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\begin{document}
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\title{Fuzzy Differential Subordinations Connected with Convolution}
\author{S. M. El-Deeb}
\address{Department of Mathematics, Faculty of Science, \\
Damietta University, New Damietta 34517, Egypt \newline
Department of Mathematics, College of Science and Arts in Badaya,\\
 Qassim University, Qassim, Saudi Arabia}
\email{shezaeldeeb@yahoo.com}
%
\author{Alina Alb Lupas}
\address{Department of Mathematics and Computer Science,\\
University of Oradea\\
str. Universitatii nr. 1, 410087 Oradea, Romania}
\email{dalb@uoradea.ro}
%
\subjclass{30C45, 30A20.}
\keywords{Fuzzy differential subordination, fuzzy best
dominant, binomial series, linear differential operator, convolution.}
\begin{abstract}
The object of the present paper is to obtain several fuzzy differential
subordinations\textbf{\ }associated with Linear operator $\mathcal{D}%
_{n,\delta ,g}^{m}f(z)=z+\sum\limits_{j=2}^{\infty }\left[ 1+\left(
j-1\right) c^{n}(\delta )\right] ^{m}a_{j}b_{j}z^{j}.\ $ Using the operator $%
\mathcal{D}_{n,\delta ,g}^{m},\ $we also introduce a class $\mathcal{H}%
_{n,m,\delta }^{F}\left( \eta ,g\right) $ of univalent analytic functions
for which we give some properties.
\end{abstract}
\maketitle

\section{Introduction}

Let $\Omega \mathcal{\subset
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
},$ $H(\Omega )$ the class of holomorphic functions on $\Omega $ and denote
by $H_{d}(\Omega )$ the class of holomorphic and univalent functions on $%
\Omega $. In this paper, we denote by $H(\Delta )$ the class of holomorphic
functions in the open unit disk $\Delta =\{z\in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
:\left\vert z\right\vert <1\}$ with $B_{\Delta }=\{z\in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
:\left\vert z\right\vert =1\}$ the boundary of the unit disk. For $\beta \in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$ and $d\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$, we denote
\begin{equation*}
H\left[ \beta ,d\right] =\left\{ f\in H(\Delta ):\ f(z)=\beta
+\sum\limits_{j=d+1}^{\infty }a_{j}z^{j},\ \ z\in \Delta \right\} ,
\end{equation*}%
\begin{equation*}
\mathbb{A}_{d}=\left\{ f\in H(\Delta ):\ f(z)=z+\sum\limits_{j=d+1}^{\infty
}a_{j}z^{j},\ \ z\in \Delta \right\} \ \ \ \ \text{with\ \ \ }\mathbb{A}_{1}=%
\mathbb{A},
\end{equation*}%
and,
\begin{equation*}
\mathcal{S=}\left\{ f\in \mathbb{A}:\ f\text{\ is a univalent function in }%
\Delta \right\} .
\end{equation*}%
We denote by
\begin{equation*}
\mathcal{C}=\left\{ f\in \mathbb{A}:\ \Re \left( 1+\frac{zf^{^{\prime \prime
}}(z)}{f^{^{\prime }}(z)}\right) >0,\text{ }z\in \Delta \right\} ,
\end{equation*}%
is convex functions in$\ \Delta $.

\begin{definition}
\label{d1}\cite{1,5}\textbf{\ }Let $f_{1}$ and $f_{2}$ are analytic function
in $\Delta $, then $f_{1}$ is subordinate to $f_{2}$, written $f_{1}\prec
f_{2}$ if there exists a Schwarz function $\emph{w}$, which is analytic in $%
\Delta $ with $\emph{w}(0)=0$ and $\left\vert \emph{w}(z)\right\vert <1\;$%
for all $z\in \Delta ,$ such that $f_{1}(z)=f_{2}(\emph{w}(z)).$
Furthermore, if the function $f_{2}$ is univalent in $\Delta ,$ then we have
the following equivalence:{\small
\begin{equation*}
f_{1}(z)\prec f_{2}(z)\Leftrightarrow f_{1}(0)=f_{2}(0)\text{ and }%
f_{1}(\Delta )\subset f_{2}(\Delta ).
\end{equation*}%
}
\end{definition}

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

\begin{definition}
\label{d2}\cite{2}\textbf{\ }Assume that the set $\mathcal{Y\ }\neq
\varnothing $. Application $\mathcal{F}:\mathcal{Y\rightarrow }\left[ 0,1%
\right] $ is fuzzy subset. A pair $\left( \mathcal{B},\mathcal{F}_{\mathcal{B%
}}\right) ,$ where $\mathcal{F}_{\mathcal{B}}:\mathcal{Y\rightarrow }\left[
0,1\right] $ and
\begin{equation}
A=\left\{ x\in \mathcal{Y}:0<\mathcal{F}_{\mathcal{B}}(x)\leq 1\right\}
=\sup \left( \mathcal{B},\mathcal{F}_{\mathcal{B}}\right) ,  \tag{1.1}
\label{1.1}
\end{equation}%
is said fuzzy subset. A function $\mathcal{F}_{\mathcal{B}}$\ is said to be
the fuzzy set $\left( \mathcal{B},\mathcal{F}_{\mathcal{B}}\right) .$
\end{definition}

\begin{proposition}
\label{p1}\cite{6}\textbf{\ }(i)\textbf{\ }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)\textbf{\ }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 $\mathcal{U}=\sup \left(
\mathcal{U},\mathcal{F}_{\mathcal{U}}\right) .$
\end{proposition}

Let $\mathcal{\ }f,g\in H(\Omega )$, we denote by
\begin{equation}
f\left( \Omega \right) =\left\{ f(z):\ 0<\mathcal{F}_{f\left( \Omega \right)
}f(z)\leq 1,\ z\in \Omega \right\} =\sup \left( f\left( \Omega \right) ,%
\mathcal{F}_{f\left( \Omega \right) }\right) ,  \tag{1.2}  \label{1.2}
\end{equation}%
and,
\begin{equation}
\emph{g}\left( \Omega \right) =\left\{ g(z):\ 0<\mathcal{F}_{g\left( \Omega
\right) }g(z)\leq 1,\ z\in \Omega \right\} =\sup \left( g\left( \Omega
\right) ,\mathcal{F}_{g\left( \Omega \right) }\right) .  \tag{1.3}
\label{1.3}
\end{equation}

\begin{definition}
\label{d3}\cite{6}\textbf{\ }Let $z_{0}\in \Omega $ be a fixed point and let
the functions $f,g\in H(\Omega ).\ $The function $f$ is said to be fuzzy
subordinate to $g$ and write $f\prec _{\mathcal{F}}g$ or $f(z)\prec _{%
\mathcal{F}}g(z),$ which is satisfied the following conditions:

$(i)\ f(z_{0})=g(z_{0})$

$(ii)\ \mathcal{F}_{f\left( \Omega \right) }f(z)\leq \mathcal{F}_{\emph{g}%
\left( \Omega \right) }g(z),\ \ \ z\in \Omega .$
\end{definition}

\begin{proposition}
\label{p2}\cite{6}\textbf{\ }Assume that $z_{0}\in \Omega $ is a fixed point
and the functions $f,\emph{g}\in H(\Omega ).\ $If $f(z)\prec _{\mathcal{F}%
}g(z),$ $z\in \Omega ,$ then

$(i)\ f(z_{0})=g(z_{0})$

$(ii)\ f\left( \Omega \right) \subseteq g\left( \Omega \right) ,\ \ \mathcal{%
F}_{f\left( \Omega \right) }f(z)\leq \mathcal{F}_{\emph{g}\left( \Omega
\right) }g(z),\ z\in \Omega ,$

\noindent where $f\left( \Omega \right) \ $and $\emph{g}\left( \Omega
\right) $ are defined by (\ref{1.2})$\ $and (\ref{1.3}), respectively.
\end{proposition}

\begin{definition}
\label{d4}\cite{7}\textbf{\ }Assume that $\Phi :\mathbb{C}^{3}\times \Delta
\rightarrow \mathbb{C}$ and $h\in \mathcal{S}$, with $\Phi \left( \alpha
,0,0;0\right) =h(0)=\alpha .$ If $p$ is analytic in $\Delta $, with $%
p(0)=\alpha $ and satisfies the second order fuzzy differential
subordination
\begin{equation*}
\mathcal{F}_{\Phi (\mathbb{C}^{3}\times \Delta \mathbb{)}}\Phi \left(
p(z),zp^{^{\prime }}(z),z^{2}p^{^{\prime \prime }}(z);z\right) \leq \mathcal{%
F}_{h\left( \Delta \right) }h(z),
\end{equation*}%
\begin{equation}
\text{i.e.\ \ \ \ }\Phi \left( p(z),zp^{^{\prime }}(z),z^{2}p^{^{\prime
\prime }}(z);z\right) \prec _{\mathcal{F}}h(z),\ \ z\in \Delta \text{.}
\tag{1.4}  \label{1.4.}
\end{equation}%
Then $p$ is said to be 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
\begin{equation*}
\mathcal{F}_{p\left( \Delta \right) }p(z)\leq \mathcal{F}_{q\left( \Delta
\right) }q(z),\ \ \ \text{i.e. \ \ }p(z)\prec _{\mathcal{F}}q(z),\ \ z\in
\Delta
\end{equation*}%
for all $p$ satisfying (\ref{1.4.}).

A fuzzy dominant $\widetilde{q}\ $that satisfies
\begin{equation*}
\mathcal{F}_{\widetilde{q}\left( \Delta \right) }\widetilde{q}(z)\leq
\mathcal{F}_{q\left( \Delta \right) }q(z),\ \ \ \text{i.e. \ \ }\widetilde{q}%
(z)\prec _{\mathcal{F}}q(z),\ \ z\in \Delta
\end{equation*}%
$\mathbb{\ }$for all fuzzy dominants $q$ of (\ref{1.4.}) is called the fuzzy
best dominant of (\ref{1.4.}).\newline
\end{definition}

Making use the binomial series%
\begin{equation*}
\left( 1-\delta \right) ^{n}=\sum\limits_{i=0}^{n}\left(
\begin{array}{c}
n \\
i%
\end{array}%
\right) \left( -1\right) ^{i}\ \delta ^{i}\ \ \ \ \ \left( n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
=\left\{ 1,2,...\right\} \right) ,
\end{equation*}%
for $f\in \mathbb{A},\ $we introduced the linear differential operator as
follows:%
\begin{equation*}
\mathcal{D}_{n,\delta ,g}^{0}f(z)=\left( f\ast g\right) (z),
\end{equation*}%
\begin{eqnarray}
\mathcal{D}_{n,\delta ,g}^{1}f(z) &=&\mathcal{D}_{n,\delta ,g}f(z)=\left(
1-\delta \right) ^{n}\left( f\ast g\right) (z)+\left[ 1-\left( 1-\delta
\right) ^{n}\right] z\left( f\ast g\right) ^{^{\prime }}(z)  \notag \\
&=&z+\sum\limits_{j=2}^{\infty }\left[ 1+\left( j-1\right) c^{n}(\delta )%
\right] a_{j}b_{j}z^{j}  \notag \\
&&.  \notag \\
&&.  \notag \\
&&.  \notag \\
\mathcal{D}_{n,\delta ,g}^{m}f(z) &=&\mathcal{D}_{n,\delta ,g}\left(
\mathcal{D}_{n,\delta ,g}^{m-1}f(z)\right)  \notag \\
&=&\left( 1-\delta \right) ^{n}\mathcal{D}_{n,\delta ,g}^{m-1}f(z)+\left[
1-\left( 1-\delta \right) ^{n}\right] z\left( \mathcal{D}_{n,\delta
,g}^{m-1}f(z)\right) ^{^{\prime }}  \notag 
\end{eqnarray}
\begin{equation}
=z+\sum\limits_{j=2}^{\infty }\left[ 1+\left( j-1\right) c^{n}(\delta )%
\right] ^{m}a_{j}b_{j}z^{j}\ \   \tag{1.5}  \label{1.5} 
\end{equation}
\begin{equation*}
\left( \delta >0,\ n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,\ m\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
_{0}=%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\cup \{0\}\right) ,  
\end{equation*}%
where%
\begin{equation*}
c^{n}(\delta )=\sum\limits_{i=1}^{n}\left(
\begin{array}{c}
n \\
i%
\end{array}%
\right) \left( -1\right) ^{i+1}\ \delta ^{i}\ \ \ \ \ \left( n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) .
\end{equation*}%
From (\ref{1.5}), we obtain that%
\begin{equation*}
c^{n}(\delta )\ z\ \left( \mathcal{D}_{n,\delta ,g}^{m}f(z)\right)
^{^{\prime }}=\mathcal{D}_{n,\delta ,g}^{m+1}f(z)-\left[ 1-c^{n}(\delta )%
\right] \mathcal{D}_{n,\delta ,g}^{m}f(z).
\end{equation*}

By specializing the parameters $n,\ \delta $ and $b_{j},$ we note that%
\newline
(i) Putting $b_{j}=1\ $(or $g(z)=\frac{z}{1-z}$)$,\ $then $\mathcal{D}%
_{n,\delta ,\frac{z}{1-z}}^{m}=\mathcal{D}_{n,\delta }^{m}$ defined by
Yousef et al.\cite{yousef};\newline
(ii) Putting $b_{j}=1\ $(or $g(z)=\frac{z}{1-z}$) and $n=1,\ $then $\mathcal{%
D}_{1,\delta ,\frac{z}{1-z}}^{m}=\mathcal{D}_{\delta }^{m}$ defined by
Al-Oboudi \cite{Alob};\newline
(iii) Putting $b_{j}=1\ $(or $g(z)=\frac{z}{1-z}$) and $n=\delta =1,\ $then $%
\mathcal{D}_{1,1,\frac{z}{1-z}}^{m}=\mathcal{D}^{m}$ defined by Salagean.%
\cite{sal};\newline
(iv)\ Putting $b_{j}=\left( \frac{\ell +1}{\ell +j}\right) ^{\alpha }$ $%
(\alpha >0,~\ell >-1)$ and $n=1,$ then $\mathcal{D}_{1,\delta ,g}^{m}=%
\mathcal{I}_{\ell ,\delta }^{m,\alpha }f(z)$ defind by El-Deeb and Lupas
\cite{deeb};\newline
(v)\ Putting $b_{j}=\left( \frac{\alpha +1}{\alpha +j}\right) ^{n}\frac{%
m^{j-1}}{\left( j-1\right) !}e^{-m}$ $(m,\alpha \geq 0,~n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
_{0})$ and $m=0,$ then $\mathcal{D}_{n,\delta ,g}^{0}=\mathcal{H}_{\alpha
,m}^{n}f(z)$ defind by El-Deeb and Oros \cite{dbor};\newline
(vi) Putting $b_{j}=\frac{(-1)^{k-1}\Gamma (\upsilon +1)}{%
4^{k-1}(k-1)!\Gamma (k+\upsilon )}\cdot \frac{\lbrack k,q]!}{[\lambda
+1,q]_{k-1}}$, $(\upsilon >0,\;\lambda >-1,\;0<q<1)$ studied by El-Deeb and
Bulboac\u{a} \cite{db190} and El-Deeb \cite{eldeeb2}, we obtain the operator
$\mathcal{N}_{\upsilon ,n,\delta }^{m,\lambda ,q}$, defined as follows:%
\begin{eqnarray*}
\mathcal{N}_{\upsilon ,n,\delta }^{m,\lambda ,q}f(z)
&=&z+\sum\limits_{j=2}^{\infty }\left[ 1+\left( j-1\right) c^{n}(\delta )%
\right] ^{m}\dfrac{(-1)^{j-1}\Gamma (\upsilon +1)}{4^{j-1}(j-1)!\Gamma
(j+\upsilon )}a_{j}z^{j}\  \\
&&\left( \lambda >-1;\;0<q<1;\ \delta ,\upsilon >0;\ n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
;\ m\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
_{0}\right) ;
\end{eqnarray*}%
(vi) Putting $b_{j}=\left( \frac{\ell +1}{\ell +j}\right) ^{\alpha }\cdot
\frac{\lbrack k,q]!}{[\lambda +1,q]_{k-1}}$, $(\alpha >0,~n\geq 0,\;\lambda
>-1,\ 0<q<1)$ studied by El-Deeb and Bulboac\u{a} \cite{db19} and Srivastava
and El-Deeb \cite{sriv5}, we obtain the operator $\mathcal{M}_{\ell
,n,\delta ,\alpha }^{m,\lambda ,q}$, defined as follows:%
\begin{eqnarray*}
\mathcal{M}_{\ell ,n,\delta ,\alpha }^{m,\lambda ,q}f(z)
&=&z+\sum\limits_{j=2}^{\infty }\left[ 1+\left( j-1\right) c^{n}(\delta )%
\right] ^{m}\left( \frac{n+1}{n+k}\right) ^{\alpha }\frac{[k,q]!}{[\lambda
+1,q]_{k-1}}a_{j}z^{j}\ \  \\
&&\left( \alpha >0;\ \lambda >-1;\ \ell \geq 0;\;0<q<1;\ \delta >0;\ n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
;\ m\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
_{0}\right) .
\end{eqnarray*}

\section{Preliminary}

To prove our results, we need the following lemmas.

\begin{lemma}
\label{l1}\cite{5}\textbf{\ }Let $\psi \in \mathbb{A}$ and $\mathcal{G}(z)=%
\frac{1}{z}\int\limits_{0}^{z}\psi (t)dt,\ z\in \Delta $. If $\Re \left\{ 1+%
\frac{z\psi ^{^{\prime \prime }}(z)}{\psi ^{^{\prime }}(z)}\right\} >\frac{-1%
}{2},\ z\in \Delta $, then $\mathcal{G\in K}$.
\end{lemma}

\begin{lemma}
\label{l2}\cite[Theorem 2.6]{8}\textbf{\ }Let $\psi $ be a convex function
with $\psi (0)=\beta $ and $\nu \in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{\ast }=%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
\backslash \{0\}$ with $\Re \left( \nu \right) \geq 0.$ If $p\in $ $H\left[
\beta ,d\right] $ with $p(0)=\beta ,\ \Phi :\mathbb{C}^{2}\times \Delta
\rightarrow \mathbb{C}$, $\Phi \left( p(z),zp^{^{\prime }}(z);z\right) =p(z)+%
\frac{1}{\nu }zp^{^{\prime }}(z)$ is analytic function in $\Delta $\ and%
\begin{equation*}
\mathcal{F}_{\Phi (\mathbb{C}^{2}\times \Delta \mathbb{)}}\left( p(z)+\frac{1%
}{\nu }zp^{^{\prime }}(z)\right) \leq \mathcal{F}_{h\left( \Delta \right)
}h(z)\ \ \rightarrow p(z)+\frac{1}{\nu }zp^{^{\prime }}(z)\prec _{\mathcal{F}%
}h(z),\ \ z\in \Delta ,
\end{equation*}%
then
\begin{equation*}
\mathcal{F}_{p\left( \Delta \right) }p(z)\leq \mathcal{F}_{q\left( \Delta
\right) }q(z)\leq \mathcal{F}_{h\left( \Delta \right) }h(z)\text{\ }%
\rightarrow \text{\ }p(z)\prec _{\mathcal{F}}q(z),\ z\in \Delta \text{,\ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }
\end{equation*}%
where
\begin{equation*}
q(z)=\frac{\nu }{dz^{\frac{\nu }{d}}}\int\limits_{0}^{z}\psi (t)t^{\frac{\nu
}{d}-1}dt,\ z\in \Delta .
\end{equation*}%
The function $q$ is convex and it is the fuzzy best dominant.
\end{lemma}

\begin{lemma}
\label{l3}\cite[Theorem 2.7]{8} Let $g\ $be$\ $a convex function in $\Delta $
and $\psi (z)=g(z)+d\gamma zg^{^{\prime }}(z),\ $where $z\in \Delta ,\ d\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$ and $\gamma >0.$ If
\begin{equation*}
p(z)=g(0)+p_{d}~z^{d}+p_{d+1}~z^{d+1}+...\
\end{equation*}%
belongs to $H(\Delta \mathbb{)}$, and
\begin{equation*}
\mathcal{F}_{p(\Delta \mathbb{)}}\left( p(z)+\gamma zp^{^{\prime
}}(z)\right) \leq \mathcal{F}_{\psi \left( \Delta \right) }\psi (z)\text{\ \
}\rightarrow \ \text{\ }p(z)+\gamma zp^{^{\prime }}(z)\prec _{\mathcal{F}%
}\psi (z),\ z\in \Delta \text{,}
\end{equation*}%
then
\begin{equation*}
\mathcal{F}_{p(\Delta \mathbb{)}}\left( p(z)\right) \leq \mathcal{F}%
_{g\left( \Delta \right) }g(z)\text{\ \ }\rightarrow \text{\ \ }p(z)\prec _{%
\mathcal{F}}g(z),\ \ z\in \Delta \text{.}
\end{equation*}%
This result is sharp.
\end{lemma}

For the general theory of fuzzy differential subordination and its
applications, we refer the reader to \cite{3,4}.

In the next section, we obtain several fuzzy differential subordinations%
\textbf{\ }associated with the diferential operator $\mathcal{D}_{n,\delta
,g}^{m}f(z)$ by using the method of fuzzy differential subordination.

\section{Main results}

Assume that $\eta \in \left[ 0,1\right) $, $\delta >0,\ n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,\ m\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
_{0},\ \lambda >0\ $and $z\in \Delta $ are mentioned through this paper:

By using the integral operator $\mathcal{D}_{n,\delta ,g}^{m},$ we define a
class of analytic functions\ and we derive several fuzzy differential
subordinations for this class.

\begin{definition}
\label{d5}\textbf{\ }Let the function\textbf{\ }$f\in \mathbb{A}$ belongs to
the class $\mathcal{H}_{n,m,\delta }^{F}\left( \eta ,g\right) $ for all $%
\eta \in \left[ 0,1\right) $, $n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
_{0},\ m>0$ and$\ \alpha \geq 0$ if it satisfies the inequality:%
\begin{equation*}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}f\right) ^{^{\prime }}\left( \Delta
\right) }\left( \mathcal{D}_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}>\eta
,\ \ \ \ \ \left( z\in \Delta \right) .
\end{equation*}
\end{definition}

\begin{theorem}
\label{t1}Let\textbf{\ }$k$ belongs to $\mathcal{C}$ in $\Delta $ and
suppose that $h(z)=k(z)+\frac{1}{\lambda +2}zk^{^{\prime }}(z).$ If $f\in
\mathcal{H}_{n,m,\delta }^{F}\left( \eta ,g\right) $ and
\begin{equation}
G(z)=I^{\lambda }f(z)=\frac{\lambda +2}{z^{\lambda +1}}\int%
\limits_{0}^{z}t^{\lambda }f(t)dt,  \tag{3.1}  \label{3.1}
\end{equation}%
then
\begin{equation}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}f\right) ^{^{\prime }}\left( \Delta
\right) }\left( \mathcal{D}_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}\leq
F_{h\left( \Delta \right) }h(z)\ \ \ \ \rightarrow \text{ \ \ \ }\left(
\mathcal{D}_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}\prec _{\mathcal{F}%
}h(z),\ \ \ \ \ \ \   \tag{3.2}  \label{3.2}
\end{equation}%
implies
\begin{equation*}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}G\right) ^{^{\prime }}\left( \Delta
\right) }\left( \mathcal{D}_{n,\delta ,g}^{m}G(z)\right) ^{^{\prime }}\leq
F_{k\left( \Delta \right) }k(z)\ \ \ \ \ \rightarrow \text{ \ \ \ }\left(
\mathcal{D}_{n,\delta ,g}^{m}G(z)\right) ^{^{\prime }}\prec _{\mathcal{F}%
}k(z),
\end{equation*}%
and this result is sharp.
\end{theorem}

\begin{proof}
Since
\begin{equation*}
z^{\lambda +1}G(z)=\left( \lambda +2\right) \int\limits_{0}^{z}t^{\lambda
}f(t)dt,
\end{equation*}%
by differentiating, we obtain%
\begin{equation*}
\left( \lambda +1\right) G(z)+zG^{^{\prime }}(z)=\left( \lambda +2\right)
f(z),
\end{equation*}%
and,%
\begin{equation}
\left( \lambda +1\right) \mathcal{D}_{n,\delta ,g}^{m}G(z)+z\left( \mathcal{D%
}_{n,\delta ,g}^{m}G(z)\right) ^{^{\prime }}=\left( \lambda +2\right)
\mathcal{D}_{n,\delta ,g}^{m}f(z),  \tag{3.3}  \label{3.3}
\end{equation}%
and also, by differentiating (\ref{3.3}) we obtain
\begin{equation}
\left( \mathcal{D}_{n,\delta ,g}^{m}G(z)\right) ^{^{\prime }}+\frac{1}{%
\left( \lambda +2\right) }z\left( \mathcal{D}_{n,\delta ,g}^{m}G(z)\right)
^{^{\prime \prime }}=\left( \mathcal{D}_{n,\delta ,g}^{m}f(z)\right)
^{^{\prime }}  \tag{3.4}  \label{3.4}
\end{equation}%
By using (\ref{3.4}), the fuzzy differential subordination (\ref{3.2}) is
\begin{equation*}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}f\right) ^{^{\prime }}\left( \Delta
\right) }\left( \left( \mathcal{D}_{n,\delta ,g}^{m}G(z)\right) ^{^{\prime
}}+\frac{1}{\left( \lambda +2\right) }z\left( \mathcal{D}_{n,\delta
,g}^{m}G(z)\right) ^{^{\prime \prime }}\right) \leq
\end{equation*}%
\begin{equation}
F_{h\left( \Delta \right) }\left( k(z)+\frac{1}{\left( \lambda +2\right) }%
zk^{^{\prime }}(z)\right) .  \tag{3.5}  \label{3.5}
\end{equation}%
We denote
\begin{equation}
q(z)=\left( \mathcal{D}_{n,\delta ,g}^{m}G(z)\right) ^{^{\prime }},\ \ \text{%
so \ \ }q\in \mathcal{H}\left[ 1,n\right] .  \tag{3.6}  \label{3.6}
\end{equation}%
Putting (\ref{3.6}) in (\ref{3.5}), we have%
\begin{equation}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}f\right) ^{^{\prime }}\left( \Delta
\right) }\left( q(z)+\frac{1}{\left( \lambda +2\right) }zq^{^{\prime
}}(z)\right) \leq F_{h\left( \Delta \right) }\left( k(z)+\frac{1}{\left(
\lambda +2\right) }zk^{^{\prime }}(z)\right) ,  \tag{3.7}  \label{3.7}
\end{equation}%
and applying Lemma\ (\ref{l3}), we have
\begin{equation*}
F_{q\left( \Delta \right) }q(z)\leq F_{k\left( \Delta \right) }k(z),\ \ \ \
\text{i.e \ \ \ }F_{\left( \mathcal{D}_{n,\delta ,g}^{m}G(z)\right)
^{^{\prime }}\left( \Delta \right) }\left( \mathcal{D}_{n,\delta
,g}^{m}G(z)\right) ^{^{\prime }}\leq F_{k\left( \Delta \right) }k(z),
\end{equation*}%
therefore $\left( \mathcal{D}_{n,\delta ,g}^{m}G(z)\right) ^{^{\prime
}}\prec _{\mathcal{F}}k(z),$ and $k$ is the fuzzy best dominant.
\end{proof}

\begin{theorem}
\label{t2} Assume that $h(z)=\frac{1+\left( 2\eta -1\right) z}{1+z},\ \eta
\in \left[ 0,1\right) ,$ $\lambda >0$ and $\mathcal{I}^{\lambda }$ is given
by (\ref{3.1}), then%
\begin{equation}
\mathcal{I}^{\lambda }\left[ \mathcal{H}_{n,m,\delta }^{F}\left( \eta
,g\right) \right] \subset \mathcal{H}_{n,m,\delta }^{F}\left( \eta ^{\ast
},g\right) ,  \tag{3.8}  \label{3.8}
\end{equation}%
where
\begin{equation}
\mathit{\eta }^{\ast }\mathit{=2\eta -1+}\left( \lambda +2\right) \left(
2-2\eta \right) \int\limits_{0}^{1}\frac{t^{\lambda +2}}{t+1}\mathit{dt.}
\tag{3.9}  \label{3.9}
\end{equation}
\end{theorem}

\begin{proof}
A function\textbf{\ }$h$ belongs to $\mathcal{C}$ and using the same
technique in the proof of Theorem \ref{t1}, we obtain from the hypothesis of
Theorem \ref{t2} that
\begin{equation*}
F_{q\left( \Delta \right) }\left( q(z)+\frac{1}{\left( \lambda +2\right) }%
zq^{^{\prime }}(z)\right) \leq F_{h\left( \Delta \right) }h(z),
\end{equation*}%
where $q(z)$ is defined in (\ref{3.6}).\ By using Lemma \ref{l2}, we obtain
\begin{equation*}
F_{q\left( \Delta \right) }q(z)\leq F_{k\left( \Delta \right) }k(z)\leq
F_{h\left( \Delta \right) }h(z),
\end{equation*}%
which implies%
\begin{equation*}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}G\right) ^{^{\prime }}\left( \Delta
\right) }\left( \mathcal{D}_{n,\delta ,g}^{m}G(z)\right) ^{^{\prime }}\leq
F_{k\left( \Delta \right) }k(z)\leq F_{h\left( \Delta \right) }h(z),
\end{equation*}%
where%
\begin{eqnarray*}
k(z) &=&\frac{\lambda +2}{z^{\lambda +2}}\int\limits_{0}^{z}t^{\lambda +1}%
\frac{1+\left( 2\eta -1\right) t}{1+t}dt \\
&=&\left( 2\eta -1\right) +\frac{\left( \lambda +2\right) \left( 2-2\eta
\right) }{z^{\lambda +2}}\int\limits_{0}^{z}\frac{t^{\lambda +1}}{1+t}dt.
\end{eqnarray*}%
$k$ belongs to $\mathcal{C}$ and $k\left( \Delta \right) $ is symmetric with
respect to the real axis, so we conclude%
\begin{equation}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}G\right) ^{^{\prime }}\left( \Delta
\right) }\left( \mathcal{D}_{n,\delta ,g}^{m}G(z)\right) ^{^{\prime }}\geq \
\underset{\left\vert z\right\vert =1}{\min }F_{k\left( \Delta \right)
}k(z)=F_{k\left( \Delta \right) }k(1),  \tag{3.10}  \label{3.10}
\end{equation}%
and $\eta ^{\ast }=k(1)=2\eta -1+\left( \lambda +2\right) \left( 2-2\eta
\right) \int\limits_{0}^{1}\frac{t^{\lambda +2}}{t+1}dt.$
\end{proof}

\begin{theorem}
\label{t3} Let $k$ belongs to $\mathcal{C}$ in $\Delta ,$ $k(0)=1,\ $and $%
h(z)=k(z)+zk^{^{\prime }}(z).$ If $f\in \mathbb{A}$ and satisfies the fuzzy
differential subordination%
\begin{equation}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}f\right) ^{^{\prime }}\left( \Delta
\right) }\left( \mathcal{D}_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}\leq
F_{h\left( \Delta \right) }h(z)\ \ \ \ \ \rightarrow \text{ \ \ \ }\left(
\mathcal{D}_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}\prec _{\mathcal{F}%
}h(z),  \tag{3.11}  \label{3.11}
\end{equation}%
holds, then%
\begin{equation}
F_{\mathcal{D}_{n,\delta ,g}^{m}f\left( \Delta \right) }\frac{\mathcal{D}%
_{n,\delta ,g}^{m}f\left( z\right) }{z}\leq F_{k\left( \Delta \right) }k(z)\
\ \ \ \ \rightarrow \text{ \ \ \ }\frac{\mathcal{D}_{n,\delta ,g}^{m}f\left(
z\right) }{z}\prec _{\mathcal{F}}k(z).  \tag{3.12}  \label{3.12}
\end{equation}%
The result is sharp.
\end{theorem}

\begin{proof}
\textbf{\ }For
\begin{eqnarray*}
q(z) &=&\frac{\mathcal{D}_{n,\delta ,g}^{m}f\left( z\right) }{z}=\frac{%
z+\sum\limits_{j=2}^{\infty }\left[ 1+\left( j-1\right) c^{n}(\delta )%
\right] ^{m}a_{j}b_{j}z^{j}}{z} \\
&=&1+\sum\limits_{j=2}^{\infty }\left[ 1+\left( j-1\right) c^{n}(\delta )%
\right] ^{m}a_{j}b_{j}z^{j-1},
\end{eqnarray*}%
we obtain that $q(z)+zq^{^{\prime }}(z)=\left( \mathcal{D}_{n,\delta
,g}^{m}f\left( z\right) \right) ^{^{\prime }},$ so%
\begin{equation*}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}f\right) ^{^{\prime }}\left( \Delta
\right) }\left( \mathcal{D}_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}\leq
F_{h\left( \Delta \right) }h(z)
\end{equation*}%
implies
\begin{equation*}
F_{q\left( \Delta \right) }\left( q(z)+zq^{^{\prime }}(z)\right) \leq
F_{h\left( \Delta \right) }h(z)=F_{k\left( \Delta \right) }\left(
k(z)+zk^{^{\prime }}(z)\right) .
\end{equation*}%
Applying Lemma \ref{l3}, we have
\begin{equation*}
F_{q\left( \Delta \right) }q(z)\leq F_{k\left( \Delta \right) }k(z)\ \ \
\rightarrow \ \ \ F_{\mathcal{D}_{n,\delta ,g}^{m}f\left( \Delta \right) }%
\frac{\mathcal{D}_{n,\delta ,g}^{m}f\left( z\right) }{z}\leq F_{k\left(
\Delta \right) }k(z),
\end{equation*}%
and we get
\begin{equation*}
\frac{\mathcal{D}_{n,\delta ,g}^{m}f\left( z\right) }{z}\prec _{\mathcal{F}%
}k(z).
\end{equation*}%
The result is sharp.
\end{proof}

\begin{theorem}
\label{t4} Consider $h\in \mathcal{H}(\Delta )$ with $h(0)=1,$ which
satisfies $\Re \left( 1+\frac{zh^{^{\prime \prime }}(z)}{h^{^{\prime }}(z)}%
\right) $ $>\frac{-1}{2}.$ If $f\in \mathbb{A}$ and the fuzzy differential
subordination%
\begin{equation}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}f\right) ^{^{\prime }}\left( \Delta
\right) }\left( \mathcal{D}_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}\leq
F_{h\left( \Delta \right) }h(z)\ \ \ \ \ \rightarrow \text{ \ \ \ }\left(
\mathcal{D}_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}\prec _{\mathcal{F}%
}h(z),  \tag{3.13}  \label{3.13}
\end{equation}%
then%
\begin{equation}
F_{\mathcal{D}_{n,\delta ,g}^{m}f\left( \Delta \right) }\frac{\mathcal{D}%
_{n,\delta ,g}^{m}f\left( z\right) }{z}\leq F_{k\left( \Delta \right) }k(z)\
\ \ \ \ \text{i.e \ \ \ }\frac{\mathcal{D}_{n,\delta ,g}^{m}f\left( z\right)
}{z}\prec _{\mathcal{F}}k(z),  \tag{3.14}  \label{3.14}
\end{equation}%
where%
\begin{equation*}
k(z)=\frac{1}{z}\int\limits_{0}^{z}\mathit{h(t)dt},
\end{equation*}%
the function $k$ is convex and it is the fuzzy best dominant.
\end{theorem}

\begin{proof}
Let
\begin{equation*}
q(z)=\frac{\mathcal{D}_{n,\delta ,g}^{m}f\left( z\right) }{z}%
=1+\sum\limits_{j=2}^{\infty }\left[ 1+\left( j-1\right) c^{n}(\delta )%
\right] ^{m}a_{j}b_{j}z^{j-1},\ \ q\in \mathcal{H}\left[ 1,1\right] ,
\end{equation*}%
\newline
where $\Re \left( 1+\frac{zh^{^{\prime \prime }}(z)}{h^{^{\prime }}(z)}%
\right) >\frac{-1}{2}.$ From Lemma \ref{l1}, we have
\begin{equation*}
k(z)=\frac{1}{z}\int\limits_{0}^{z}h(t)dt
\end{equation*}%
belongs to the class $\mathcal{C},$ which satisfies the fuzzy differential
subordination (\ref{3.13}). Since
\begin{equation*}
k(z)+zk^{^{\prime }}(z)=h(z),
\end{equation*}%
it is the fuzzy best dominant.

We have $q(z)+zq^{^{\prime }}(z)=\left( \mathcal{D}_{n,\delta ,g}^{m}f\left(
z\right) \right) ^{^{\prime }},$ then (\ref{3.13}) becomes%
\begin{equation*}
F_{q\left( \Delta \right) }\left( q(z)+zq^{^{\prime }}(z)\right) \leq
F_{h\left( \Delta \right) }h(z).
\end{equation*}%
Applying Lemma \ref{l3}, we have
\begin{equation*}
F_{q\left( \Delta \right) }q(z)\leq F_{k\left( \Delta \right) }k(z),\ \ \
\text{i.e.}\ \ \ F_{\mathcal{D}_{n,\delta ,g}^{m}f\left( \Delta \right) }%
\frac{\mathcal{D}_{n,\delta ,g}^{m}f\left( z\right) }{z}\leq F_{k\left(
\Delta \right) }k(z),
\end{equation*}%
then
\begin{equation*}
\frac{\mathcal{D}_{n,\delta ,g}^{m}f\left( z\right) }{z}\prec _{\mathcal{F}%
}k(z).
\end{equation*}
\end{proof}

Putting $h(z)=\frac{1+\left( 2\beta -1\right) z}{1+z}$ in Theorem \ref{t4},
we obtain the following corollary:

\begin{corollary}
\label{c1} Let $h=\frac{1+\left( 2\beta -1\right) z}{1+z}$ a convex function
in $\Delta ,$ with $h(0)=1,$ $0\leq \beta <1.$ If $f\in \mathbb{A}$ and
verifies the fuzzy differential subordination%
\begin{equation*}
F_{\left( \mathcal{D}_{n,\delta ,g}^{m}f\right) ^{^{\prime }}\left( \Delta
\right) }\left( \mathcal{D}_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}\leq
F_{h\left( \Delta \right) }h(z),\ \ \ \text{i.e\ \ \ }\left( \mathcal{D}%
_{n,\delta ,g}^{m}f(z)\right) ^{^{\prime }}\prec _{\mathcal{F}}h(z),
\end{equation*}%
then%
\begin{equation*}
k(z)=2\beta -1+\frac{2\left( 1-\beta \right) }{z}\ln \left( 1+z\right) ,
\end{equation*}%
the function $k$ is convex and it is the fuzzy best dominant.
\end{corollary}

\textbf{Concluding, }all the above results give us information about fuzzy
differential subordinations\textbf{\ }for the operator $\mathcal{D}%
_{n,\delta ,g}^{m}$, we give some properties for the class $\mathcal{H}%
_{\alpha ,m}^{F}\left( n,\eta \right) $ of univalent analytic functions.

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\end{document}