Fuzzy Differential Subordination for Meromorphic Function

Abstract

This paper is related to notions adapted from fuzzy set theory to the field of complex analysis, namely fuzzy differential subordinations. This work aims to present new fuzzy differential subordinations for which the fuzzy best dominant and fuzzy best subordinate are given, respectively. The original theorems proved in the paper generate interesting corollaries for particular choices of functions acting as fuzzy best dominant. Here, in this article, fuzzy differential subordination results are obtained using a new integral operator introduced in this paper for meromorphic function, such that the newly-defined integral operator is starlike and convex, respectively.

1. Introduction and Definitions

The introduction of the fuzzy set concept by Zadeh, in the paper “Fuzzy Sets” [1] in 1965, did not suggest the extraordinary evolution of the concept that followed. Received with distrust at first, the concept is very popular nowadays, having been adapted to several research topics. Mathematicians were also interested in embedding the concept of fuzzy set in their research and it was indeed included in many mathematical approaches. The review paper included in the present special issue, devoted to the celebration of the 100th anniversary of Zadeh’s birth [2], shows how fuzzy set theory has evolved related to certain branches of science, and points out the contribution of one of Zadeh’s disciples, Professor Dzitac, to the development of soft computing methods connected with fuzzy set theory. Professor Dzitac has celebrated his friendship with the multidisciplinary scientist, Zadeh, by writing the introductory paper of a special issue on fuzzy logic dedicated to the centenary of Zadeh’s birth [3]. As far as complex analysis is concerned, fuzzy set theory has been included in studies related to geometric function theory in 2011, when the first paper appeared introducing the notion of subordination in fuzzy set theory [4] which has had its inspiration in the classical aspects of subordination introduced by Miller and Mocanu [5,6]. The next papers published followed the line of research set by Miller and Mocanu and referred to fuzzy differential subordination, adapting notions from the already well-established theory of differential subordination [7,8,9]. The idea was soon picked up by researchers in geometric function theory and all the classical lines of research on this topic was adapted to the new fuzzy aspects. A review paper, published in 2017 [10], included in its references the first published papers related to this topic, validating its development. The dual notion of fuzzy differential subordination was also introduced in 2017 [11]. An important topic in geometric function theory is conducting studies which involve operators. Such studies for obtaining new fuzzy subordination results were published shortly after the concept was introduced, in 2013 [12], and continued in the following years [13,14,15,16] and later adding super-ordination results [17,18,19]. During the last years, many papers were published which show that the research on this topic is in continuous development process and we mention only a few here [15,20,21,22,23].

Fuzzy differential subordination involving fractional calculus has had tremendous development in recent years proving to have applications in many research domains such as physics, engineering, turbulence, electric networks, biological systems with memory, computer graphics, etc. As an For example, the Korteweg–de Vries equation, developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves, is studied using a new integral transform where the fractional derivative is proposed in the Caputo sense [24]. Related to biological systems, examples can be given as the new study on the mathematical modeling of the human liver with the Caputo–Fabrizio fractional derivative proposed in [25] and fractional calculus analysis of the transmission dynamics of the dengue infection seen in [26]. For more applications see [15,19,27,28].

Following this line of research, a new integral operator is introduced in this paper using a meromorphic function and having, as inspiration, the operator studied by Jung, Kim and Srivastava [29] in 1993, by taking specific values for parameters involved in its definition. Fuzzy differential subordinations are obtained, and the fuzzy best dominants are given, which facilitate obtaining sufficient conditions for the univalence of this operator.

2. Preliminaries

Let $\mathcal{H}\left(\mathcal{U}\right)$ be the class of functions analytic in $\mathcal{U}=\left\{\dot{\zeta }:\dot{\zeta }\in \mathbb{C}\mathrm{and}\left|\dot{\zeta }\right|<1\right\}$ and $\mathcal{H}\left[a,n\right]$ be the subclass of $\mathcal{H}\left(\mathcal{U}\right)$ consisting of functions of the form

$$\mathcal{H}\left[a,n\right]=\left\{\vartheta :\vartheta \in \mathcal{H}\left(\mathcal{U}\right)\mathrm{and}\vartheta \left(\dot{\zeta }\right)=a+a_n{\dot{\zeta }}^n+a_{n+1}{\dot{\zeta }}^{n+1}+\cdots ,\left(\dot{\zeta }\in \mathcal{U}\right)\right\}.$$

(1)

with $\mathcal{H}=\mathcal{H}\left[1,1\right].$

Let $\mathbb{C}$ and $\mathbb{N}$ denote the set of complex numbers and the set of all positive integers, respectively. For $\mathsf{\Omega }\subset \mathbb{C}$, we denote by $\mathcal{H}\left(\mathsf{\Omega }\right)$ the class of meromorphic function in $\mathsf{\Omega }$. For $d\in \mathbb{N},$ we denote by ${\mathcal{M}}_d$ the class of meromorphic function defined by

$${\mathcal{M}}_d:=\left\{\vartheta :\vartheta \in \mathcal{H}\left(\mathcal{U}\right)\mathrm{and}\vartheta \left(\dot{\zeta }\right)=\frac{1}{\dot{\zeta }}+\sum _{n=d+1}^{\infty }{a}_n{\dot{\zeta }}^n\left(\dot{\zeta }\in {\mathcal{U}}^{*}=\mathcal{U}\setminus \left\{0\right\};d\in \mathbb{N}\right).\right\}$$

(2)

where ${\mathcal{U}}^{*}$ is a punctured disc defined by

$${\mathcal{U}}^{*}:=\left\{\dot{\zeta }:\dot{\zeta }\in \mathbb{C}\mathrm{and}0<\left|\dot{\zeta }\right|<1\right\}.$$

In particular, we write $\mathcal{M}:={\mathcal{M}}_1.$

And we denote by ${\mathcal{MS}}^{*}$ and $\mathcal{MC}$ the classes of $\mathcal{M}$, which are respectively, meromorphic starlike and meromorphic convex in ${\mathcal{U}}^{*}$, so that, by definition, we have

$${\mathcal{MS}}^{*}:=\left\{\vartheta :\vartheta \in \mathcal{M}\mathrm{and}-\mathrm{Re}\left(\frac{\dot{\zeta }{\vartheta }^{\prime }\left(\dot{\zeta }\right)}{\vartheta \left(\dot{\zeta }\right)}\right)>0\left(\dot{\zeta }\in {\mathcal{U}}^{*}\right)\right\},$$

and

$$\mathcal{MC}:=\left\{\vartheta :\vartheta \in \mathcal{M}\mathrm{and}-\mathrm{Re}\left(1+\frac{\dot{\zeta }{\vartheta }^{\prime \prime }\left(\dot{\zeta }\right)}{{\vartheta }^{\prime }\left(\dot{\zeta }\right)}\right)>0\left(\dot{\zeta }\in {\mathcal{U}}^{*}\right)\right\}.$$

Definition 1

([30,31]). Let ${\vartheta }_1$ and ${\vartheta }_2$ be members of $\mathcal{H}\left(\mathcal{U}\right)$. The function ${\vartheta }_1$ is said to be subordinate to ${\vartheta }_2$, or ${\vartheta }_2$ is said to be superordinate to ${\vartheta }_1,$ if there exist a function w analytic in $\mathcal{U}$ with $w\left(0\right)=0$ and $|w(\dot{\zeta })|<1 (\dot{\zeta }\in \mathcal{U})$, such that ${\vartheta }_1(\dot{\zeta })={\vartheta }_2(w(\dot{\zeta })).$ In such case we write ${\vartheta }_1(\dot{\zeta })\prec {\vartheta }_2(\dot{\zeta }).$ If ${\vartheta }_2$ is univalent, then ${\vartheta }_1\prec {\vartheta }_2,$ if and only if ${\vartheta }_1(0)={\vartheta }_2(0)$ and ${\vartheta }_1\left(\mathcal{U}\right)\subset {\vartheta }_2\left(\mathcal{U}\right).$

To introduce the notion of fuzzy differential subordination, we use the following definitions and propositions:

Definition 2

([32]). Assume that $\mathcal{Y}$ be a non-empty set. An application $\mathcal{F}$$:\mathcal{Y}$$\to [0,1]$ is called fuzzy subset. A pair $\left(\mathcal{B},{\mathcal{F}}_{\mathcal{B}}\right)$, where ${\mathcal{F}}_{\mathcal{B}}$:$\mathcal{Y}$$\to [0,1]$ and

$$\mathcal{B}=\left\{x\in \mathcal{Y}:0<{\mathcal{F}}_{\mathcal{B}\left(x\right)}\le 1\right\}=sup\left(\mathcal{B},{\mathcal{F}}_{\mathcal{B}}\right)$$

(3)

is called a fuzzy set. The function ${\mathcal{F}}_{\mathcal{B}}$ is called membership function of the fuzzy set $\left(\mathcal{B},{\mathcal{F}}_{\mathcal{B}}\right)$.

Definition 3

([13]). Let $\mathcal{F}:\mathbb{C}\to {\mathbb{R}}_{+}$ be a function such that

$${\mathcal{F}}_{\mathbb{C}}(\dot{\zeta })=|\mathcal{F}(\dot{\zeta })|(\dot{\zeta }\in \mathbb{C}).$$

Denote by

$${\mathcal{F}}_{\mathbb{C}}\left(\mathbb{C}\right)=\{\dot{\zeta }:\dot{\zeta }\in \mathbb{C}\mathrm{and}0<|\mathcal{F}(\dot{\zeta })|\le 1\}:=Supp(\mathbb{C},{\mathcal{F}}_{\mathbb{C}}(\dot{\zeta })),$$

the fuzzy subset of the set $\mathbb{C}$ of complex numbers. We call the following subset:

$${\mathcal{F}}_{\mathbb{C}}(\mathbb{C})=\{\dot{\zeta }:\dot{\zeta }\in \mathbb{C}\mathrm{and}0<|\mathcal{F}(\dot{\zeta })|\le 1\}:={\mathcal{U}}_{\mathcal{F}}(0,1),$$

the fuzzy unit disk. It is observed that $(\mathbb{C},{\mathcal{F}}_{\mathbb{C}}(\dot{\zeta }))$ is the same as its fuzzy unit disk ${\mathcal{U}}_{\mathcal{F}}\left(0,1\right).$

Proposition 1

([4]). (i)    If $\left(\mathcal{B},{\mathcal{F}}_{\mathcal{B}}\right)=\left(\mathcal{U},{\mathcal{F}}_{\mathcal{U}}\right)$, then we have $\mathcal{B}=\mathcal{U}$, where $\mathcal{B}=sup\left(\mathcal{B},{\mathcal{F}}_{\mathcal{B}}\right)$ and $\mathcal{U}=sup\left(\mathcal{U},{\mathcal{F}}_{\mathcal{U}}\right).$

(ii) 

If $\left(\mathcal{B},{\mathcal{F}}_{\mathcal{B}}\right)\subseteq \left(\mathcal{U},{\mathcal{F}}_{\mathcal{U}}\right)$, then we have $\mathcal{B}\subseteq \mathcal{U}$, where $\mathcal{B}=sup\left(\mathcal{B},{\mathcal{F}}_{\mathcal{B}}\right)$ and $\mathcal{U}=sup\left(\mathcal{U},{\mathcal{F}}_{\mathcal{U}}\right).$

Let $\vartheta ,\mathfrak{g}\in \mathcal{H}\left(\mathcal{U}\right)$. We denote

$$\vartheta \left(\mathcal{U}\right)=\left\{\vartheta \left(\dot{\zeta }\right):0<\left|{\mathcal{F}}_{\vartheta \left(\mathcal{U}\right)}\vartheta \left(\dot{\zeta }\right)\right|\le 1,\dot{\zeta }\in \mathcal{U}\right\}=sup\left(\vartheta \left(\mathcal{U}\right),{\mathcal{F}}_{\vartheta \left(\mathcal{U}\right)}\right)$$

(4)

and

$$\mathfrak{g}\left(\mathcal{U}\right)=\left\{\mathfrak{g}\left(\dot{\zeta }\right):0<\left|{\mathcal{F}}_{\mathfrak{g}\left(\mathcal{U}\right)}\mathfrak{g}\left(\dot{\zeta }\right)\right|\le 1,\dot{\zeta }\in \mathcal{U}\right\}=sup\left(\mathfrak{g}\left(\mathcal{U}\right),{\mathcal{F}}_{\mathfrak{g}\left(\mathcal{U}\right)}\right).$$

(5)

Definition 4

([4]). Let ${\dot{\zeta }}_0\in \mathcal{U}$ and $\vartheta ,\mathfrak{g}\in \mathcal{H}\left(\mathcal{U}\right)$. The function ϑ is said to be fuzzy subordinate to $\mathfrak{g},$ written as $\vartheta {\prec }_{\mathcal{F}}\mathfrak{g}$ or $\vartheta (\dot{\zeta }){\prec }_{\mathcal{F}}\mathfrak{g}(\dot{\zeta }),$ when the following conditions are satisfied:

$$\begin{array}{ccc}\hfill \left(i\right)\vartheta \left({\dot{\zeta }}_0\right)& =& \mathfrak{g}\left({\dot{\zeta }}_0\right)\hfill \\ \hfill \left(ii\right)\left|{\mathcal{F}}_{\vartheta \left(\mathcal{U}\right)}\vartheta \left(\dot{\zeta }\right)\right|& \le & \left|{\mathcal{F}}_{\mathfrak{g}\left(\mathcal{U}\right)}\mathfrak{g}\left(\dot{\zeta }\right)\right|,\dot{\zeta }\in \mathcal{U}.\hfill \end{array}$$

Proposition 2

([4]). Assume that ${\dot{\zeta }}_0\in \mathcal{U}$ and $\vartheta ,\mathfrak{g}\in \mathcal{H}\left(\mathcal{U}\right).$ If $\vartheta (\dot{\zeta }){\prec }_{\mathcal{F}}\mathfrak{g}(\dot{\zeta }),$$\dot{\zeta }\in \mathcal{U}$, then

$$\begin{array}{ccc}\hfill \left(i\right)\vartheta \left({\dot{\zeta }}_0\right)& =& \mathfrak{g}\left({\dot{\zeta }}_0\right)\hfill \\ \hfill \left(ii\right)\vartheta \left(\mathcal{U}\right)& \subseteq & \mathfrak{g}\left(\mathcal{U}\right)\mathrm{and}\left|{\mathcal{F}}_{\vartheta \left(\mathcal{U}\right)}\vartheta \left(\dot{\zeta }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{g}\left(\mathcal{U}\right)}\mathfrak{g}\left(\dot{\zeta }\right)\right|,\dot{\zeta }\in \mathcal{U},\hfill \end{array}$$

where $\vartheta \left(\mathcal{U}\right)$ and $\mathfrak{g}\left(\mathcal{U}\right)$ are defined by (4) and (5), respectively.

Definition  5

([7]). Assume that $\mathsf{\Phi }:{\mathbb{C}}^3\times \mathcal{U}\to \mathbb{C}$ and $\mathcal{H}$ be an analytic function with $\mathsf{\Phi }\left(a,0,0,0\right)=\mathcal{H}\left(0\right)=a$. If p is analytic in $\mathcal{U}$ with $p\left(0\right)=a$ and satisfies the second-order fuzzy differential subordination

$$\left|{\mathcal{F}}_{\mathsf{\Phi }\left({\mathbb{C}}^3\times \mathcal{U}\right)}\mathsf{\Phi }\left(\left(p\left(\dot{\zeta }\right),\dot{\zeta }{p}^{\prime }\left(\dot{\zeta }\right),{\dot{\zeta }}^2{p}^{″}\left(\dot{\zeta }\right);\dot{\zeta }\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left(\mathcal{U}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|,$$

(6)

i.e.,

$$\mathsf{\Phi }\left(\left(p(\dot{\zeta }),\dot{\zeta }{p}^{\prime }(\dot{\zeta }),{\dot{\zeta }}^2{p}^{″}(\dot{\zeta });\dot{\zeta }\right)\right){\prec }_{\vartheta }\mathcal{H}(\dot{\zeta }).$$

then p is called a fuzzy solution of the fuzzy differential subordination. The univalent function q is called a fuzzy dominant of the fuzzy solutions for the fuzzy differential subordination if

$$\left|{\mathcal{F}}_{p\left(\mathcal{U}\right)}p(\dot{\zeta })\right|\le \left|{\mathcal{F}}_{q\left(\mathcal{U}\right)}q(\dot{\zeta })\right|,i.e.,p(\dot{\zeta }){\prec }_{\vartheta }q(\dot{\zeta }),\dot{\zeta }\in \mathcal{U}.$$

for all p satisfying (6). A fuzzy dominant $\tilde{q}$ that satisfies

$$\left|{\mathcal{F}}_{\tilde{q}\left(\mathcal{U}\right)}\tilde{q}(\dot{\zeta })\right|\le \left|{\mathcal{F}}_{q\left(\mathcal{U}\right)}q(\dot{\zeta })\right|,i.e.,\tilde{q}(\dot{\zeta }){\prec }_{\vartheta }q(\dot{\zeta }),\dot{\zeta }\in \mathcal{U}.$$

for all fuzzy dominant q of (6) is said to be the fuzzy best dominant of (6).

The set of all analytic and injective functions $q(\dot{\zeta })$ denoted by Q on $\tilde{\mathcal{U}}\setminus E\left(q\right)$, where

$$E\left(q\right)=\left\{\zeta \in \partial \mathcal{U}:\underset{\dot{\zeta }\to \zeta }{lim}q(\dot{\zeta })=\infty \right\},$$

and are such that $q^{\prime }\left(\zeta \right)\ne 0$ for $\zeta \in \partial \mathcal{U}\setminus E\left(q\right).$ Further let the subclass of Q for which $q\left(0\right)=a$ be denoted by $Q\left(a\right)$ with $Q\left(1\right)\equiv Q_1.$

The research presented in this paper is done in the general environment known in the theory of differential subordination combined with fuzzy set notions introduced in [4,5,6,7].

In our investigation, we need the following lemmas which are proved by Miller and Mocanu [30,31].

Lemma 1

([30]). Let $\mathsf{\Psi }\in {\mathsf{\Psi }}_n\left[\mathsf{\Omega },q\right]$ with $q\left(0\right)=a.$ If the analytic function $\mathfrak{g}\left(\dot{\zeta }\right)=a+a_n{\dot{\zeta }}^n+a_{n+1}{\dot{\zeta }}^{n+1}+\cdots ,$ satisfies

$$\mathsf{\Psi }\left(\mathfrak{g}\left(\dot{\zeta }\right),\dot{\zeta }{\mathfrak{g}}^{\prime }\left(\dot{\zeta }\right),{\dot{\zeta }}^2{\mathfrak{g}}^{″}\left(\dot{\zeta }\right);\dot{\zeta }\right)\in \mathsf{\Omega }$$

then $\mathfrak{g}\left(\dot{\zeta }\right)\prec q\left(\dot{\zeta }\right).$

Lemma 2

([31]). Let $\mathsf{\Psi }\in {\mathsf{\Psi }}_n^{\prime }\left[\mathsf{\Omega },q\right]$ with $q\left(0\right)=a.$ If $\mathfrak{g}\left(\dot{\zeta }\right)\in Q\left(\dot{\zeta }\right)$ and

$$\mathsf{\Psi }\left(\mathfrak{g}\left(\dot{\zeta }\right),\dot{\zeta }{\mathfrak{g}}^{\prime }\left(\dot{\zeta }\right),{\dot{\zeta }}^2{\mathfrak{g}}^{″}\left(\dot{\zeta }\right);\dot{\zeta }\right)$$

is univalent in $\mathcal{U}$ then

$$\mathsf{\Omega }\subset \left\{\mathsf{\Psi }\left(\mathfrak{g}\left(\dot{\zeta }\right),\dot{\zeta }{\mathfrak{g}}^{\prime }\left(\dot{\zeta }\right),{\dot{\zeta }}^2{\mathfrak{g}}^{″}\left(\dot{\zeta }\right);\dot{\zeta }\right):\dot{\zeta }\in \mathcal{U}\right\},$$

implies $\mathfrak{g}\left(\dot{\zeta }\right)\prec q\left(\dot{\zeta }\right).$

Various families of linear or convolution operators are known to play important roles in the Geometric Function Theory of Complex Analysis and its related fields. One can indeed express derivative and integral operators as convolutions of some families of meromorphic functions. This kind of formalism makes further mathematical investigation much easier and also aids in a better understanding of the geometric properties of the operators involved.

Analogous to the operators defined by Jung, Kim, and Srivastava [29] on the normalized analytic functions, we now define the following integral operators ${\mathfrak{P}}_{\beta }^{\alpha }:\mathcal{M}\to \mathcal{M}$

$${\mathfrak{P}}_{\beta }^{\alpha }=\frac{{\beta }^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)}\frac{1}{{\dot{\zeta }}^{\beta +1}}{\int }_0^{\dot{\zeta }}{\left(ln\frac{\dot{\zeta }}{t}\right)}^{\alpha -1}{t}^{\beta }\vartheta \left(t\right)dt\left(\alpha >0,\beta >0;\dot{\zeta }\in {\mathcal{U}}^{*}\right)$$

(7)

and

$${\mathcal{J}}_{\beta }^{\alpha }=\frac{\beta }{{\dot{\zeta }}^{\beta +1}}{\int }_0^{\dot{\zeta }}{t}^{\beta }\vartheta \left(t\right)dt\left(\beta >0;\dot{\zeta }\in {\mathcal{U}}^{\ast }\right)$$

(8)

where $\mathsf{\Gamma }\left(\alpha \right)$ is the familiar Gamma function. Moreover, by making use of the Hadamard product (or convolution), for

$$\vartheta \left(\dot{\zeta }\right)=\frac{1}{\dot{\zeta }}+\sum _{n=1}^{\infty }{a}_n{\dot{\zeta }}^n\left(\dot{\zeta }\in {\mathcal{U}}^{*}=\mathcal{U}\setminus \left\{0\right\}\right),$$

we define

$${\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right):={\mathfrak{P}}_{\beta }^{\alpha }\left(\dot{\zeta }\right)\ast \vartheta \left(\dot{\zeta }\right)=\frac{1}{\dot{\zeta }}+\sum _{n=1}^{\infty }{\left(\frac{\beta }{n+\beta +1}\right)}^{\alpha }{a}_n{\dot{\zeta }}^n\left(\alpha >0,\beta >0\right).$$

(9)

and

$${\mathcal{J}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right):={\mathcal{J}}_{\beta }^{\alpha }\left(\dot{\zeta }\right)\ast \vartheta \left(\dot{\zeta }\right)=\frac{1}{\dot{\zeta }}+\sum _{n=1}^{\infty }\frac{\beta }{n+\beta +1}{a}_n{\dot{\zeta }}^n\left(\beta >0\right),$$

(10)

By virtue of (9) and (10), we see that

$${\mathcal{J}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)={\mathfrak{P}}_{\beta }^1\vartheta \left(\dot{\zeta }\right),$$

$$\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }=\beta {\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)-\left(\beta +1\right){\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\left(\alpha >0,\beta >0\right)$$

(11)

We need the following Lemma to investigate our main results.

Lemma 3

([30]). Let $\mathfrak{H}\in \mathcal{M}$ and suppose that

$$\mathfrak{K}\left(\dot{\zeta }\right)=\frac{1}{\dot{\zeta }}{\int }_0^{\dot{\zeta }}\mathfrak{H}\left(t\right)dt\left(\dot{\zeta }\in {\mathcal{U}}^{*}\right).$$

If

$$-Re\left(1+\frac{\dot{\zeta }{\mathfrak{H}}^{\prime }\left(\dot{\zeta }\right)}{{\mathfrak{H}}^{\prime }\left(\dot{\zeta }\right)}\right)>-\frac{1}{2}\left(\dot{\zeta }\in {\mathcal{U}}^{*}\right),$$

then $\mathfrak{K}\in \mathbb{C}.$

Lemma 4

([33]). Suppose that the convex function $\mathfrak{H}$ satisfies $\mathfrak{H}\left(0\right)=a,$ let $\mu \in {\mathbb{C}}^{*}=\mathbb{C}\setminus \left\{0\right\}$ such that Re$\left(\mu \right)\ge 0.$ If $\mathcal{P}\in \mathcal{H}\left[a,n\right]$ with $\mathcal{P}\left(0\right)=a$ and $\mathsf{\Psi }:\left({\mathbb{C}}^2\times \mathcal{U}\right)\to \mathbb{C},$$\mathsf{\Psi }\left(\mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)\right)=\mathcal{P}\left(\dot{\zeta }\right)+\frac{1}{\mu }\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)$ is holomorphic in $\mathcal{U}$, then

$$\left|{\mathcal{F}}_{\mathsf{\Psi }\left({\mathbb{C}}^2\times \mathcal{U}\right)}\left(\mathcal{P}\left(\dot{\zeta }\right)+\frac{1}{\mu }\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left(\mathcal{U}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|⟹\mathcal{P}\left(\dot{\zeta }\right)+\frac{1}{\mu }\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right){\prec }_{\vartheta }\mathfrak{H}\left(\dot{\zeta }\right)\left(\dot{\zeta }\in \mathcal{U}\right),$$

implies

$${\mathcal{F}}_{\mathcal{P}\left(\mathcal{U}\right)}\mathcal{P}\left(\dot{\zeta }\right)\le {\mathcal{F}}_{q\left(\mathcal{U}\right)}q\left(\dot{\zeta }\right)\le {\mathcal{F}}_{\mathfrak{H}\left(\mathcal{U}\right)}\mathfrak{H}\left(\dot{\zeta }\right).$$

i.e.,

$$\mathcal{P}\left(\dot{\zeta }\right){\prec }_{\vartheta }q\left(\dot{\zeta }\right),$$

where

$$q\left(\dot{\zeta }\right)=\frac{\mu }{n{\dot{\zeta }}^{\mu /n}}{\int }_0^{\dot{\zeta }}{t}^{\frac{\mu }{n}-1}\mathfrak{H}\left(t\right)dt,$$

is convex and best dominant.

Lemma 5

([33]). Suppose that q is a convex function in $\mathcal{U}$, let $\mathfrak{H}\left(\dot{\zeta }\right)=q\left(\dot{\zeta }\right)+nv\dot{\zeta }{q}^{\prime }\left(\dot{\zeta }\right),$$v>0$ and $n\in \mathbb{N}$. If $\mathcal{P}\in \mathcal{H}\left[q\left(0\right),n\right]$ and $\mathsf{\Psi }:\left({\mathbb{C}}^2\times \mathcal{U}\right)\to \mathbb{C},$$\mathsf{\Psi }\left(\mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)\right)=\mathcal{P}\left(\dot{\zeta }\right)+v\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)$ is holomorphic in $\mathcal{U}$, then

$$\left|{\mathcal{F}}_{\mathcal{P}\left(\mathcal{U}\right)}\left(\left(\mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left(\mathcal{U}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|⟹\mathcal{P}\left(\dot{\zeta }\right)+v\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right){\prec }_{\vartheta }q\left(\dot{\zeta }\right),$$

then

$$\left|{\mathcal{F}}_{\mathcal{P}\left(\mathcal{U}\right)}\mathcal{P}\left(\dot{\zeta }\right)\right|\le \left|{\mathcal{F}}_{q\left(\mathcal{U}\right)}q\left(\dot{\zeta }\right)\right|,\dot{\zeta }\in \mathcal{U}$$

i.e.,

$$\mathcal{P}\left(\dot{\zeta }\right){\prec }_{\vartheta }q\left(\dot{\zeta }\right).$$

and q is the best fuzzy dominant.

Recently, Srivastava [15], Lupaş [27,33], Oros [21,22,34], and Wanas [35,36] have obtained fuzzy differential subordination results for certain classes of holomorphic functions.

3. Main Results

Theorem 1.

Suppose the convex function $\mathfrak{K}$ in ${\mathcal{U}}^{*},$ satisfies $\mathfrak{K}\left(0\right)=1.$

$$\mathfrak{H}=\mathfrak{K}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathfrak{K}}^{\prime }\left(\dot{\zeta }\right)\dot{\zeta }\in {\mathcal{U}}^{*}$$

Let $\vartheta \in \mathcal{M}$, and us satisfy the following fuzzy differential subordination:

$$\left|{\mathcal{F}}_{\mathsf{\Psi }\left({\mathbb{C}}^2\times {\mathcal{U}}^{*}\right)}\left[\beta {\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)-\left(\beta +1\right){\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)+\dot{\zeta }{\left(\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right)}^{\prime }\right]\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|,$$

(12)

implies

$$\left[\beta {\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)-\left(\beta +1\right){\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)+\dot{\zeta }{\left(\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right)}^{\prime }\right]{\prec }_{\vartheta }\mathfrak{H}\left(\dot{\zeta }\right)$$

then

$$\left|{\mathcal{F}}_{\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}\left(\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|⟹\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }{\prec }_{\vartheta }\mathfrak{K}\left(\dot{\zeta }\right).$$

This result is sharp.

Proof. 

Let us assume

$$\mathcal{P}\left(\dot{\zeta }\right)=\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }.$$

(13)

Therefore in the view of (11) and (13), we have

$$\begin{array}{ccc}\hfill \mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)& =& \beta \left(\frac{1}{\dot{\zeta }}+\sum _{n=1}^{\infty }{\left(\frac{\beta }{n+\beta +1}\right)}^{\alpha -1}{a}_n{\dot{\zeta }}^n\right)-\left(\beta +1\right)\left(\frac{1}{\dot{\zeta }}+\sum _{n=1}^{\infty }{\left(\frac{\beta }{n+\beta +1}\right)}^{\alpha }{a}_n{\dot{\zeta }}^n\right)\hfill \\ & & +\left(\frac{1}{\dot{\zeta }}+\sum _{n=1}^{\infty }{n}^2{\left(\frac{\beta }{n+\beta +1}\right)}^{\alpha }{a}_n{\dot{\zeta }}^n\right)\hfill \\ & =& \beta {\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)-\left(\beta +1\right){\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)+\dot{\zeta }{\left(\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right)}^{\prime }.\hfill \end{array}$$

(14)

According to (12) and (14), we deduce that

$$\left|{\mathcal{F}}_{\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}\left(\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|,$$

implies

$$\left|{\mathcal{F}}_{\mathsf{\Psi }\left({\mathbb{C}}^2\times {\mathcal{U}}^{*}\right)}\left(\mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{K}\left({\mathcal{U}}^{*}\right)}\left(\mathfrak{K}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathfrak{K}}^{\prime }\left(\dot{\zeta }\right)\right)\right|.$$

Thus by applying Lemma 4 with $\mu =1,$ we obtain

$$\left|{\mathcal{F}}_{\mathcal{P}\left({\mathcal{U}}^{*}\right)}\mathcal{P}\left(\dot{\zeta }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{K}\left(\mathcal{U}\right)}\mathfrak{K}\left(\dot{\zeta }\right)\right|⟹\left|{\mathcal{F}}_{\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}\left(\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{K}\left({\mathcal{U}}^{*}\right)}\mathfrak{K}\left(\dot{\zeta }\right)\right|$$

i.e.,

$$\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }{\prec }_{\vartheta }\mathfrak{K}\left(\dot{\zeta }\right),$$

It has sharp and equal boundaries for the Möbius function,${\vartheta }_0\left(\dot{\zeta }\right)=\frac{1+\dot{\zeta }}{1-\dot{\zeta }}$ or any other suitable one. The proof is complete. □

Theorem 2.

Suppose the convex function $\mathfrak{K}$ in ${\mathcal{U}}^{*},$ satisfies $\mathfrak{K}\left(0\right)=1.$

$$\mathfrak{H}=\mathfrak{K}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathfrak{K}}^{\prime }\left(\dot{\zeta }\right)\dot{\zeta }\in {\mathcal{U}}^{*}$$

Let $\vartheta \in \mathcal{M}$ and us satisfy the following fuzzy differential subordination:

$$\left|{\mathcal{F}}_{{\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}{\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|⟹{\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }{\prec }_{\vartheta }\mathfrak{H}\left(\dot{\zeta }\right)$$

(15)

then

$$\left|{\mathcal{F}}_{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|⟹{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right){\prec }_{\vartheta }\mathfrak{K}\left(\dot{\zeta }\right).$$

The result is sharp, that is, the assertion holds true for a suitably specified function.

Proof. 

Let us assume

$$\mathcal{P}\left(\dot{\zeta }\right)={\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)$$

(16)

Therefore in the view of (11) and (16), we have

$$\begin{array}{ccc}\hfill \mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)& =& \left(\frac{1}{\dot{\zeta }}+\sum _{n=1}^{\infty }{\left(\frac{\beta }{n+\beta +1}\right)}^{\alpha }{a}_n{\dot{\zeta }}^n\right)+\left(-\frac{1}{\dot{\zeta }}+\sum _{n=1}^{\infty }n{\left(\frac{\beta }{n+\beta +1}\right)}^{\alpha }{a}_n{\dot{\zeta }}^n\right)\hfill \\ & =& \sum _{n=1}^{\infty }\left(n+1\right){\left(\frac{\beta }{n+\beta +1}\right)}^{\alpha }{a}_n{\dot{\zeta }}^n.\hfill \end{array}$$

(17)

We find that

$$\mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)={\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }.$$

We, thus, see the following inequality:

$$\left|{\mathcal{F}}_{{\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}\left({\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|,$$

implies that

$$\left|{\mathcal{F}}_{\mathsf{\Psi }\left({\mathbb{C}}^2\times {\mathcal{U}}^{*}\right)}\left(\mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{K}\left({\mathcal{U}}^{*}\right)}\left(\mathfrak{K}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathfrak{K}}^{\prime }\left(\dot{\zeta }\right)\right)\right|.$$

Thus by applying Lemma 5, we obtain

$$\left|{\mathcal{F}}_{\mathcal{P}\left({\mathcal{U}}^{*}\right)}\mathcal{P}\left(\dot{\zeta }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{K}\left({\mathcal{U}}^{*}\right)}\mathfrak{K}\left(\dot{\zeta }\right)\right|⟹\left|{\mathcal{F}}_{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{K}\left({\mathcal{U}}^{*}\right)}\mathfrak{K}\left(\dot{\zeta }\right)\right|,$$

which implies that

$${\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right){\prec }_{\vartheta }\mathfrak{K}\left(\dot{\zeta }\right).$$

The result is easily seen to be sharp, that is, the result holds true for a suitably specified function. The proof of Theorem 2 is, thus, completed. □

Theorem 3.

Let $\mathfrak{H}\in \mathcal{H}\left({\mathcal{U}}^{*}\right)$ with $\mathfrak{H}\left(0\right)=1,$ such that

$$-Re\left(1+\frac{\dot{\zeta }{\mathfrak{H}}^{\prime }\left(\dot{\zeta }\right)}{{\mathfrak{H}}^{\prime }\left(\dot{\zeta }\right)}\right)>-\frac{1}{2}\left(\dot{\zeta }\in {\mathcal{U}}^{*}\right)$$

If $\vartheta \in \mathcal{M}$ and the following fuzzy differential subordination holds true:

$$\left|{\mathcal{F}}_{{\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}{\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|⟹{\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }{\prec }_{\vartheta }\mathfrak{H}\left(\dot{\zeta }\right)$$

(18)

then

$$\left|{\mathcal{F}}_{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|⟹{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right){\prec }_{\vartheta }\mathfrak{K}\left(\dot{\zeta }\right).$$

where the function $\mathfrak{K}\left(\dot{\zeta }\right),$ given by

$$\mathfrak{K}\left(\dot{\zeta }\right)=\frac{1}{\dot{\zeta }}{\int }_0^{\dot{\zeta }}\mathfrak{H}\left(t\right)dt$$

is convex and is the fuzzy best dominant.

Proof. 

Assume that

$$\mathcal{P}\left(\dot{\zeta }\right)={\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right).$$

(19)

It is clear that $\mathcal{P}\left(\dot{\zeta }\right)\in \mathcal{H}\left[1,1\right]$. Suppose that $\mathfrak{H}\in \mathcal{H}\left({\mathcal{U}}^{*}\right)$ with $\mathfrak{H}\left(0\right)=1,$ such that

$$-\mathrm{Re}\left(1+\frac{\dot{\zeta }{\mathfrak{H}}^{\prime }\left(\dot{\zeta }\right)}{{\mathfrak{H}}^{\prime }\left(\dot{\zeta }\right)}\right)>-\frac{1}{2}\left(\dot{\zeta }\in {\mathcal{U}}^{*}\right)$$

From Lemma 3, we have

$$\mathfrak{K}\left(\dot{\zeta }\right)=\frac{1}{\dot{\zeta }}{\int }_0^{\dot{\zeta }}\mathfrak{H}\left(t\right)dt$$

which is convex and satisfies the fuzzy differential subordination (18) $.$ Since

$$\mathfrak{H}\left(\dot{\zeta }\right)=\mathfrak{K}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathfrak{K}}^{\prime }\left(\dot{\zeta }\right)\left(\dot{\zeta }\in {\mathcal{U}}^{*}\right)$$

It is the fuzzy best dominant.

We next observe that

$$\begin{array}{ccc}\hfill \mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)& =& \sum _{n=1}^{\infty }\left(n+1\right){\left(\frac{\beta }{n+\beta +1}\right)}^{\alpha }{a}_n{\dot{\zeta }}^n\hfill \\ & =& {\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }.\hfill \end{array}$$

(20)

In the view of (20), the fuzzy differential subordination (18) becomes

$$\left|{\mathcal{F}}_{\mathcal{P}\left(\mathcal{U}\right)}\left(\mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left(\mathcal{U}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|.$$

Thus by applying Lemma 5 with $v=1,$ we obtain

$$\left|{\mathcal{F}}_{\mathcal{P}\left({\mathcal{U}}^{*}\right)}\mathcal{P}\left(\dot{\zeta }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{K}\left({\mathcal{U}}^{*}\right)}\mathfrak{K}\left(\dot{\zeta }\right)\right|.$$

This completes the proof. □

Upon setting

$$\mathfrak{H}\left(\dot{\zeta }\right)=\frac{1+\left(2\beta -1\right)\dot{\zeta }}{1+\dot{\zeta }}\left(\dot{\zeta }\in {\mathcal{U}}^{*}\right)$$

in Theorem 3, we can deduce the following corollary.

Corollary 1.

Let

$$\mathfrak{H}\left(\dot{\zeta }\right)=\frac{1+\left(2\beta -1\right)\dot{\zeta }}{1+\dot{\zeta }}\left(\dot{\zeta }\in {\mathcal{U}}^{*}\right)$$

be normalized convex function in ${\mathcal{U}}^{*}$ with $\mathfrak{H}\left(0\right)=1$ and $0\le \beta <1.$ If the function $\vartheta \in \mathcal{M}$ satisfies the following fuzzy differential subordination:

$$\begin{array}{ccc}\hfill \left|{\mathcal{F}}_{{\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}{\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }\right|& \le & \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|\hfill \\ & ⟹& {\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }{\prec }_{\vartheta }\mathfrak{H}\left(\dot{\zeta }\right),\hfill \end{array}$$

(21)

then the function $\mathfrak{K}\left(\dot{\zeta }\right)$ is given by

$$\mathfrak{K}\left(\dot{\zeta }\right)=2\beta -1+\frac{2\left(1-\beta \right)}{\dot{\zeta }}log\left(1+\dot{\zeta }\right),$$

is convex and is the fuzzy best dominant.

Theorem 4.

Suppose $\mathfrak{K}$ is convex function in ${\mathcal{U}}^{*}.$ Such that $\mathfrak{K}\left(0\right)=1,$

$$\mathfrak{H}\left(\dot{\zeta }\right)=\mathfrak{K}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathfrak{K}}^{\prime }\left(\dot{\zeta }\right).$$

Let $\vartheta \in \mathcal{M}$, and ${\left(\frac{\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)}^{\prime }$ is holomorphic in ${\mathcal{U}}^{*}.$ If

$$\left|{\mathcal{F}}_{{\left(\frac{\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)}^{\prime }}{\left(\frac{\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)}^{\prime }\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|⟹{\left(\frac{\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)}^{\prime }{\prec }_{\vartheta }\mathfrak{H}\left(\dot{\zeta }\right),$$

(22)

then

$$\left|{\mathcal{F}}_{\left(\frac{{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)}\left(\frac{{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|,$$

i.e.,

$$\frac{{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}{\prec }_{\vartheta }\mathfrak{K}\left(\dot{\zeta }\right).$$

This result is sharp.

Proof. 

Assume that

$$\mathcal{P}\left(\dot{\zeta }\right)=\frac{{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}.$$

(23)

Therefore we note that $\mathcal{P}\left(\dot{\zeta }\right)\in \mathcal{H}\left[1,1\right].$

Differentiating both sides of (23) with respect to $\dot{\zeta }$, it yields

$${\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)=\frac{{\left({\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}-\mathcal{P}\left(\dot{\zeta }\right)\frac{{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}.$$

Then

$$\begin{array}{ccc}\hfill \mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)& =& \frac{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\left[\dot{\zeta }{\left({\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }+{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right]-\left(\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)\right){\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^{\prime }}{{\left({\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)\right)}^2}\hfill \\ & =& {\left(\frac{\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)}^{\prime }.\hfill \end{array}$$

(24)

Utilizing (24) in (22), we can get

$$\left|{\mathcal{F}}_{{\left(\frac{\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)}^{\prime }}{\left(\frac{\dot{\zeta }{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)}^{\prime }\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|,$$

implies

$$\left|{\mathcal{F}}_{\mathcal{P}\left(\mathcal{U}\right)}\left(\mathcal{P}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathcal{P}}^{\prime }\left(\dot{\zeta }\right)\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{H}\left({\mathcal{U}}^{*}\right)}\mathfrak{H}\left(\dot{\zeta }\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{K}\left({\mathcal{U}}^{*}\right)}\left(\mathfrak{K}\left(\dot{\zeta }\right)+\dot{\zeta }{\mathfrak{K}}^{\prime }\left(\dot{\zeta }\right)\right)\right|.$$

Thus by applying Lemma 5 with $v=1,$ we obtain

$$\left|{\mathcal{F}}_{\left(\frac{{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)}\left(\frac{{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}\right)\right|\le \left|{\mathcal{F}}_{\mathfrak{K}\left({\mathcal{U}}^{*}\right)}\mathfrak{K}\left(\dot{\zeta }\right)\right|,$$

i.e.,

$$\frac{{\mathfrak{P}}_{\beta }^{\alpha -1}\vartheta \left(\dot{\zeta }\right)}{{\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)}{\prec }_{\vartheta }\mathfrak{K}\left(\dot{\zeta }\right).$$

It has sharpness consider Möbius function,${\vartheta }_0\left(\dot{\zeta }\right)=\frac{1+\dot{\zeta }}{1-\dot{\zeta }}$ or any other suitable one. The proof of Theorem 4 is complete. □

4. Conclusions

In our present investigation of applications of fuzzy differential subordination in Geometric Function Theory of Complex Analysis, we successfully made use of integral operator ${\mathfrak{P}}_{\beta }^{\alpha }\vartheta \left(\dot{\zeta }\right)$ for meromorphic function. Another avenue for further research on this subject is provided by the fact that, in the theory of differential subordinations and differential super-ordinations, there are differential subordinations and differential subordinations of the third and higher orders as well (see, for details, [30]; see also [37] or recent developments on this subject). In this presentation, we only used and explored the second-order differential subordinations and differential super-ordinations. This differential subordination is very helpful in computer graphics, etc.