New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution

Abstract

Based upon the Pascal distribution series ${\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right):=\zeta +{\displaystyle \sum _{j=m+1}^{\infty }}\left(\genfrac{}{}{0pt}{}{j+r-2}{r-1}\right)\left[1+\lambda (j-1)\right]q^{j-1}{(1-q)}^r{a}_j{\zeta }^j$, we can obtain a set of fuzzy differential subordinations in this investigation. We also newly obtain class ${\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ of univalent analytic functions defined by the operator ${\mathcal{N}}_{q,\lambda }^{r,m}$, give certain properties for the class ${\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ and also obtain some applications connected with a special case for the operator. New research directions can be taken on fuzzy differential subordinations associated with symmetry operators.

1. Introduction

Let ${\mathcal{H}}_m\left(\varpi \right)$ represent the class of holomorphic and univalent functions on $\varpi$ such that $\varpi \subset \mathbb{C}$ and let $\mathcal{H}\left(\varpi \right)$ denote the class of holomorphic functions on $\varpi$. The class of holomorphic functions in the open unit disk of the complex plane $\Lambda =\{\zeta \in \mathbb{C}:\left|\zeta \right|<1\}$ is denoted in this study by a note $\mathcal{H}(\Lambda )$, with $B_{\Lambda }=\{\zeta \in \mathbb{C}:\left|\zeta \right|=1\}$ standing as the unit disk’s boundary. For $m\in \mathbb{N}=\left\{1,2,\dots \right\}$, we define

$${\mathcal{H}}_m\left[\gamma \right]=\left\{\mathsf{{\rm Y}}\in \mathcal{H}(\Lambda ):\mathsf{{\rm Y}}\left(\zeta \right)=\gamma +\sum _{j=m+1}^{\infty }{a}_j{\zeta }^j,\zeta \in \Lambda \right\},$$

$${\mathcal{A}}_m=\left\{\mathsf{{\rm Y}}\in \mathcal{H}(\Lambda ):\mathsf{{\rm Y}}\left(\zeta \right)=\zeta +\sum _{j=m+1}^{\infty }{a}_j{\zeta }^j,\zeta \in \Lambda \right\}\mathrm{with}{\mathcal{A}}_1=\mathcal{A},$$

(1)

and

$$\mathcal{S}=\left\{\mathsf{{\rm Y}}\in {\mathcal{A}}_m:\mathsf{{\rm Y}}\mathrm{is}\mathrm{a}\mathrm{univalent}\mathrm{function}\mathrm{in}\Lambda \right\}.$$

We denote by

$$\mathcal{C}=\left\{\mathsf{{\rm Y}}\in {\mathcal{A}}_m:\Re \left(1+\frac{\zeta {\mathsf{{\rm Y}}}^{{}^{″}}\left(\zeta \right)}{{\mathsf{{\rm Y}}}^{{}^{\prime }}\left(\zeta \right)}\right)>0,\zeta \in \Lambda \right\},$$

which is the set of convex functions on $\Lambda$.

Let ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ be analytic in $\Lambda .$ Then ${\mathsf{{\rm Y}}}_1$ is subordinate to ${\mathsf{{\rm Y}}}_2$ written as ${\mathsf{{\rm Y}}}_1\prec {\mathsf{{\rm Y}}}_2$ if there exists a Schwarz function $\varphi ,$ which is analytic in $\Lambda$ with $\varphi \left(0\right)=0$ and $\left|\varphi \left(\zeta \right)\right|<1$ for all $\zeta \in \Lambda$ such that ${\mathsf{{\rm Y}}}_1\left(\zeta \right)=$ ${\mathsf{{\rm Y}}}_2\left(\varphi \left(\zeta \right)\right).$ Furthermore, if the function ${\mathsf{{\rm Y}}}_2$ is univalent in $\Lambda ,$ then we have the following equivalence (see [1,2]):

$${\mathsf{{\rm Y}}}_1\left(\zeta \right)\prec {\mathsf{{\rm Y}}}_2\left(\zeta \right)\iff {\mathsf{{\rm Y}}}_1\left(0\right)={\mathsf{{\rm Y}}}_2\left(0\right)\mathrm{and}{\mathsf{{\rm Y}}}_1(\Lambda )\subset {\mathsf{{\rm Y}}}_2(\Lambda ).$$

In order to introduce the notion of fuzzy differential subordination, we use the following definitions and propositions:

Definition 1

([3]). Assume that $\mathcal{T}\ne ⌀$ is a Fuzzy subset and $\mathcal{F}:\mathcal{T}\to \left[0,1\right]$ is an application. A pair of $\left(\Lambda ,{\mathcal{F}}_{\Lambda }\right),$ where ${\mathcal{F}}_{\Lambda }:\mathcal{T}\to \left[0,1\right],$ and

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

a fuzzy subset. The fuzzy set $\left(\Lambda ,{\mathcal{F}}_{\Lambda }\right)$ is called a function ${\mathcal{F}}_{\Lambda }$.

Let $\mathsf{{\rm Y}},g\in \mathcal{H}\left(\varpi \right)$ be denoted by

$$\mathsf{{\rm Y}}\left(\varpi \right)=\left\{\mathsf{{\rm Y}}\left(\zeta \right):0<{\mathcal{F}}_{\mathsf{{\rm Y}}\left(\varpi \right)}\mathsf{{\rm Y}}\left(\zeta \right)\le 1,\zeta \in \varpi \right\}=sup\left(\mathsf{{\rm Y}}\left(\varpi \right),{\mathcal{F}}_{\mathsf{{\rm Y}}\left(\varpi \right)}\right),$$

(2)

and

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

(3)

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 $\mathcal{U}=sup\left(\mathcal{U},{\mathcal{F}}_{\mathcal{U}}\right);\left(ii\right)$ 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 $\mathcal{U}=sup\left(\mathcal{U},{\mathcal{F}}_{\mathcal{U}}\right).$

Definition 2

([4]). Let ${\zeta }_0\in \varpi$ be a fixed point and let the functions $\mathsf{{\rm Y}},g\in \mathcal{H}\left(\varpi \right).$ The function Υ is said to be fuzzy subordinate to g, and we write $\mathsf{{\rm Y}}{\prec }_{\mathcal{F}}g$ or $\mathsf{{\rm Y}}\left(\zeta \right){\prec }_{\mathcal{F}}g\left(\zeta \right),$ which satisfies the following conditions:

(i) 

$\mathsf{{\rm Y}}\left({\zeta }_0\right)=g\left({\zeta }_0\right);$

(ii) 

${\mathcal{F}}_{\mathsf{{\rm Y}}\left(\varpi \right)}\mathsf{{\rm Y}}\left(\zeta \right)\le {\mathcal{F}}_{\mathrm{g}\left(\varpi \right)}g\left(\zeta \right),\zeta \in \varpi .$

Proposition 2

([4]). Assume that ${\zeta }_0\in \omega$ is a fixed point and the functions $\mathsf{{\rm Y}},g\in H\left(\omega \right).$ If $\mathsf{{\rm Y}}\left(\zeta \right){\prec }_{\mathcal{F}}g\left(\zeta \right),$ $\zeta \in \omega ,$ then

(i) 

$\mathsf{{\rm Y}}\left({\zeta }_0\right)=g\left({\zeta }_0\right)$

(ii) 

$\mathsf{{\rm Y}}\left(\omega \right)\subseteq g\left(\omega \right),{\mathcal{F}}_{\mathsf{{\rm Y}}\left(\omega \right)}\mathsf{{\rm Y}}\left(\zeta \right)\le {\mathcal{F}}_{\mathrm{g}\left(\omega \right)}g\left(\zeta \right),\zeta \in \omega ,$

where $\mathsf{{\rm Y}}\left(\omega \right)$ and $\mathrm{g}\left(\omega \right)$ are defined by (2) and (3), respectively.

Definition 3

([5]). Assume that $h\in \mathcal{S}$ and $\Phi :{\mathbb{C}}^3\times \Lambda \to \mathbb{C}$, $\Phi \left(\alpha ,0,0;0\right)=h\left(0\right)=\alpha$. If p satisfies the requirements of the second-order fuzzy differential subordination and is analytic in Λ, with $p\left(0\right)=\alpha$,

$${\mathcal{F}}_{\Phi ({\mathbb{C}}^3\times \Lambda )}\Phi \left(p\left(\zeta \right),\zeta p^{{}^{\prime }}\left(\zeta \right),{\zeta }^2{p}^{{}^{″}}\left(\zeta \right);\zeta \right)\le {\mathcal{F}}_{h\left(\Lambda \right)}h\left(\zeta \right).$$

(4)

If q is a fuzzy dominant of the fuzzy differential subordination solutions, then p is said to be a fuzzy solution of the fuzzy differential subordination and satisfies

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

for each and every p satisfying (4).

Definition 4.

A fuzzy dominant $\tilde{q}$ that satisfies

$${\mathcal{F}}_{\tilde{q}\left(\Lambda \right)}\tilde{q}\left(\zeta \right)\le {\mathcal{F}}_{q\left(\Lambda \right)}q\left(\zeta \right),$$

then

$$\tilde{q}\left(\zeta \right){\prec }_{\mathcal{F}}q\left(\zeta \right),\zeta \in \Lambda .$$

The fuzzy best dominant of (4) is referred to for all fuzzy dominants.

Assume the function $\Omega \in {\mathcal{A}}_m$ is given by

$$\Omega \left(\zeta \right):=\zeta +\sum _{j=m+1}^{\infty }{\psi }_j{\zeta }^j,\zeta \in \Lambda .$$

The Hadamard (or convolution) product of $\mathsf{{\rm Y}}$ and $\Omega$ is defined as

$$(\mathsf{{\rm Y}}\ast \Omega )\left(\zeta \right):=\zeta +\sum _{j=m+1}^{\infty }{a}_j{\psi }_j{\zeta }^j,\zeta \in \Lambda .$$

A variable x is said to have the Pascal distribution if it takes the values $0,1,2,3,\dots$ with the probabilities ${(1-q)}^r$, $\frac{qr{(1-q)}^r}{1!}$, $\frac{q^{2}r(r+1){(1-q)}^r}{2!}$, $\frac{q^{3}r(r+1)(r+2){(1-q)}^r}{3!}$, …, respectively, where q and r are called the parameters, and thus we have the probability formula

$$P(X=k)=\left(\genfrac{}{}{0pt}{}{k+r-1}{r-1}\right)q^k{(1-q)}^r,k\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}.$$

Now we present a power series whose coefficients are Pascal distribution probabilities, i.e.,

$$\begin{array}{c}{\mathrm{Q}}_{q,m}^r\left(\zeta \right):=\zeta +\sum _{j=m+1}^{\infty }\left(\genfrac{}{}{0pt}{}{j+r-2}{r-1}\right)q^{j-1}{(1-q)}^r{\zeta }^j,\zeta \in \Lambda ,\\ \left(m\in \mathbb{N},r\ge 1,0\le q\le 1\right).\end{array}$$

We easily determine from the ratio test that the radius of convergence of the above power series is at least ${\displaystyle \frac{1}{q}}\ge 1$; hence, ${\mathrm{Q}}_{q,m}^r\in {\mathcal{A}}_m$.

We define the functions

$$\begin{array}{c}{\mathcal{M}}_{q,\lambda }^{r,m}\left(\zeta \right):=(1-\lambda ){\mathrm{Q}}_{q,m}^r\left(\zeta \right)+\lambda z{\left({\mathrm{Q}}_{q,m}^r\left(\zeta \right)\right)}^{\prime }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=\zeta +\sum _{j=m+1}^{\infty }\left(\genfrac{}{}{0pt}{}{j+r-2}{r-1}\right)\left[1+\lambda (j-1)\right]q^{j-1}{(1-q)}^r{\zeta }^j,\zeta \in \Lambda ,\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(m\in \mathbb{N},r\ge 1,0\le q\le 1,\lambda \ge 0\right).\hfill \end{array}$$

El-Deeb and Bulboacă [6] introduced the linear operator ${\mathcal{N}}_{q,\lambda }^{r,m}:{\mathcal{A}}_m\to {\mathcal{A}}_m$ defined by

$$\begin{array}{c}{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right):={\mathcal{M}}_{q,\lambda }^{r,m}\left(\zeta \right)\ast \mathsf{{\rm Y}}\left(\zeta \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}=\zeta +\sum _{j=m+1}^{\infty }\left(\genfrac{}{}{0pt}{}{j+r-2}{r-1}\right)\left[1+\lambda (j-1)\right]q^{j-1}{(1-q)}^r{a}_j{\zeta }^j,\zeta \in \Lambda ,\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(m\in \mathbb{N},r\ge 1,0\le q\le 1,\lambda \ge 0\right),\hfill \end{array}$$

where $\mathsf{{\rm Y}}$ is given by (1), and the symbol “*” stands for the Hadamard (or convolution) product.

Remark 1.

(i) For $m=1,$ the operator ${\mathcal{N}}_{q,\lambda }^{r,m}$ reduces to ${\mathcal{I}}_{q,\lambda }^r:={\mathcal{N}}_{q,\lambda }^{r,1}$, introduced and studied by El-Deeb et al. [7]; (ii) for $m=1$ and $\lambda =0,$ the operator ${\mathrm{Q}}_q^r$ reduces to ${\mathrm{Q}}_q^r:={\mathcal{N}}_{q,0}^{r,1}$, introduced and studied by El-Deeb et al. [7].

Using the operator ${\mathcal{N}}_{q,\lambda }^{r,m}$, we create a class of analytical functions and derive several fuzzy differential subordinations for this class.

Definition 5.

If the function $\mathsf{{\rm Y}}\in \mathcal{A}$ belongs to the class ${\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ for all $\eta \in \left[0,1\right)$ and satisfies the inequality

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}>\eta ,\left(\zeta \in \Lambda \right).$$

2. Preliminary

The following lemmas are needed to show our results.

Lemma 1

([2]). Assume that $\mathsf{Ϝ}\in \mathcal{A}$ and $\mathcal{G}\left(\zeta \right)=\frac{1}{\zeta }\underset{0}{\overset{\zeta }{\int }}\mathsf{Ϝ}\left(t\right)dt,\zeta \in \Lambda$. If $\Re \left\{1+\frac{\zeta {\mathsf{Ϝ}}^{″}\left(\zeta \right)}{{\mathsf{Ϝ}}^{\prime }\left(\zeta \right)}\right\}>\frac{-1}{2},\zeta \in \Lambda$, then $\mathcal{G}\in \mathcal{C}$.

Lemma 2

(Theorem 2.6 in [8]). If $\mathsf{Ϝ}$ is a convex function such that $\mathsf{Ϝ}\left(0\right)=\gamma$, $\nu \in {\mathbb{C}}^{\ast }=\mathbb{C}\setminus \left\{0\right\}$ with $\Re \left(\nu \right)\ge 0.$ If $p\in$ ${\mathcal{H}}_m\left[\gamma \right]$ such that $p\left(0\right)=\gamma ,\Phi :{\mathbb{C}}^2\times \Lambda \to \mathbb{C}$, $\Phi \left(p\left(\zeta \right),\zeta p^{{}^{\prime }}\left(\zeta \right);\zeta \right)=p\left(\zeta \right)+\frac{1}{\nu }\zeta p^{{}^{\prime }}\left(\zeta \right)$ is an analytic function in Λ and

$${\mathcal{F}}_{\Phi ({\mathbb{C}}^2\times \Lambda )}\left(p\left(\zeta \right)+\frac{1}{\nu }\zeta p^{{}^{\prime }}\left(\zeta \right)\right)\le {\mathcal{F}}_{h\left(\Lambda \right)}h\left(\zeta \right)\to p\left(\zeta \right)+\frac{1}{\nu }\zeta p^{{}^{\prime }}\left(\zeta \right){\prec }_{\mathcal{F}}h\left(\zeta \right),\zeta \in \Lambda ,$$

then

$${\mathcal{F}}_{p\left(\Lambda \right)}p\left(\zeta \right)\le {\mathcal{F}}_{q\left(\Lambda \right)}q\left(\zeta \right)\le {\mathcal{F}}_{h\left(\Lambda \right)}h\left(\zeta \right)\to p\left(\zeta \right){\prec }_{\mathcal{F}}q\left(\zeta \right),\zeta \in \Lambda ,$$

where

$$q\left(\zeta \right)=\frac{\nu }{m{\zeta }^{\frac{\nu }{m}}}\underset{0}{\overset{\zeta }{\int }}\psi \left(t\right)t^{\frac{\nu }{m}-1}dt,\zeta \in \Lambda .$$

The function q is convex, and it is the fuzzy best dominant.

Lemma 3

(Theorem 2.7 in [8]). Let $g$ be a convex function in Λ and $\mathsf{Ϝ}\left(\zeta \right)=g\left(\zeta \right)+m\gamma \zeta g^{{}^{\prime }}\left(\zeta \right),$ where $\zeta \in \Lambda ,m\in \mathbb{N}$ and $\gamma >0,$ if

$$p\left(\zeta \right)=g\left(0\right)+p_m{\zeta }^m+p_{m+1}{\zeta }^{m+1}+...\in \mathcal{H}(\Lambda ),$$

and

$${\mathcal{F}}_{p(\Lambda )}\left(p\left(\zeta \right)+\gamma \zeta p^{{}^{\prime }}\left(\zeta \right)\right)\le {\mathcal{F}}_{\psi \left(\Lambda \right)}\psi \left(\zeta \right)\to p\left(\zeta \right)+\gamma \zeta p^{{}^{\prime }}\left(\zeta \right){\prec }_{\mathcal{F}}\psi \left(\zeta \right),\zeta \in \Lambda .$$

Then

$${\mathcal{F}}_{p(\Lambda )}\left(p\left(\zeta \right)\right)\le {\mathcal{F}}_{g\left(\Lambda \right)}g\left(\zeta \right)\to p\left(\zeta \right){\prec }_{\mathcal{F}}g\left(\zeta \right),\zeta \in \Lambda .$$

This result is sharp.

We define the fuzzy differential subordination general theory and its applications (see [9,10,11,12,13]). The method of fuzzy differential subordination is applied in the next section to obtain a set of fuzzy differential subordinations related to the operator ${\mathcal{N}}_{q,\lambda }^{r,m}$.

3. Main Results

Assume that $\eta \in \left[0,1\right)$, $m\in \mathbb{N},r\ge 1,0\le q\le 1,\lambda \ge 0$ and $\zeta \in \Lambda$ are mentioned throughout this paper.

Theorem 1.

Let k belong to $\mathcal{C}$ in Λ, and $h\left(\zeta \right)=k\left(\zeta \right)+\frac{1}{\rho +2}\zeta k^{{}^{\prime }}\left(\zeta \right).$ If $\mathsf{{\rm Y}}\in {\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ and

$$\mathcal{G}\left(\zeta \right)={\mathcal{I}}^{\rho }\mathsf{{\rm Y}}\left(\zeta \right)=\frac{\rho +2}{{\zeta }^{\rho +1}}\underset{0}{\overset{\zeta }{\int }}{t}^{\rho }\mathsf{{\rm Y}}\left(t\right)dt,$$

(5)

then

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{h\left(\Lambda \right)}h\left(\zeta \right)\to {\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}{\prec }_{\mathcal{F}}h\left(\zeta \right),$$

(6)

implies

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{k\left(\Lambda \right)}k\left(\zeta \right)\to {\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}{\prec }_{\mathcal{F}}k\left(\zeta \right).$$

Proof. 

Since

$${\zeta }^{\rho +1}\mathcal{G}\left(\zeta \right)=\left(\rho +2\right)\underset{0}{\overset{\zeta }{\int }}{t}^{\rho }\mathsf{{\rm Y}}\left(t\right)dt,$$

by differentiating, we obtain

$$\left(\rho +1\right)\mathcal{G}\left(\zeta \right)+\zeta {\mathcal{G}}^{{}^{\prime }}\left(\zeta \right)=\left(\rho +2\right)\mathsf{{\rm Y}}\left(\zeta \right),$$

and

$$\left(\rho +1\right){\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)+\zeta {\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}=\left(\rho +2\right){\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right),$$

(7)

and also, by differentiating (7), we obtain

$${\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}+\frac{1}{\left(\rho +2\right)}\zeta {\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{″}}={\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}.$$

(8)

The fuzzy differential subordination (6) technique is used

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}\left({\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}+\frac{1}{\left(\rho +2\right)}\zeta {\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{″}}\right)\le$$

$$F_{h\left(\Lambda \right)}\left(k\left(\zeta \right)+\frac{1}{\left(\rho +2\right)}\zeta k^{{}^{\prime }}\left(\zeta \right)\right).$$

(9)

We denote

$$q\left(\zeta \right)={\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }},\mathrm{so}q\in {\mathcal{H}}_1\left[n\right].$$

(10)

Putting (10) in (9), we have

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}\left(q\left(\zeta \right)+\frac{1}{\left(\rho +2\right)}\zeta q^{{}^{\prime }}\left(\zeta \right)\right)\le F_{h\left(\Lambda \right)}\left(k\left(\zeta \right)+\frac{1}{\left(\rho +2\right)}\zeta k^{{}^{\prime }}\left(\zeta \right)\right).$$

(11)

Using Lemma (3), we obtain

$$F_{q\left(\Lambda \right)}q\left(\zeta \right)\le F_{k\left(\Lambda \right)}k\left(\zeta \right),\mathrm{i}.\mathrm{e}.F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{k\left(\Lambda \right)}k\left(\zeta \right),$$

and therefore, ${\left({\mathcal{N}}_{q,\lambda }^{r,m}G\left(\zeta \right)\right)}^{{}^{\prime }}{\prec }_{\mathcal{F}}k\left(\zeta \right),$ where k is the fuzzy best dominant. □

Putting $m=1$ and $\lambda =0$ in Theorem 1, we obtain the following example since the operator ${\mathrm{Q}}_q^r$ reduces to ${\mathrm{Q}}_q^r:={\mathcal{N}}_{q,0}^{r,1}$.

Example 1.

Let k be an element of $\mathcal{C}$ in Λ and $h\left(\zeta \right)=k\left(\zeta \right)+\frac{1}{\rho +2}\zeta k^{{}^{\prime }}\left(\zeta \right).$ If $\mathsf{{\rm Y}}\in {\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ and $\mathcal{G}$ is given by (5), then

$$F_{{\left({\mathrm{Q}}_q^r\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathrm{Q}}_q^r\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{h\left(\Lambda \right)}h\left(\zeta \right)\to {\left({\mathrm{Q}}_q^r\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}{\prec }_{\mathcal{F}}h\left(\zeta \right),$$

implies

$$F_{{\left({\mathrm{Q}}_q^r\mathcal{G}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathrm{Q}}_q^r\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{k\left(\Lambda \right)}k\left(\zeta \right)\to {\left({\mathrm{Q}}_q^r\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}{\prec }_{\mathcal{F}}k\left(\zeta \right).$$

Theorem 2.

Assume that $h\left(\zeta \right)=\frac{1+\left(2\eta -1\right)\zeta }{1+\zeta },\eta \in \left[0,1\right),$ $\lambda >0$ and ${\mathcal{I}}^{\rho }$ is given by (5), then

$${\mathcal{I}}^{\rho }\left[{\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)\right]\subset {\mathit{P}}_{q,\lambda }^{F,r,m}\left({\eta }^{\ast }\right),$$

(12)

where

$${\eta }^{\ast }=2\eta -1+\left(\rho +2\right)\left(2-2\eta \right)\underset{0}{\overset{1}{\int }}\frac{t^{\rho +2}}{t+1}\mathit{dt}.$$

(13)

Proof. 

A function h belongs to $\mathcal{C}$, and we obtain from the hypothesis of Theorem 2 using the same technique as that in the proof of Theorem 1 that

$$F_{q\left(\Lambda \right)}\left(q\left(\zeta \right)+\frac{1}{\left(\rho +2\right)}\zeta q^{{}^{\prime }}\left(\zeta \right)\right)\le F_{h\left(\Lambda \right)}h\left(\zeta \right),$$

where $q\left(\zeta \right)$ is defined in (10). By using Lemma 2, we obtain

$$F_{q\left(\Lambda \right)}q\left(\zeta \right)\le F_{k\left(\Lambda \right)}k\left(\zeta \right)\le F_{h\left(\Lambda \right)}h\left(\zeta \right),$$

which implies

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{k\left(\Lambda \right)}k\left(\zeta \right)\le F_{h\left(\Lambda \right)}h\left(\zeta \right),$$

where

$$\begin{array}{ccc}\hfill k\left(\zeta \right)& =& \frac{\rho +2}{{\zeta }^{\rho +2}}\underset{0}{\overset{\zeta }{\int }}{t}^{\rho +1}\frac{1+\left(2\eta -1\right)t}{1+t}dt\hfill \\ & =& \left(2\eta -1\right)+\frac{\left(\rho +2\right)\left(2-2\eta \right)}{{\zeta }^{\rho +2}}\underset{0}{\overset{\zeta }{\int }}\frac{t^{\rho +1}}{1+t}dt\in \mathcal{C},\hfill \end{array}$$

where $k\left(\Lambda \right)$ is symmetric with respect to the real axis, so we have

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathcal{G}\left(\zeta \right)\right)}^{{}^{\prime }}\ge \underset{\left|\zeta \right|=1}{min}{F}_{k\left(\Lambda \right)}k\left(\zeta \right)=F_{k\left(\Lambda \right)}k\left(1\right),$$

(14)

and ${\eta }^{\ast }=k\left(1\right)=2\eta -1+\left(\rho +2\right)\left(2-2\eta \right)\underset{0}{\overset{1}{\int }}{\displaystyle \frac{t^{\rho +2}}{t+1}}dt.$ □

Theorem 3.

Assume that k belongs to $\mathcal{C}$ in Λ, that $k\left(0\right)=1,$ and that $h\left(\zeta \right)=k\left(\zeta \right)+\zeta k^{{}^{\prime }}\left(\zeta \right).$ When $\mathsf{{\rm Y}}\in \mathcal{A}$ and the fuzzy differential subordination is satisfied,

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{h\left(\Lambda \right)}h\left(\zeta \right)\to {\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}{\prec }_{\mathcal{F}}h\left(\zeta \right),$$

(15)

holds, then

$$F_{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\Lambda \right)}\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }\le F_{k\left(\Lambda \right)}k\left(\zeta \right)\to \frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }{\prec }_{\mathcal{F}}k\left(\zeta \right).$$

(16)

Proof. 

Let

$$\begin{array}{ccc}\hfill q\left(\zeta \right)& =& \frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }=\frac{\zeta +{\displaystyle \sum _{j=m+1}^{\infty }}\left(\genfrac{}{}{0pt}{}{j+r-2}{r-1}\right)\left[1+\lambda (j-1)\right]q^{j-1}{(1-q)}^r{a}_j{\zeta }^j}{\zeta }\hfill \\ & =& 1+\sum _{j=m+1}^{\infty }\left(\genfrac{}{}{0pt}{}{j+r-2}{r-1}\right)\left[1+\lambda (j-1)\right]q^{j-1}{(1-q)}^r{a}_j{\zeta }^{j-1},\hfill \end{array}$$

and we obtain that

$$q\left(\zeta \right)+\zeta q^{{}^{\prime }}\left(\zeta \right)={\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }},$$

so

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{h\left(\Lambda \right)}h\left(\zeta \right)$$

implies

$$F_{q\left(\Lambda \right)}\left(q\left(\zeta \right)+\zeta q^{{}^{\prime }}\left(\zeta \right)\right)\le F_{h\left(\Lambda \right)}h\left(\zeta \right)=F_{k\left(\Lambda \right)}\left(k\left(\zeta \right)+\zeta k^{{}^{\prime }}\left(\zeta \right)\right).$$

Using the Lemma 3, we obtain

$$F_{q\left(\Lambda \right)}q\left(\zeta \right)\le F_{k\left(\Lambda \right)}k\left(\zeta \right)\to F_{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\Lambda \right)}\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }\le F_{k\left(\Lambda \right)}k\left(\zeta \right),$$

and we obtain

$$\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }{\prec }_{\mathcal{F}}k\left(\zeta \right).$$

□

Theorem 4.

Consider $h\in \mathcal{H}(\Lambda )$, which satisfies $\Re \left(1+\frac{\zeta h^{{}^{″}}\left(\zeta \right)}{h^{{}^{\prime }}\left(\zeta \right)}\right)$ $>\frac{-1}{2}$ when $h\left(0\right)=1$. If the fuzzy differential subordination

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{h\left(\Lambda \right)}h\left(\zeta \right)\to {\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}{\prec }_{\mathcal{F}}h\left(\zeta \right),$$

(17)

then

$$F_{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\Lambda \right)}\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }\le F_{k\left(\Lambda \right)}k\left(\zeta \right)i.e.\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }{\prec }_{\mathcal{F}}k\left(\zeta \right),$$

(18)

where

$$k\left(\zeta \right)=\frac{1}{\zeta }\underset{0}{\overset{\zeta }{\int }}\mathit{h}\left(\mathit{t}\right)\mathit{dt},$$

the function k is convex, and it is the fuzzy best dominant.

Proof. 

Let

$$q\left(\zeta \right)=\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }=1+\sum _{j=d+1}^{\infty }\frac{{\left[j\right]}_q!}{{[\lambda +1]}_{q,j-1}}{a}_j{\psi }_j{\zeta }^{j-1},q\in {\mathcal{H}}_1\left[1\right],$$

where $\Re \left(1+\frac{\zeta h^{{}^{″}}\left(\zeta \right)}{h^{{}^{\prime }}\left(\zeta \right)}\right)>\frac{-1}{2}.$ From Lemma 1, we have

$$k\left(\zeta \right)=\frac{1}{\zeta }\underset{0}{\overset{\zeta }{\int }}h\left(t\right)dt\in$$

belongs to the class $\mathcal{C},$ which satisfies the fuzzy differential subordination (17). Since

$$k\left(\zeta \right)+\zeta k^{{}^{\prime }}\left(\zeta \right)=h\left(\zeta \right),$$

it is the fuzzy best dominant. We have

$$q\left(\zeta \right)+\zeta q^{{}^{\prime }}\left(\zeta \right)={\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }},$$

then (17) becomes

$$F_{q\left(\Lambda \right)}\left(q\left(\zeta \right)+\zeta q^{{}^{\prime }}\left(\zeta \right)\right)\le F_{h\left(\Lambda \right)}h\left(\zeta \right).$$

By using Lemma 3, we obtain

$$F_{q\left(\Lambda \right)}q\left(\zeta \right)\le F_{k\left(\Lambda \right)}k\left(\zeta \right),\mathrm{i}.\mathrm{e}.F_{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\Lambda \right)}\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }\le F_{k\left(\Lambda \right)}k\left(\zeta \right),$$

then

$$\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }{\prec }_{\mathcal{F}}k\left(\zeta \right).$$

□

Putting $h\left(\zeta \right)=\frac{1+\left(2\upsilon -1\right)\zeta }{1+\zeta }$ in Theorem 4. As a result, we have the following corollary:

Corollary 1.

Let $h=\frac{1+\left(2\upsilon -1\right)\zeta }{1+\zeta }$ be a convex function in Λ, with $h\left(0\right)=1,$ $0\le \beta <1.$ If $\mathsf{{\rm Y}}\in \mathcal{A}$ and verifies the fuzzy differential subordination

$$F_{{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{h\left(\Lambda \right)}\left(\frac{1+\left(2\upsilon -1\right)\zeta }{1+\zeta }\right),i.e.{\left({\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}{\prec }_{\mathcal{F}}\left(\frac{1+\left(2\upsilon -1\right)\zeta }{1+\zeta }\right),$$

then

$$F_{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\Lambda \right)}\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }\le F_{k\left(\Lambda \right)}k\left(\zeta \right),$$

(19)

then

$$\frac{{\mathcal{N}}_{q,\lambda }^{r,m}\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }{\prec }_{\mathcal{F}}k\left(\zeta \right),$$

where

$$k\left(\zeta \right)=2\upsilon -1+\frac{2\left(1-\upsilon \right)}{\zeta }ln\left(1+\zeta \right),$$

the function k is convex and it is the fuzzy best dominant.

Putting $m=1$ and $\lambda =0$ in Corollary 1, we obtain the following example.

Example 2.

Let $h=\frac{1+\left(2\upsilon -1\right)\zeta }{1+\zeta }$ be a convex function in Λ, with $h\left(0\right)=1,$ $0\le \beta <1.$ If $f\in \mathcal{A}$ and verifies the fuzzy differential subordination

$$F_{{\left({\mathrm{Q}}_q^r\mathsf{{\rm Y}}\right)}^{{}^{\prime }}\left(\Lambda \right)}{\left({\mathrm{Q}}_q^r\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}\le F_{h\left(\Lambda \right)}\left(\frac{1+\left(2\upsilon -1\right)\zeta }{1+\zeta }\right),i.e.{\left({\mathrm{Q}}_q^r\mathsf{{\rm Y}}\left(\zeta \right)\right)}^{{}^{\prime }}{\prec }_{\mathcal{F}}\left(\frac{1+\left(2\upsilon -1\right)\zeta }{1+\zeta }\right)$$

then

$$F_{{\mathrm{Q}}_q^r\mathsf{{\rm Y}}\left(\Lambda \right)}\frac{{\mathrm{Q}}_q^r\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }\le F_{k\left(\Lambda \right)}k\left(\zeta \right)i.e.\frac{{\mathrm{Q}}_q^r\mathsf{{\rm Y}}\left(\zeta \right)}{\zeta }{\prec }_{\mathcal{F}}k\left(\zeta \right),$$

(20)

where

$$k\left(\zeta \right)=2\upsilon -1+\frac{2\left(1-\upsilon \right)}{\zeta }ln\left(1+\zeta \right).$$

4. Conclusions

All of the above results provide information about fuzzy differential subordinations for the operator ${\mathcal{N}}_{q,\lambda }^{r,m}$; we also provide certain properties for the class ${\mathit{P}}_{q,\lambda }^{F,r,m}\left(\eta \right)$ of univalent analytic functions. Using these classes and operators, we can create some simple applications.