prec }_F\mathrm{Y}\left(\tau \right),$$

then

$$\left|F_{\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}\left(\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right)\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\Rightarrow \tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right){\prec }_F\mu \left(\tau \right).$$

Proof. 

Assume that

$$\omega \left(\tau \right)=\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right).$$

(11)

From (11) and (6), we have

$$\begin{array}{ccc}\hfill \omega \left(\tau \right)+\tau {\omega }^{\prime }\left(\tau \right)& =& \tau \left[\varkappa \left({\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +1}}\right)}^{\varrho }{\displaystyle \frac{{\left(n\right)}_{\kappa +1}}{{\left(m\right)}_{\kappa +1}}}{a}_{\kappa }{\tau }^{\kappa }\right)\right.-\hfill \\ & & \left.(\varkappa +1)\left({\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +1}}\right)}^{\varrho +1}{\displaystyle \frac{{\left(n\right)}_{\kappa +1}}{{\left(m\right)}_{\kappa +1}}}{a}_{\kappa }{\tau }^{\kappa }\right)\right]\hfill \\ & & +2\tau \left({\displaystyle \frac{1}{\tau }}+\sum _{\kappa =0}^{\infty }{\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +1}}\right)}^{\varrho +1}{\displaystyle \frac{{\left(n\right)}_{\kappa +1}}{{\left(m\right)}_{\kappa +1}}}{a}_{\kappa }{\tau }^{\kappa }\right)\hfill \\ & =& \tau \left[\varkappa {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)-(\varkappa +1){\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right]+2\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right).\hfill \\ & =& \varkappa \left[\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)-\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right]+\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right).\hfill \end{array}$$

(12)

We observe from (1) and (12), that

$$\left|F_{\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}\left(\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right)\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|,$$

implies

$$\left|F_{\Omega ({\mathbb{C}}^2\times {\mathfrak{D}}^{\ast })}(\omega \left(\tau \right)+\tau {\omega }^{\prime }\left(\tau \right))\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\le \left|F_{\mu \left(\mathfrak{D}\right)}(\mu \left(\tau \right)+\tau {\mu }^{\prime }\left(\tau \right))\right|.$$

Consequently, using Lemma 2 with $\epsilon =1$, yields

$$\left|F_{\omega \left(\mathfrak{D}\right)}\left(\omega \left(\tau \right)\right)\right|\le \left|F_{\mu \left(\mathfrak{D}\right)}\left(\mu \left(\tau \right)\right)\right|$$

$$\Rightarrow \left|F_{\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}\left(\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right)\right|\le \left|F_{\mu \left(\mathfrak{D}\right)}\left(\mu \left(\tau \right)\right)\right|,$$

i.e.,

$$\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right){\prec }_F\mu \left(\tau \right).$$

This completes the proof. □

Theorem 2. 

Let μ be a convex function in $\mathfrak{D}$, $\mu \left(0\right)=1,$ and

$$\mathrm{Y}\left(\tau \right)=\mu \left(\tau \right)+\tau {\mu }^{\prime }\left(\tau \right)(\tau \in \mathfrak{D}).$$

If for $\mathfrak{f}\in \Sigma$, satisfying

$$\left|F_{\Omega ({\mathbb{C}}^2\times {\mathfrak{D}}^{\ast })}\left[n\left[\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)-\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right]+\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right]\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|,$$

implies

$$\left[n\left[\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)-\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right]+\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right]{\prec }_F\mathrm{Y}\left(\tau \right),$$

then

$$\left|F_{\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}\left(\tau \left({\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right)\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\Rightarrow \tau \left({\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right){\prec }_F\mu \left(\tau \right).$$

Proof. 

Since the proof is similar to the Theorem 1 proof employing (7), we decided to leave it out. □

Theorem 3. 

Let μ be a convex function in $\mathfrak{D}$, $\mu \left(0\right)=1,$ and

$$\mathrm{Y}\left(\tau \right)=\mu \left(\tau \right)+\tau {\mu }^{\prime }\left(\tau \right)\tau \in \mathfrak{D}.$$

If for $\mathfrak{f}\in \Sigma$, satisfying

$$\left|F_{{\left({\tau }^2{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}{\left({\tau }^2{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\Rightarrow {\left({\tau }^2{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }{\prec }_F\mathrm{Y}\left(\tau \right),$$

then

$$\left|F_{{\left({\tau }^2{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}{\left({\tau }^2{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\Rightarrow \tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right){\prec }_F\mu \left(\tau \right).$$

Proof. 

Let

$$\omega \left(\tau \right)=\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right).$$

(13)

From (13) and (5), we have

$$\begin{array}{ccc}\hfill \omega \left(\tau \right)+\tau {\omega }^{\prime }\left(\tau \right)& =& 2\left(1+\sum _{\kappa =0}^{\infty }{\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +1}}\right)}^{\varrho +1}{\displaystyle \frac{{\left(n\right)}_{\kappa +1}}{{\left(m\right)}_{\kappa +1}}}{a}_{\kappa }{\tau }^{\kappa }\right)\hfill \\ & & +\left(-1+\sum _{\kappa =0}^{\infty }\kappa {\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +1}}\right)}^{\varrho +1}{\displaystyle \frac{{\left(n\right)}_{\kappa +1}}{{\left(m\right)}_{\kappa +1}}}{a}_{\kappa }{\tau }^{\kappa }\right)\hfill \\ & =& 1+\sum _{n=0}^{\infty }(\kappa +2){\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +1}}\right)}^{\varrho +1}{\displaystyle \frac{{\left(n\right)}_{\kappa +1}}{{\left(m\right)}_{\kappa +1}}}{a}_{\kappa }{\tau }^{\kappa +1}.\hfill \end{array}$$

We find that

$$\omega \left(\tau \right)+\tau {\omega }^{\prime }\left(\tau \right)={\left({\tau }^2{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }.$$

We get that

$$\left|F_{{\left({\tau }^2{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}\left[{\left({\tau }^2{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }\right]\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|,$$

implies

$$\left|F_{\Omega ({\mathbb{C}}^2\times {\mathfrak{D}}^{\ast })}(\omega \left(\tau \right)+\tau {\omega }^{\prime }\left(\tau \right))\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\le \left|F_{\mu \left(\mathfrak{D}\right)}(\mu \left(\tau \right)+\tau {\mu }^{\prime }\left(\tau \right))\right|.$$

Consequently, using Lemma 3 with $n=\upsilon =1$, we have

$$\begin{array}{ccc}\hfill \left|F_{\omega \left(\mathfrak{D}\right)}\left(\omega \left(\tau \right)\right)\right|& \le & \left|F_{\mu \left(\mathfrak{D}\right)}\left(\mu \left(\tau \right)\right)\right|\Rightarrow \left|F_{\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}\left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right|\hfill \\ & \le & \left|F_{\mu \left(\mathfrak{D}\right)}\left(\mu \left(\tau \right)\right)\right|,\hfill \end{array}$$

which implies that

$$\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right){\prec }_F\mu \left(\tau \right).$$

This completes the proof. □

Theorem 4. 

Assume that $\mathrm{Y}\in \mathcal{H}\left(\mathfrak{D}\right)$ with $\mathrm{Y}\left(0\right)=1,$ and

$$Re\left(1+{\displaystyle \frac{\tau {\mathrm{Y}}^{″}\left(\tau \right)}{{\mathrm{Y}}^{\prime }\left(\tau \right)}}\right)>-{\displaystyle \frac{1}{2}}(\tau \in \mathfrak{D}),$$

If $\mathfrak{f}\in \Sigma$, and the fuzzy differential subordination stated below is valid:

$$\begin{array}{ccc}& & \left|F_{\left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}\left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\hfill \\ & \Rightarrow & \left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right){\prec }_F\mathrm{Y}\left(\tau \right),\hfill \end{array}$$

(14)

then

$$\begin{array}{ccc}\hfill \left|F_{\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}\left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right|& \le & \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\hfill \\ & \Rightarrow & \tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right){\prec }_F\mu \left(\tau \right),\hfill \end{array}$$

where $\mu \left(\tau \right)$, defined as

$$\mu \left(\tau \right)={\displaystyle \frac{1}{\tau }}\underset{0}{\overset{\tau }{\int }}\mathrm{Y}\left(t\right)dt,$$

is the fuzzy best dominant and convex.

Proof. 

Suppose that:

$$\omega \left(\tau \right)=\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right).$$

(15)

We have $\omega \left(\tau \right)\in \mathcal{H}[1,1].$ Let $\mathrm{Y}\in \mathcal{H}\left(\mathfrak{D}\right)$ with $\mathrm{Y}\left(0\right)=1,$ and

$$Re\left(1+{\displaystyle \frac{\tau {\mathrm{Y}}^{″}\left(\tau \right)}{{\mathrm{Y}}^{\prime }\left(\tau \right)}}\right)>-{\displaystyle \frac{1}{2}}(\tau \in \mathfrak{D}).$$

Lemma 1 gives

$$\mu \left(\tau \right)={\displaystyle \frac{1}{\tau }}\underset{0}{\overset{\tau }{\int }}\mathrm{Y}\left(t\right)dt,$$

which is convex and satisfies (14), and

$$\mathrm{Y}\left(\tau \right)=\mu \left(\tau \right)+\tau {\mu }^{\prime }\left(\tau \right)(\tau \in \mathfrak{D})$$

is the fuzzy best dominant.

Next,

$$\begin{array}{ccc}\hfill \omega \left(\tau \right)+\tau {\omega }^{\prime }\left(\tau \right)& =& 1+\sum _{n=0}^{\infty }(\kappa +2){\left({\displaystyle \frac{\varkappa }{\kappa +\varkappa +1}}\right)}^{\varrho +1}{\displaystyle \frac{{\left(n\right)}_{\kappa +1}}{{\left(m\right)}_{\kappa +1}}}{a}_{\kappa }{\tau }^{\kappa +1}\hfill \\ & =& {\left({\tau }^2{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }.\hfill \end{array}$$

(16)

From (16), the fuzzy differential subordination (14) involves

$$\left|F_{\omega \left(\mathfrak{D}\right)}(\omega \left(\tau \right)+\tau {\omega }^{\prime }\left(\tau \right))\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\mathrm{Y}\left(\tau \right)\right|.$$

Consequently, using Lemma 3 with $\upsilon =1$, we find

$$\left|F_{\omega \left(\mathfrak{D}\right)}\left(\omega \left(\tau \right)\right)\right|\le \left|F_{\mu \left(\mathfrak{D}\right)}\left(\mu \left(\tau \right)\right)\right|.$$

This completes the proof. □

By taking $-1\le A<\mathcal{B}\le 1,$ and

$$\mathrm{Y}\left(\tau \right)={\displaystyle \frac{1+A\tau }{1+\mathcal{B}\tau }}(\tau \in \mathfrak{D}),$$

the next result can be deduced from Theorem 4.

Corollary 1. 

Assume $-1\le A<\mathcal{B}\le 1,$

$$\mathrm{Y}\left(\tau \right)={\displaystyle \frac{1+A\tau }{1+\mathcal{B}\tau }}(\tau \in \mathfrak{D}),$$

with $\mathrm{Y}\left(0\right)=1.$

If for $\mathfrak{f}\in \Sigma$, satisfying

$$\begin{array}{ccc}& & \left|F_{\left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}\left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\hfill \\ & \Rightarrow & \left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right){\prec }_F\mu \left(\tau \right),\hfill \end{array}$$

(17)

where $\mu \left(\tau \right)$ is defined as

$$\mu \left(\tau \right)={\displaystyle \frac{A}{\mathcal{B}}}+{\displaystyle \frac{1-\frac{A}{\mathcal{B}}}{\mathcal{B}\tau }}log(1+\mathcal{B}\tau ),$$

is the fuzzy best dominant and convex.

Example 1. 

Assume:

$\left(i\right)$$A=2\gamma -1,0\le \gamma <1,\mathcal{B}=1$

$$\mathrm{Y}\left(\tau \right)={\displaystyle \frac{1+(2\gamma -1)\tau }{1+\tau }}(\tau \in \mathfrak{D}),$$

with $\mathrm{Y}\left(0\right)=1.$ If for $\mathfrak{f}\in \Sigma$, satisfying

$$\begin{array}{ccc}& & \left|F_{\left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}\left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\hfill \\ & \Rightarrow & \left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right){\prec }_F\mu \left(\tau \right),\hfill \end{array}$$

where $\mu \left(\tau \right)$ is defined as

$$\mu \left(\tau \right)=2\gamma -1+{\displaystyle \frac{2(1-\gamma )}{\tau }}log(1+\tau ),$$

and is the fuzzy best dominant and convex;

$\left(ii\right)\gamma =0$ then

$$\mathrm{Y}\left(\tau \right)={\displaystyle \frac{1-\tau }{1+\tau }}(\tau \in \mathfrak{D}),$$

where

$$\mu \left(\tau \right)=-1+{\displaystyle \frac{2}{\tau }}log(1+\tau ),$$

is the fuzzy best dominant and convex.

Theorem 5. 

Let μ be a convex in $\mathfrak{D},$$\mu \left(0\right)=1,$ and

$$\mathrm{Y}\left(\tau \right)=\mu \left(\tau \right)+\tau {\mu }^{\prime }\left(\tau \right).$$

Let $\mathfrak{f}\in \Sigma$, and ${\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }$ is analytic in $\mathfrak{D}$. If

$$\begin{array}{ccc}& & \left|F_{{\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }}{\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\hfill \\ & \Rightarrow & {\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }{\prec }_F\mathrm{Y}\left(\tau \right),\hfill \end{array}$$

(18)

then

$$\left|F_{{\left({\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }}{\left({\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|.$$

i.e.,

$${\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}{\prec }_F\mu \left(\tau \right).$$

Proof. 

Let

$$\omega \left(\tau \right)={\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}.$$

(19)

Then $\omega \left(\tau \right)\in \mathcal{H}[1,1].$ Differentiating (19),

$${\omega }^{\prime }\left(\tau \right)={\displaystyle \frac{{\left({\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}-\omega \left(\tau \right){\displaystyle \frac{{\left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}.$$

Then

$$\omega \left(\tau \right)+\tau {\omega }^{\prime }\left(\tau \right)={\displaystyle \frac{\left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\left[\tau {\left({\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }+\left({\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)\right]-}{{\left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^2}}$$

$${\displaystyle \frac{\left(\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)\right){\left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^{\prime }}{{\left({\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)\right)}^2}}$$

$$\begin{array}{ccc}& =& {\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }.\hfill \end{array}$$

(20)

Utilizing (20) in (18), we can get

$$\left|F_{{\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }}{\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|,$$

implying that

$$\begin{array}{ccc}\hfill \left|F_{\omega \left(\mathfrak{D}\right)}(\omega \left(\tau \right)+\tau {\omega }^{\prime }\left(\tau \right))\right|& \le & {|}_{\mathrm{Y}\left({\mathfrak{D}}^{\ast }\right)}\left(\mathrm{Y}\left(\tau \right)\right)|\le \hfill \\ & & \left|F_{\mu \left({\mathfrak{D}}^{\ast }\right)}(\mu \left(\tau \right)+\tau {\mu }^{\prime }\left(\tau \right))\right|.\hfill \end{array}$$

Consequently, using Lemma 3 with $\upsilon =1$, we get

$$\left|F_{\left({\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}\left({\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)\right|\le \left|F_{\mu \left(\mathfrak{D}\right)}\left(\mu \left(\tau \right)\right)\right|,$$

i.e.,

$${\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}{\prec }_F\mu \left(\tau \right).$$

This completes the proof. □

Theorem 6. 

Let μ be a convex in $\mathfrak{D}$ such that $\mu \left(0\right)=1,$ and

$$\mathrm{Y}\left(\tau \right)=\mu \left(\tau \right)+\tau {\mu }^{\prime }\left(\tau \right).$$

Let $\mathfrak{f}\in \Sigma$, and ${\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }$ is holomorphic in $\mathfrak{D}$. If

$$\begin{array}{ccc}& & \left|F_{{\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }}{\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|\hfill \\ & \Rightarrow & {\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }{\prec }_F\mathrm{Y}\left(\tau \right),\hfill \end{array}$$

then

$$\left|F_{{\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }}{\left({\displaystyle \frac{\tau {\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}}\right)}^{\prime }\right|\le \left|F_{\mathrm{Y}\left(\mathfrak{D}\right)}\left(\mathrm{Y}\left(\tau \right)\right)\right|,$$

i.e.,

$${\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}}{\prec }_F\mu \left(\tau \right).$$

Proof. 

Since the proof is similar to the Theorem 5 proof, we decided to leave it out.

□

4. Conclusions

The use of fuzzy set theory in differential subordination has allowed researchers to explore new methods of research and to develop more sophisticated mathematical models. By incorporating fuzzy sets into the existing framework of differential subordination, researchers have been able to overcome limitations in traditional approaches. Moving forward, continued research in this area is likely to yield further insights and advancements involving both fuzzy set theory and the study of differential subordination. The operator ${\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )$ provided by relation (5) has been effectively employed for obtaining new fuzzy differential subordinations for which the best fuzzy dominants were also found. El-Ashwah [36] examined the operator for the meromorphic function using the Hurwitz–Lerch Zeta function $\phi (\tau ,\varrho ,\varkappa )$. We are now investigating fuzzy differential subordination applications in (GFT). Fuzzy differential subordinations are given in the first three theorems, for which the fuzzy best dominants are found. Theorem 4 provides an integral representation for the best dominant and is followed by a corollary and an example of how the new outcome can be used. Fuzzy differential subordinations were obtained for ${\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho +1}(n,m;\tau )\mathfrak{f}\left(\tau \right)}}$ and ${\displaystyle \frac{{\mathfrak{k}}_{\varkappa }^{\varrho }(n+1,m;\tau )\mathfrak{f}\left(\tau \right)}{{\mathfrak{k}}_{\varkappa }^{\varrho }(n,m;\tau )\mathfrak{f}\left(\tau \right)}}$ through Theorems 5 and 6.

The outcome contained in this paper is new and will encourage more research in the field of GFT considering that the second-order fuzzy differential subordinations obtained here could be extended to third-order fuzzy differential subordinations taking into account the very recent results presented by [39]. Also, the dual notion of fuzzy differential superordination can be applied following the pattern set here for the new integral operator. New operators could also be introduced by associating Hurwitz–Lerch-Zeta functions and meromorphic functions, which could be further applied to obtain new fuzzy differential subordination results following the present research.