Fuzzy Treatment for Meromorphic Classes of Admissible Functions Connected to Hurwitz–Lerch Zeta Function

Abstract

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are related to a class of complex parameter operators. Complex analysis ideas from geometric function theory are used to derive fuzzy differential subordination conclusions. Due to the compositional structure of the operator, some pertinent classes of admissible functions are studied through the application of fuzzy differential subordination.

1. Introduction

In 2011, researchers established a connection between fuzzy set theory and complex analysis, focusing on the geometric properties of analytic functions. [1] Miller and Mocanu looked at the idea of unequal subordination for the first time in [2,3]. In 2011, Oros and Oros [1] conducted research on fuzzy subordination, and in 2012 [4], they first introduced fuzzy differential subordination. An excellent summary of the history of fuzzy set theory and its relationships with numerous scientific and technological domains can be found in this 2017 publication [5]. It also contains citations for the fuzzy differential subordination theory research that had been carried out up till then. Research in this area would not have been able to continue without the initial results, which changed the traditional differential subordination hypothesis to include the unique features of fuzzy differential subordination and provided ways to find the dominants and best dominants of fuzzy differential subordinations [6].

Motivated by the thought that various mathematical domains have been attempting to embrace the fuzzy set notion established by Zadeh [7], a new sort of differential subordination arose. The first published studies on the creation of fuzzy differential subordination theories were developed with the use of several types of operators [8,9,10].

Several new results are currently being produced through the use of various operators in fuzzy differential subordination theories, as demonstrated by recently published articles such as [11,12,13,14]. As evidenced by citing works like [15,16,17], contemporary studies of fuzzy differential subordination have also focused on the class of meromorphic functions. Motivated by the aforementioned research and taking into account the novelty of the field of study pertaining to the differential subordination of admissible functions, the current study employs the Hurwitz–Lerch Zeta function and the Gaussian hypergeometric function to examine a novel linear operator introduced by the convolution applied to the operator ${\mathcal{L}}_{\varrho }^b(p,o;\zeta )$ introduced by El-Ashawh [18]. It is demonstrated that the new operator is a generalization of operators that have already been studied.

The aim of this work is to use this novel operator to demonstrate second-order fuzzy differential subordination features of meromorphic functions. The following parts present novel fuzzy differential subordination for acceptable functions discovered, while the next section recalls the previously established results employed for this analysis.

Consider that $\mathfrak{U}=\{\zeta :\zeta \in \mathbb{C}$ and $\left|\zeta \right|<1\}$ describes the unit disk in the complex plane, and let $\mathcal{H}\left(\mathfrak{U}\right)$ be the space of analytic functions symbolized by $\mathfrak{U};$ then

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

and

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

for $a\in \mathbb{C}$, $\kappa \in \mathbb{N}=\{1,2,3,\cdots \}$, and $\mathcal{H}[1,1]=\mathcal{H}$. Normalized convex function classes in $\mathfrak{U}$ are represented by

$$\mathfrak{K}=\left\{\mathfrak{f}\in \mathcal{H}\left(\mathfrak{U}\right):Re\left({\displaystyle \frac{\zeta {\mathfrak{f}}^{″}\left(\zeta \right)}{{\mathfrak{f}}^{\prime }\left(\zeta \right)}}+1\right)>0,{\mathfrak{f}}^{\prime }\left(0\right)\ne 0,\tau \in \mathfrak{U}\right\}.$$

Let $\Sigma$ be a class of meromorphic functions formed by

$$\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{1}{\zeta }}+\sum _{\kappa =0}^{\infty }{a}_{\kappa }{\zeta }^{\kappa },(\zeta \in {\mathfrak{U}}^{*}=\mathfrak{U}\setminus \left\{0\right\}),$$

(1)

where ${\mathfrak{U}}^{*}$ is the punctured unit disc defined by

$$\mathfrak{U}=\{\zeta :\zeta \in C\mathrm{and}0<\left|\zeta \right|<1\}.$$

If $\mathfrak{f}\in \Sigma$, as in (1), and $\mathfrak{g}$ is defined by

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

the definition of the Hadamard product (convolution) $*:\Sigma \to \Sigma$ is

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

In recent studies, meromorphic functions have been associated with fuzzy differential subordination theory, so they are useful tools for developing this theory and will be used in this investigation. The Hurwitz–Lerch Zeta function has been successfully linked to meromorphic functions in studies utilizing new operators, including [19,20].

The general Hurwitz–Lerch Zeta function, $\varphi (\zeta ,b,\varrho )$, is defined by (see [21])

$$\varphi (\zeta ,b,\varrho )=\sum _{\kappa =0}^{\infty }{\displaystyle \frac{{\zeta }^{\kappa }}{{(\kappa +\varrho )}^b}},$$

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

Several interesting features and characteristics of the Hurwitz–Lerch Zeta function, $\varphi (\zeta ,b,\varrho )$ [22,23,24,25], have been discovered in research by several authors. Motivated by the intriguing outcomes of using this function to define new operators and for research on differential subordination theory, the following function, written as

$${\mathfrak{g}}_{\varrho }^b\left(\zeta \right)(\varrho \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};b\in \mathbb{C}),$$

is defined by

$${\mathfrak{g}}_{\varrho }^b\left(\zeta \right)={\displaystyle \frac{{\varrho }^b}{\zeta }}\varphi (\zeta ,b,\varrho )(\zeta \in {\mathfrak{U}}^{*}).$$

(2)

The linear operator, denoted by ${\mathcal{L}}_{\varrho }^{\mathfrak{b}}\mathfrak{f}\left(\zeta \right):\Sigma \to \Sigma$, is defined by

$${\mathcal{L}}_{\varrho }^b\mathfrak{f}\left(\zeta \right)={\mathfrak{g}}_{\varrho }^b\left(\zeta \right)\ast \mathfrak{f}\left(\zeta \right),(\varrho \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};b\in \mathbb{C};\zeta \in {\mathfrak{U}}^{*}).$$

Then,

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

We note that

(i) 

For $b\in {\mathbb{R}}^{+}$ and $\varrho =1,$ the operator ${\mathcal{L}}_1^b\mathfrak{f}\left(\zeta \right)$ reduces to $P^b\mathfrak{f}\left(\zeta \right)$, as introduced by Aqlan et al. (see [26], with $p=1$);

(ii) 

For $\varrho ,b\in {\mathbb{R}}^{+},$ the operator ${\mathcal{L}}_{\varrho }^b\mathfrak{f}\left(\zeta \right)$ reduces to $P_{\varrho }^b\mathfrak{f}\left(\zeta \right)$, as introduced by Lashin (see [27]);

(iii) 

For $b=1$ and $\varrho \in {\mathbb{R}}^{+},$ the operator ${\mathcal{L}}_{\varrho }^1\mathfrak{f}\left(\zeta \right)$ reduces to ${\mathfrak{F}}_{\varrho }\mathfrak{f}\left(\zeta \right)$, as introduced by Miller and Mocanu (see [28], p. 389).

For $\mathfrak{f}\left(\zeta \right)\in \Sigma ,\zeta ,t_i\in {\mathfrak{U}}^{*}(i=1,2,3,\dots ,\kappa ),\kappa \in \mathbb{N}$, and $\varrho \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-},$ we have

$$\begin{array}{ccc}\hfill {\mathcal{L}}_1^{\mathfrak{0}}\mathfrak{f}\left(\zeta \right)& =& \mathfrak{f}\left(\zeta \right)and{\mathcal{L}}_{\varrho }^{\mathfrak{0}}\mathfrak{f}\left(\zeta \right)=\mathfrak{f}\left(\zeta \right);\hfill \\ \hfill {\mathcal{L}}_1^1\mathfrak{f}\left(\zeta \right)& =& {\displaystyle \frac{1}{{\zeta }^2}}{\int }_0^{\zeta }{t}_1\mathfrak{f}\left(t_1\right)d{t}_1,(\mathfrak{f}\in \Sigma ;\zeta \in {\mathfrak{U}}^{*});\hfill \\ \hfill {\mathcal{L}}_1^2\mathfrak{f}\left(\zeta \right)& =& {\displaystyle \frac{1}{{\zeta }^2}}{\int }_0^{\zeta }{\displaystyle \frac{1}{t_1}}{\int }_0^{t_1}{t}_2\mathfrak{f}\left(t_2\right)d{t}_{2}d{t}_1,(\mathfrak{f}\in \Sigma ;\zeta \in {\mathfrak{U}}^{*});\hfill \\ & & ⋮\hfill \\ \hfill {\mathcal{L}}_1^{\kappa }\mathfrak{f}\left(\zeta \right)& =& {\displaystyle \frac{1}{{\zeta }^2}}{\int }_0^{\zeta }{\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 }\mathfrak{f}\left(t_{\kappa }\right)d{t}_{\kappa }d{t}_{\kappa -1}\dots .d{t}_{2}d{t}_1,(\mathfrak{f}\in \Sigma ;\zeta \in {\mathfrak{U}}^{*});\hfill \\ \hfill {\mathcal{L}}_{\varrho }^1\mathfrak{f}\left(\zeta \right)& =& {\displaystyle \frac{\varrho }{{\zeta }^{\varrho +1}}}{\int }_0^{\zeta }{t}^{\varrho }\mathfrak{f}\left(t\right)dt,(\mathfrak{f}\in \Sigma ;\zeta \in {\mathfrak{U}}^{*});\hfill \\ \hfill {\mathcal{L}}_{\varrho }^2\mathfrak{f}\left(\zeta \right)& =& {\displaystyle \frac{{\varrho }^2}{{\zeta }^{\varrho +1}}}{\int }_0^{\zeta }{\displaystyle \frac{1}{t_1}}{\int }_0^{t_1}{t}_2^{\varrho }\mathfrak{f}\left(t_2\right)d{t}_{2}d{t}_1,(\mathfrak{f}\in \Sigma ;\zeta \in {\mathfrak{U}}^{*});\hfill \\ & & ⋮\hfill \\ \hfill {\mathcal{L}}_{\varrho }^{\kappa }\mathfrak{f}\left(\zeta \right)& =& {\displaystyle \frac{{\varrho }^{\kappa }}{{\zeta }^{\varrho +1}}}{\int }_0^{\zeta }{\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 }^{\varrho }\mathfrak{f}\left(t_{\kappa }\right)d{t}_{\kappa }d{t}_{\kappa -1}\dots .d{t}_{2}d{t}_1,(\mathfrak{f}\in \Sigma ;\zeta \in {\mathfrak{U}}^{*}).\hfill \end{array}$$

Also, we can show that

$${\mathcal{L}}_{\varrho }^{b+1}\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{\varrho }{{\zeta }^{\varrho +1}}}{\int }_0^{\zeta }{t}^{\varrho }{\mathcal{L}}_{\varrho }^b\mathfrak{f}\left(t\right)dt,(\mathfrak{f}\in \Sigma ;\zeta \in {\mathfrak{U}}^{*}).$$

Let us specify the function

$$\Psi (p,o;\zeta )={\displaystyle \frac{1}{\zeta }}+\sum _{\kappa =0}^{\infty }{\displaystyle \frac{{\left(p\right)}_{\kappa +1}}{{\left(o\right)}_{\kappa +1}}}{\zeta }^{\kappa },(p\in {\mathbb{C}}^{*}=\mathbb{C}\setminus \left\{0\right\};o\notin {\mathbb{Z}}_0^{-};\zeta \in {\mathfrak{U}}^{*}),$$

(3)

where ${\left(\pi \right)}_{\kappa }$ is defined by

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

We note that

$$\Psi (p,o;\zeta )={\displaystyle \frac{1}{\zeta }}{}_2\mathsf{ϝ}_1(p,1;o;\zeta ),$$

where

$${}_2\mathsf{ϝ}_1(p,\upsilon ;o;\zeta )=\sum _{\kappa =0}^{\infty }{\displaystyle \frac{{\left(p\right)}_{\kappa }{\left(\upsilon \right)}_{\kappa }}{{\left(o\right)}_{\kappa }{\left(1\right)}_{\kappa }}}{\zeta }^{\kappa },(p,\upsilon ,o\in \mathbb{C}ando\notin {\mathbb{Z}}_0^{-};\zeta \in \mathfrak{U}),$$

is the Gaussian hypergeometric function.

If we set

$${\mathcal{L}}_{\varrho }^b\ast {\mathcal{L}}_{\varrho }^b\left(\zeta \right)={\displaystyle \frac{1}{\zeta (1-\zeta )}},$$

we have

$${\mathcal{L}}_{\varrho }^b\left(\zeta \right)={\displaystyle \frac{1}{\zeta }}+\sum _{\kappa =0}^{\infty }{\left({\displaystyle \frac{\kappa +\varrho +1}{\varrho }}\right)}^b{\zeta }^{\kappa }.$$

By using the operator ${\mathcal{L}}_{\varrho }^b\left(\zeta \right)$, we define a new operator ${\mathcal{L}}_{\varrho }^b(p,o;\zeta )$ as follows:

$${\mathcal{L}}_{\varrho }^b\left(\zeta \right)\ast {\mathcal{L}}_{\varrho }^b(p,o;\zeta )=\Psi (p,o;\zeta )(\zeta \in {\mathfrak{U}}^{*}).$$

(4)

The linear operator ${\mathcal{L}}_{\varrho }^b(p,o;\zeta ):\Sigma \to \Sigma$ is defined by

$${\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)={\mathcal{L}}_{\varrho }^b(p,o;\zeta )\ast \mathfrak{f}\left(\zeta \right),(\varrho ,o\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};p\in {\mathbb{C}}^{*};b\in \mathbb{C},\zeta \in {\mathfrak{U}}^{*}),$$

whose series expansion for $\varrho ,o\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-};p\in {\mathbb{C}}^{*};b\in \mathbb{C},\zeta \in {\mathfrak{U}}^{*}$; and $\mathfrak{f}$, as in (1), is given by

$${\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{1}{\zeta }}+\sum _{\kappa =0}^{\infty }{\left({\displaystyle \frac{\varrho }{\kappa +\varrho +1}}\right)}^b{\displaystyle \frac{{\left(p\right)}_{\kappa +1}}{{\left(o\right)}_{\kappa +1}}}{a}_{\kappa }{\zeta }^{\kappa }.$$

(5)

We note that this class of operators, ${\mathcal{L}}_{\varrho }^b(p,o;\zeta )$, was introduced by El-Ashawh [18].

From (5), we can obtain

$$\zeta {\left({\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)}^{\prime }=\varrho {\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)-(\varrho +1){\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right),$$

(6)

and

$$\zeta {\left({\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)}^{\prime }=p{\mathcal{L}}_{\varrho }^b(p+1,o;\zeta )\mathfrak{f}\left(\zeta \right)-(p+1){\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)(p\in \mathbb{C}\setminus \{-1\}).$$

(7)

We observe that ${\mathcal{L}}_{\varrho }^b(\mu ,1;\zeta )\mathfrak{f}\left(\zeta \right)=I_{\varrho ,\mu }^b\mathfrak{f}\left(\zeta \right)(\varrho ,\mu \in {\mathbb{R}}^{+}({\mathbb{R}}^{+}$ is a positive real number) $,b\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\})$ (see Cho et al. [29]).

The purpose of this investigation is to obtain fuzzy differential subordinations associated with the Hurwitz–Lerch Zeta function through the operator ${\mathcal{L}}_{\varrho }^b(p,o;\zeta )$. For each fuzzy differential subordination obtained, we also obtain fuzzy best dominants.

The paper is organized as follows: Two known classes of admissible functions and some results related to these classes are presented in Section 2, Definitions and Preliminaries. Section 3, entitled Main Results, contains the main results of the paper and presents some subordination results, involving the operator ${\mathcal{L}}_{\varrho }^b(p,o;\zeta )$, which is also obtained. The conclusions are outlined in Section 4.

2. Definitions and Preliminaries

The following definitions and lemmas are used as tools to prove the new results included in the following section.

Let ℘ be the set of all functions $\chi$ that are holomorphic and univalent on $\overline{\mathfrak{U}}\setminus E\left(\chi \right)$, where

$$E\left(\chi \right)=\{\varsigma :\varsigma \in \partial \mathfrak{U}\mathrm{and}\underset{\zeta \to \varsigma }{lim}f\left(\zeta \right)=\infty \},$$

and ${\chi }^{\prime }\left(\varsigma \right)\ne 0$ for $\varsigma \in \partial \mathfrak{U}\setminus E\left(\chi \right)$. We also denote by $\wp \left(\mathfrak{a}\right)$ the subclass of ℘ for which $\chi \left(0\right)=\mathfrak{a},$ and let

$$\wp \left(0\right)={\wp }_0\mathrm{and}\wp \left(1\right)={\wp }_1.$$

Definition 1

([1]). A fuzzy set is a pair $\left(S,F\right)$, where S is a set $S\ne \varphi$ and $F:S\to \left[0,1\right]$ is a membership function.

The fuzzy subset is likewise shown as follows:

Definition 2

([1]). A fuzzy subset of S is a pair $\left(\mathcal{L},F_{\mathcal{L}}\right)$, where the support of the fuzzy set $\left(\mathcal{L},F_{\mathcal{L}}\right)$ is defined as $\mathcal{L}=\left\{x\in \mathcal{S}:0<F_{\mathcal{L}}\left(x\right)\le 1\right\}=sup\left(\mathcal{L},F_{\mathcal{L}}\right)$ and $F_{\mathcal{L}}:S\to \left[0,1\right]$ belongs to $\left(\mathcal{L},F_{\mathcal{L}}\right).$

Definition 3

([1]). Fuzzy subsets $\left(Y_{\mathfrak{1}},F_{Y_{\mathfrak{1}}}\right)$ and $\left(Y_{\mathfrak{2}},F_{Y_{\mathfrak{2}}}\right)$ of S are equal if $Y_{\mathfrak{1}}=Y_{\mathfrak{2}}$, whereas $\left(Y_{\mathfrak{1}},F_{Y_{\mathfrak{1}}}\right)\subseteq \left(Y_{\mathfrak{2}},F_{Y_{\mathfrak{2}}}\right)$ if $F_{Y_{\mathfrak{1}}}\left(\eta \right)\le F_{Y_{\mathfrak{2}}}\left(\eta \right)$, $\eta \in S$.

Definition 4

([1]). Let $\mathfrak{U}\subset \mathbb{C}$ and ${\zeta }_0$ be a fixed point in $\mathfrak{U}$ and let the functions $\mathfrak{f}$, $\mathfrak{h}\in \mathcal{H}\left(\mathfrak{U}\right)$. $\mathfrak{f}$ is said to be fuzzy subordinate to $\mathfrak{h}$ and we can write $\mathfrak{f}{\prec }_F\mathfrak{h}$ or $\mathfrak{f}\left(\zeta \right){\prec }_F\mathfrak{h}\left(\zeta \right)$ if

$$\mathfrak{f}\left({\zeta }_0\right)=\mathfrak{h}\left({\zeta }_0\right)\mathit{and}{F}_{\mathfrak{f}\left(\mathfrak{U}\right)}\left(\mathfrak{f}\left(\zeta \right)\right)\le F_{\mathfrak{h}\left(\mathfrak{U}\right)}\mathfrak{h}\left(\zeta \right)),\zeta \in \mathfrak{U},$$

where

$$\mathfrak{f}\left(\mathfrak{U}\right)=sup(\mathfrak{f}\left(\mathfrak{U}\right),F_{\mathfrak{f}\left(\mathfrak{U}\right)})=\{\mathfrak{f}\left(\zeta \right):0<F_{\mathfrak{f}\left(\mathfrak{U}\right)}\left(\mathfrak{f}\left(\zeta \right)\right)\le 1,\zeta \in \mathfrak{U}$$

and

$$\mathfrak{h}\left(\mathfrak{U}\right)=sup(\mathfrak{h}\left(\mathfrak{U}\right),F_{\mathfrak{h}\left(\mathfrak{U}\right)})=\{\mathfrak{h}\left(\zeta \right):0<F_{\mathfrak{h}\left(\mathfrak{U}\right)}\left(\mathfrak{h}\left(\zeta \right)\right)\le 1,\zeta \in \mathfrak{U}.$$

Definition 5

([4]). Let $\psi :{\mathbb{C}}^3\times \mathfrak{U}\to \mathbb{C}$ and let $\mathfrak{h}$ be univalent in $\mathfrak{U}$. If ω is analytic in $\mathfrak{U}$ and satisfies the (second-order) fuzzy differential subordination

$$F_{\psi ({\mathbb{C}}^3\times \mathfrak{U})}\left(\psi (\omega \left(\zeta \right),\zeta {\omega }^{\prime }\left(\zeta \right),{\zeta }^2{\omega }^{″}\left(\zeta \right);\zeta )\right)\le F_{\mathfrak{h}\left(\mathfrak{U}\right)}\left(\mathfrak{h}\left(\zeta \right)\right),$$

(8)

i.e.,

$$\psi (\omega \left(\zeta \right),\zeta {\omega }^{\prime }\left(\zeta \right),{\zeta }^2{\omega }^{″}\left(\zeta \right);\zeta ){\le }_F\left(\mathfrak{h}\left(\zeta \right)\right),\zeta \in \mathfrak{U},$$

then ω is called a fuzzy solution of (8) and is called a fuzzy dominant if $\omega \left(\zeta \right){\prec }_F\chi \left(\zeta \right)$ for all ω satisfying (8). A fuzzy dominant $\tilde{\chi }$ that satisfies $\tilde{\chi }\left(\zeta \right){\prec }_F\chi \left(\zeta \right)$ for all fuzzy dominants χ of (8) is said to be the fuzzy best dominant of (8).

Definition 6

([4]). Let Ω be a set in $\mathbb{C}$, $\chi \in \wp$, and $\kappa \in \mathbb{N}$. The class ${\Psi }_{\kappa }[\Omega ,\chi ]$ of admissible functions consists of the functions $\psi :{\mathbb{C}}^3\times \mathfrak{U}\to \mathbb{C}$ that satisfy $F_{\Omega }\left(\psi (\tau ,s,t;\zeta )\right)=0$ whenever

$$\tau =\chi \left(\varsigma \right),s=k\varsigma {\chi }^{\prime }\left(\varsigma \right)\mathit{and}\Re \left({\displaystyle \frac{t}{s}}+1\right)\geqq k\Re \left(1+{\displaystyle \frac{\varsigma {\chi }^{″}\left(\varsigma \right)}{{\chi }^{\prime }\left(\varsigma \right)}}\right),$$

where $\zeta \in \mathfrak{U},$$\varsigma \in \partial \mathfrak{U}\setminus E\left(\chi \right)$, and $\mathrm{k}\geqq \kappa$. For simplicity, we write ${\Psi }_1[\Omega ,\chi ]$ as $\Psi [\Omega ,\chi ]$.

Lemma 1

([4]). Let $\psi \in {\Psi }_{\kappa }[\Omega ,\chi ]$ with $\chi \left(0\right)=a$. If $\omega \in \mathcal{H}[a_0,\kappa ]$ satisfies

$$F_{\psi ({\mathbb{C}}^3\times \mathfrak{U})}\left(\psi (\omega \left(\zeta \right),\zeta {\omega }^{\prime }\left(\zeta \right),{\zeta }^2{\omega }^{″}\left(\zeta \right);\zeta )\right)\le F_{\Omega }\left(\zeta \right),\zeta \in \mathfrak{U},$$

then $F_{\omega \left(\mathfrak{U}\right)}\left(\omega \left(\zeta \right)\right)\le F_{\chi \left(\mathfrak{U}\right)}\left(\chi \left(\zeta \right)\right),i.e.,$$\omega \left(\zeta \right){\prec }_F\chi \left(\zeta \right).$

3. Main Results

Throughout this paper, unless otherwise mentioned, we suppose that $\varrho ,p,o\in \mathbb{R}\setminus {\mathbb{Z}}_0^{-}$, $b\in \mathbb{R},and\zeta \in {\mathfrak{U}}^{*}.$

Definition 7.

Let Ω be a set in $\mathbb{C}$ and $\chi \in {\wp }_0\cap \mathcal{H}$. The class $\Psi [\Omega ,\chi ]$ of admissible functions consists of $\phi :{\mathbb{C}}^3\times \mathfrak{U}\to \mathbb{C}$, which satisfies

$$F_{\Omega }\left(\phi (\alpha ,\beta ,\gamma ;\zeta )\right)=0,$$

whenever

$$\alpha =\chi \left(\varsigma \right),\beta ={\displaystyle \frac{k\varsigma {\chi }^{\prime }\left(\varsigma \right)+\varrho \chi \left(\varsigma \right)}{\varrho }},$$

and

$$\begin{array}{cc}\hfill & \Re \left\{\varrho \left({\displaystyle \frac{\gamma -\alpha }{\beta -\alpha }}-2\right)\right\}\hfill \\ \hfill & \geqq k\Re \left({\displaystyle \frac{\varsigma {\chi }^{″}\left(\varsigma \right)}{{\chi }^{\prime }\left(\varsigma \right)}}+1\right),(k>0),\hfill \end{array}$$

(9)

where $\zeta \in \mathfrak{U},$$\varsigma \in \partial \mathfrak{U}\setminus E\left(\chi \right)$, and $k\geqq 1$.

The following results are provided and validate our first finding.

Theorem 1.

Suppose that $\phi \in \Psi [\Omega ,\chi ]$. If $\mathfrak{f}\in \Sigma$ satisfies

$$F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}\left(\phi (\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right);\zeta )\right)\le F_{\Omega }\left(\zeta \right),$$

(10)

then

$$F_{\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\right)\left(\mathfrak{U}\right)}\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\le F_{\chi \left(\mathfrak{U}\right)}\left(\chi \left(\zeta \right)\right),$$

i.e.,

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right).$$

Proof. 

Assume that

$$\omega \left(\zeta \right)=\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right).$$

(11)

Differentiating (11) and using (6), we obtain

$$\zeta {\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{\zeta {\omega }^{\prime }\left(\zeta \right)+\varrho \omega \left(\zeta \right)}{\varrho }}.$$

(12)

Further computations show that

$$\begin{array}{cc}\hfill & \zeta {\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)=\hfill \\ \hfill & ={\displaystyle \frac{{\zeta }^2{\omega }^{″}\left(\zeta \right)+\left(1+2\varrho \right)\zeta {\omega }^{\prime }\left(\zeta \right)+{\varrho }^2\omega \left(\zeta \right)}{{\varrho }^2}}.\hfill \end{array}$$

(13)

For $\phi :{\mathbb{C}}^3\times \mathfrak{U}\to \mathbb{C}$, the following transformations are now defined:

$$\alpha (\tau ,s,t)=\tau ,\beta (\tau ,s,t)={\displaystyle \frac{s+\varrho \tau }{\varrho }}$$

and

$$\gamma (\tau ,s,t)={\displaystyle \frac{t+\left(1+2\varrho \right)s+{\varrho }^2\tau }{{\varrho }^2}}.$$

(14)

We also set

$$\begin{array}{cc}\hfill \psi (\tau ,s,t;\zeta )& =\phi (\alpha ,\beta ,\gamma ;\zeta )\hfill \\ \hfill & =\phi \left(\tau ,{\displaystyle \frac{s+\varrho \tau }{\varrho }},{\displaystyle \frac{t+\left(1+2\varrho \right)s+{\varrho }^2\tau }{{\varrho }^2}};\zeta \right).\hfill \end{array}$$

(15)

Then, by using Equations (11)–(15), we obtain

$$\begin{array}{cc}\hfill & \psi \left(\omega \left(\zeta \right),\zeta {\omega }^{\prime }\left(\zeta \right),{\zeta }^2{\omega }^{″}\left(\zeta \right);\zeta \right)\hfill \\ \hfill & =\phi \left(\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right);\zeta \right)\right).\hfill \end{array}$$

Thus, clearly, Equation (10) becomes

$$F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}\left(\psi \left(\omega \left(\zeta \right),\zeta {\omega }^{\prime }\left(\zeta \right),{\zeta }^2{\omega }^{″}\left(\zeta \right);\zeta \right)\right)\le F_{\Omega }\left(\zeta \right).$$

A computation using (14) yields

$${\displaystyle \frac{t}{s}}+1=\varrho \left({\displaystyle \frac{\gamma -\alpha }{\beta -\alpha }}-2\right).$$

Thus, the admissibility condition for $\phi \in \Psi [\Omega ,\chi ]$ is equivalent to the admissibility condition for $\psi$, as given in Definition 6. So $\psi \in \Psi [\Omega ,\chi ]$ and, from Lemma 1,

$$F_{\omega \left(\mathfrak{U}\right)}\omega \left(\zeta \right)\le F_{\chi \left(\mathfrak{U}\right)}\left(\chi \left(\zeta \right)\right),$$

or equivalently,

$$F_{\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\right)\left(\mathfrak{U}\right)}\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\le F_{\chi \left(\mathfrak{U}\right)}\left(\chi \left(\zeta \right)\right),$$

i.e.,

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right).$$

Theorem 1 has thus been proven. □

A simply connected domain $\Omega =\mathfrak{h}\left(\mathfrak{U}\right)$ for every conformal mapping $\mathfrak{h}\left(\zeta \right)$ of $\mathfrak{U}$ onto $\Omega$ exists when $\Omega \ne \mathbb{C}$. The class $\Psi \left[\mathfrak{h}\right(\mathfrak{U}),\chi ]$ is represented as $\Psi [\mathfrak{h},\chi ]$ in this instance.

Theorem 2.

Let $\phi \in \Psi [\mathfrak{h},\chi ]$. If $\mathfrak{f}\in \Sigma$,

$$\phi \left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right);\zeta \right)$$

(16)

is analytic in $\mathfrak{U}$, and

$$\begin{array}{ccc}& & F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}\left(\phi \left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right);\zeta \right)\right)\hfill \\ & \le & F_{\mathfrak{h}\left(\mathfrak{U}\right)}\left(\mathfrak{h}\left(\zeta \right)\right),\hfill \end{array}$$

(17)

then

$$F_{\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\right)\left(\mathfrak{U}\right)}\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\le F_{\chi \left(\mathfrak{U}\right)}\left(\chi \left(\zeta \right)\right),$$

i.e.,

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right).$$

Proof. 

Since this proof is similar to Theorem 1, we decided to leave it out. □

Taking $\phi (\alpha ,\beta ,\gamma ;\zeta )=1+{\displaystyle \frac{\beta }{\alpha }}$ in Theorem 2, we obtain the following:

Corollary 1.

Assume that $\phi \in \Psi [\mathfrak{h},\chi ]$. If $\mathfrak{f}\in \Sigma$,

$$2+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{\zeta \left({\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\prime }{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}\right)$$

is analytic in $\mathfrak{U}$, and

$$2+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{\zeta \left({\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\prime }{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}\right){\prec }_F\mathfrak{h}\left(\zeta \right),$$

(18)

then

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right).$$

Example 1.

By taking $b=-1$ and $p=o$ in Corollary 1, we can see that

$$\phi (\alpha ,\beta ,\gamma ;\zeta )=2+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{\zeta {\mathfrak{f}}^{\prime }\left(\zeta \right)}{\mathfrak{f}\left(\zeta \right)}}\right)$$

is analytic in $\mathfrak{U}$ and

$$2+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{\zeta {\mathfrak{f}}^{\prime }\left(\zeta \right)}{\mathfrak{f}\left(\zeta \right)}}\right){\prec }_F\mathfrak{h}\left(\zeta \right),$$

so

$$\zeta \mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right).$$

Our result extends Theorem 1 to the case of unclear behavior of $\chi$ on the boundary of $\mathfrak{U}$.

Corollary 2.

Consider that $\Omega \subset \mathbb{C}$ and $\chi \left(\zeta \right)$ is univalent in $\mathfrak{U}$ with $\chi \left(0\right)=0$. Also, suppose that $\phi \in \Psi [\Omega ,{\chi }_{\rho }]$ for some $\rho \in (0,1),$ where

$${\chi }_{\rho }\left(\zeta \right)=\chi \left(\rho \zeta \right).$$

If $\mathfrak{f}\in \Sigma$ satisfies

$$\begin{array}{ccc}& & F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}\left(\phi \left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right),\zeta {\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right);\zeta \right)\right)\hfill \\ & \le & F_{\Omega }\left(\zeta \right),\hfill \end{array}$$

(19)

then

$$F_{\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\right)\left(\mathfrak{U}\right)}\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\le F_{\chi \left(\mathfrak{U}\right)}\left(\chi \left(\zeta \right)\right),$$

i.e.,

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right).$$

Proof. 

From Theorem 1, we obtain

$$F_{\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\right)\left(\mathfrak{U}\right)}\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\le F_{{\chi }_{\rho }\left(\mathfrak{U}\right)}\left({\chi }_{\rho }\left(\zeta \right)\right).$$

Since

$${\chi }_{\rho }\left(\zeta \right)\prec \chi \left(\rho \zeta \right),$$

we have

$$F_{{\chi }_{\rho }\left(\mathfrak{U}\right)}\left({\chi }_{\rho }\left(\zeta \right)\right)=F_{{\chi }_{\left(\rho \mathfrak{U}\right)}}\left(\chi \left(\rho \zeta \right)\right)and{\chi }_{\rho }\left(0\right)=\chi \left(0\right).$$

Hence,

$$F_{\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\right)\left(\mathfrak{U}\right)}\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\le F_{{\chi }_{\left(\rho \mathfrak{U}\right)}}\left(\chi \left(\rho \zeta \right)\right),$$

i.e.,

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\rho \zeta \right).$$

By letting $\rho \to 1,$ we obtain

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right).$$

□

In the next theorem, we prove that the fuzzy properties are satisfied if $\phi \in \Psi [\mathfrak{h},{\chi }_{\rho }]$ and $\phi \in \Psi [{\mathfrak{h}}_{\rho },{\chi }_{\rho }]$ for all $\rho \in (0,1).$

Theorem 3.

Assume that $\mathfrak{h}$ and χ are univalent in $\mathfrak{U}$ with $\chi \left(0\right)=0$ and set

$${\chi }_{\rho }\left(\zeta \right)=\chi \left(\rho \zeta \right)\mathit{and}{\mathfrak{h}}_{\rho }\left(\zeta \right)=\mathfrak{h}\left(\rho \zeta \right).$$

Let $\phi :{\mathbb{C}}^3\times \mathfrak{U}\to \mathbb{C}$ satisfy the following conditions:

(1) 

$\phi \in \Psi [\mathfrak{h},{\chi }_{\rho }]$ for some $\rho \in (0,1);$

(2) 

There exists ${\rho }_0\in (0,1)$ such that $\phi \in \Psi [{\mathfrak{h}}_{\rho },{\chi }_{\rho }]$ for all $\rho \in ({\rho }_0,1)$.

If $\mathfrak{f}\in \Sigma$ satisfies condition (17) then

$$F_{\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\right)\left(\mathfrak{U}\right)}\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\le F_{{\chi }_{\left(\mathfrak{U}\right)}}\left(\chi \left(\zeta \right)\right),$$

i.e.,

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right).$$

and $\chi \left(\zeta \right)$ is the fuzzy best dominant.

Proof. 

Case (1): We exclude the proof because it is similar to Theorem 2.

Case (2): Suppose

$$\omega \left(\zeta \right)=\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)and{\omega }_{\rho }\left(\zeta \right)=\omega \left(\rho \zeta \right).$$

Then,

$$\begin{array}{ccc}& & F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}\left(\phi ({\omega }_{\rho }\left(\zeta \right),\zeta {\omega }_{\rho }^{\prime }\left(\zeta \right),{\zeta }^2{\omega }_{\rho }^{″}\left(\zeta \right);\rho \zeta )\right)\hfill \\ & =& F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}\left(\phi (\omega \left(\rho \zeta \right),\zeta {\omega }^{\prime }\left(\rho \zeta \right),{\zeta }^2{\omega }^{″}\left(\rho \zeta \right);\rho \zeta )\right)\hfill \\ & \le & F_{{\mathfrak{h}}_{\rho }\left(\mathfrak{U}\right)}({\mathfrak{h}}_{\rho }(\zeta \left)\right).\hfill \end{array}$$

Making use of Theorem 1 and the statement related to

$$F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}\left(\phi (\omega \left(\zeta \right),\zeta {\omega }^{\prime }\left(\zeta \right),{\zeta }^2{\omega }^{″}\left(\zeta \right);\omega \left(\zeta \right))\right)\le F_{\Omega }\left(\zeta \right),$$

where $\omega$ is any function mapping $\mathfrak{U}$ to $\mathfrak{U}$, with $\omega \left(\zeta \right)=\rho \zeta$, we obtain

$${\omega }_{\rho }\left(\zeta \right){\prec }_F{\chi }_{\rho }\left(\zeta \right),$$

for $\rho \in (0,1).$ By letting $\rho \to 1,$ we obtain $\omega \left(\zeta \right){\prec }_F\chi \left(\zeta \right).$ Then,

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right).$$

□

The fuzzy differential subordination’s best dominant (17) is acquired by applying the next theorem.

Theorem 4.

Assume that $\mathfrak{h}$ is univalent in $\mathfrak{U}$ and let $\phi :{\mathbb{C}}^3\times \mathfrak{U}\to \mathbb{C}$. Assume that the differential equation

$$\begin{array}{ccc}& & \phi \left(\omega \left(\zeta \right),{\displaystyle \frac{\zeta {\omega }^{\prime }\left(\zeta \right)+\varrho \omega \left(\zeta \right)}{\varrho }},\right.\hfill \\ & & \left.{\displaystyle \frac{{\zeta }^2{\omega }^{″}\left(\zeta \right)+\left(1+2\varrho \right)\zeta {\omega }^{\prime }\left(\zeta \right)+{\varrho }^2\omega \left(\zeta \right)}{{\varrho }^2}};\zeta \right)\hfill \\ & =& \mathfrak{h}\left(\zeta \right)\hfill \end{array}$$

(20)

has a solution $\chi \left(\zeta \right),$ with $\chi \left(0\right)=0,$ satisfying the following conditions:

(1) 

$\chi \left(\zeta \right)\in {\wp }_0$ and $\phi \in \Psi [\mathfrak{h},\chi ];$

(2) 

$\chi \left(\zeta \right)$ is univalent in $\mathfrak{U}$ and $\phi \in \Psi [\mathfrak{h},{\chi }_{\rho }]$ for some $\rho \in (0,1);$

(3) 

$\chi \left(\zeta \right)$ is univalent in $\mathfrak{U}$ and there exists ${\rho }_0\in (0,1)$ such that $\phi \in \Psi [{\mathfrak{h}}_{\rho },{\chi }_{\rho }]$ for all $\rho \in ({\rho }_0,1)$.

If $\mathfrak{f}\in \Sigma$ satisfies (17) then

$$F_{\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\right)\left(\mathfrak{U}\right)}\left(\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)\right)\le F_{{\chi }_{\left(\mathfrak{U}\right)}}\left(\chi \left(\zeta \right)\right),$$

i.e.,

$$\zeta {\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right){\prec }_F\chi \left(\zeta \right),$$

and $\chi \left(\zeta \right)$ is the fuzzy best dominant.

Proof. 

Applying Theorems 2 and 3, we conclude that $\chi \left(\zeta \right)$ is a fuzzy dominant of (17). Since $\chi \left(\zeta \right)$ satisfies (20), it is also a solution of (17), and hence $\chi \left(\zeta \right)$ is dominated by all fuzzy dominants of (17) and so it is the fuzzy best dominant of (17). □

Finally, in Definition 8 and Theorem 5, we discuss the fuzzy properties for another admissible function $\phi :{\mathbb{C}}^3\times \mathfrak{U}\to \mathbb{C}$.

Definition 8.

Let Ω be a set in $\mathbb{C}$ and $\chi \left(\zeta \right)\in {\wp }_0\cap \mathcal{H}$. The class $\tilde{\Psi }$$[\Omega ,\chi ]$ of admissible functions consists of the functions $\phi :{\mathbb{C}}^3\times \mathfrak{U}\to \mathbb{C}$ that satisfy

$$F_{\Omega }\left(\phi (\alpha ,\beta ,\gamma ;\zeta )\right)=0,$$

whenever

$$\alpha =\chi \left(\xi \right),\beta =\chi \left(\xi \right)+{\displaystyle \frac{1}{\varrho }}{\displaystyle \frac{k\xi {\chi }^{\prime }\left(\xi \right)}{\chi \left(\xi \right)}},$$

and

$$\begin{array}{cc}\hfill & \Re \left({\displaystyle \frac{\varrho \left(\gamma \beta -\alpha (3\beta -2\alpha )\right)}{(\beta -\alpha )}}\right)\hfill \\ \hfill & \geqq k\Re \left(1+{\displaystyle \frac{\xi {\chi }^{″}\left(\xi \right)}{{\chi }^{\prime }\left(\xi \right)}}\right),\hfill \end{array}$$

where $\zeta \in \mathfrak{U},$ ξ $\in \partial \mathfrak{U}\setminus E\left(\chi \right)$, and $k\geqq 1$.

Theorem 5.

Let $\phi \in \tilde{\Psi }[\Omega ,\chi ]$. If $\mathfrak{f}\in \Sigma$ satisfies

$$\begin{array}{ccc}& & F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}\left\{\phi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}},{\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}},{\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-2}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}};\zeta \right)\right\}\hfill \\ & \le & F_{\Omega }\left(\zeta \right),\hfill \end{array}$$

then

$$F_{\left({\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}\right)\left(\mathfrak{U}\right)}\left({\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}\right)\le F_{{\chi }_{\left(\mathfrak{U}\right)}}\left(\chi \left(\zeta \right)\right),$$

i.e.,

$${\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}{\prec }_F\chi \left(\zeta \right).$$

Proof. 

Let

$$g\left(\zeta \right)={\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}.$$

(21)

Using (7) and (21), we obtain

$${\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}=g\left(\zeta \right)+{\displaystyle \frac{1}{\varrho }}{\displaystyle \frac{\zeta g^{\prime }\left(\zeta \right)}{g\left(\zeta \right)}}.$$

(22)

Further computations show that

$${\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-2}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}=g\left(\zeta \right)+{\displaystyle \frac{1}{\varrho }}\left[{\displaystyle \frac{\zeta g^{\prime }\left(\zeta \right)}{g\left(\zeta \right)}}+{\displaystyle \frac{\varrho \zeta g^{\prime }\left(\zeta \right)+\frac{{\zeta }^2{g}^{″}\left(\zeta \right)}{g\left(\zeta \right)}+\frac{\zeta g^{\prime }\left(\zeta \right)}{g\left(\zeta \right)}-{\left(\frac{\zeta g^{\prime }\left(\zeta \right)}{g\left(\zeta \right)}\right)}^2}{\varrho g\left(\zeta \right)+\frac{\zeta g^{\prime }\left(\zeta \right)}{g\left(\zeta \right)}}}\right].$$

Our next transformations are now defined for $\phi :{\mathbb{C}}^3\times \mathfrak{U}\to \mathbb{C}$

$$\alpha =\tau ,\beta =\tau +{\displaystyle \frac{s}{\varrho \tau }},$$

and

$$\gamma =\tau +{\displaystyle \frac{1}{\varrho }}\left[{\displaystyle \frac{s}{\tau }}+{\displaystyle \frac{\varrho s+\frac{s+t}{\tau }-{\left(\frac{s}{\tau }\right)}^2}{\varrho \tau +\frac{s}{\tau }}}\right].$$

Also, let

$$\begin{array}{cc}\hfill & \psi (\tau ,s,t;\zeta )=\phi (u,v,w;\zeta )\hfill \\ \hfill & =\phi (\tau ,\tau +{\displaystyle \frac{s}{\varrho \tau }},\hfill \\ \hfill & \tau +{\displaystyle \frac{1}{\varrho }}\left[{\displaystyle \frac{s}{\tau }}+{\displaystyle \frac{\varrho s+\frac{s+t}{\tau }-{\left(\frac{s}{\tau }\right)}^2}{\varrho \tau +\frac{s}{\tau }}}\right];\zeta ).\hfill \end{array}$$

(23)

Thus, by using Equations (21)–(23), we obtain

$$\begin{array}{cc}\hfill & \psi \left(g\left(\zeta \right),\zeta g^{\prime }\left(\zeta \right),{\zeta }^2{g}^{″}\left(\zeta \right);\zeta \right)\hfill \\ \hfill & =\phi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}},{\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}},{\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-2}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}};\zeta \right).\hfill \end{array}$$

Hence, Equation (21) implies that

$$F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}(\psi \left(g\left(\zeta \right),\zeta g^{\prime }\left(\zeta \right),{\zeta }^2{g}^{″}\left(\zeta \right);\zeta \right)\le F_{\Omega }\left(\zeta \right).$$

The proof of Theorem 5 will be finished if it can be demonstrated that the admissibility condition for $\phi \in \tilde{\Psi }[\Omega ,\chi ]$ is equivalent to the admissibility condition for $\psi$, as given in Definition 6. For this purpose, we note that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{s}{\tau }}& =& \varrho (\beta -\alpha ),\hfill \\ \hfill {\displaystyle \frac{t}{\tau }}& =& {\varrho }^2\beta (\gamma -\beta )-{\displaystyle \frac{s}{\tau }}\left[\varrho \tau -{\displaystyle \frac{s}{\tau }}+1\right],\hfill \end{array}$$

and

$${\displaystyle \frac{t}{s}}+1={\displaystyle \frac{\varrho (\gamma \beta -\alpha (3\beta -2\alpha \left)\right)}{(\beta -\alpha )}}.$$

Hence, $\psi \in \Psi [\Omega ,\chi ]$. Thus, Lemma 1 leads us to the conclusion that

$$F_{g\left(\mathfrak{U}\right)}\left(g\left(\zeta \right)\right)\le F_{\chi \left(\mathfrak{U}\right)}\left(\chi \left(\zeta \right)\right)$$

or equivalently,

$$F_{\left({\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}\right)\left(\mathfrak{U}\right)}\left({\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}\right)\le F_{{\chi }_{\left(\mathfrak{U}\right)}}\left(\chi \left(\zeta \right)\right),$$

i.e.,

$${\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}{\prec }_F\chi \left(\zeta \right).$$

Theorem 5 has been proven. □

By taking $\phi (\alpha ,\beta ,\gamma ;\zeta )=1+\alpha$ in Theorem 5, we obtain the following:

Example 2.

If we set $b=-1$ and $p=o$ and let $\mathfrak{f}\in \Sigma$ and $\phi \in \tilde{\Psi }[\Omega ,\chi ],$ we find that

$$\phi (\alpha ,\beta ,\gamma ;\zeta )=2+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{\zeta {\mathfrak{f}}^{\prime }\left(\zeta \right)}{\mathfrak{f}\left(\zeta \right)}}+1\right)$$

is analytic in $\mathfrak{U}$ and

$$2+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{\zeta {\mathfrak{f}}^{\prime }\left(\zeta \right)}{\mathfrak{f}\left(\zeta \right)}}+1\right){\prec }_F\mathfrak{h}\left(\zeta \right).$$

Thus,

$$1+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{\zeta {\mathfrak{f}}^{\prime }\left(\zeta \right)}{\mathfrak{f}\left(\zeta \right)}}+1\right){\prec }_F\chi \left(\zeta \right).$$

Let $\Omega =\mathfrak{h}\left(\mathfrak{U}\right)$ for some conformal mapping $\mathfrak{h}\left(\zeta \right)$ of $\mathfrak{U}$ onto $\Omega$ and $\Omega \ne \mathbb{C}$ be a simply connected domain. $\tilde{\Psi }[\mathfrak{h}\left(\mathfrak{U}\right),\chi ]$ is shown as $\tilde{\Psi }[\mathfrak{h},\chi ]$ in this case.

The following outcome is immediately obtained by applying Theorem 5.

Theorem 6.

Suppose that $\phi \in \tilde{\Psi }[\mathfrak{h},\chi ]$. If $\mathfrak{f}\in \Sigma$ and

$$\begin{array}{ccc}& & F_{\phi ({\mathbb{C}}^3\times \mathfrak{U})}\left\{\phi \left({\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}},{\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}},{\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-2}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}};\zeta \right)\right\}\hfill \\ & \le & F_{\mathfrak{h}\left(\mathfrak{U}\right)}\left(\mathfrak{h}\left(\zeta \right)\right),\hfill \end{array}$$

then

$$F_{\left({\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}\right)\left(\mathfrak{U}\right)}\left({\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}\right)\le F_{{\chi }_{\left(\mathfrak{U}\right)}}\left(\chi \left(\zeta \right)\right),$$

i.e.,

$${\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}{\prec }_F\chi \left(\zeta \right).$$

By taking $\phi (\alpha ,\beta ,\gamma ;\zeta )=\alpha \beta$ in Theorem 6, we obtain

Corollary 3.

Suppose that $\phi \in \tilde{\Psi }[\mathfrak{h},\chi ]$. If $\mathfrak{f}\in \Sigma ,$${\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}$ is analytic in $\mathfrak{U}$, and

$${\displaystyle \frac{{\mathcal{L}}_{\varrho }^{b-1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}{\prec }_F\mathfrak{h}\left(\zeta \right),$$

(24)

then

$${\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}{\prec }_F\chi \left(\zeta \right).$$

4. Conclusions

By applying fuzzy set theory to differential subordination, academics have been able to create increasingly complex mathematical models and investigate novel research avenues. Researchers have been able to overcome the drawbacks of conventional methods by incorporating fuzzy sets into the current differential subordination framework. Ongoing research is anticipated to yield further insights into the intersection of fuzzy set theory and differential subordination. Relation (5)’s operator ${\mathcal{L}}_{\varrho }^b(p,o;\zeta )$ has been successfully used to create novel fuzzy differential subordination, whereby the ideal fuzzy dominants were also found.

El-Ashwah [18] used the Hurwitz–Lerch Zeta function $\varphi (\zeta ,b,\varrho )$ to study the operator for the meromorphic function. We are now looking at applications of fuzzy differential subordination in geometric function theory (GFT). The fuzzy best dominants for fuzzy differential subordinations can be found in Theorems 1–3. A corollary and an illustration of how to apply the new result, as shown in Theorems 5 and 6, were used to establish fuzzy differential subordinations for ${\displaystyle \frac{{\mathcal{L}}_{\varrho }^b(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}{{\mathcal{L}}_{\varrho }^{b+1}(p,o;\zeta )\mathfrak{f}\left(\zeta \right)}}.$

Since the relatively recent findings in [30] suggest that the second-order fuzzy differential subordinations found here could be extended to third-order fuzzy differential subordinations, the new discovery in this study will encourage more GFT research. The dual notion of fuzzy differential superordination can also be used following the pattern developed here for the new integral operator. Following the current study, new operators may be created by combining Hurwitz–Lerch Zeta functions and meromorphic functions. These operators may then be utilized to provide new fuzzy differential subordination results.