Properties of Differential Subordination and Superordination for Multivalent Functions Associated with the Convolution Operators

Abstract

Using convolution (or Hadamard product), we define the El-Ashwah and Drbuk linear operator, which is a multivalent function in the unit disk $U=\left[w:\left|w\right|<1 \mathrm{and} w\in ₵\right]$, and satisfy its specific relationship to derive the subordination, superordination, and sandwich results for this operator by using properties of subordination and superordination concepts.

1. Introduction and Definitions

The set $\mathsf{\Omega }\left(U\right)$ denotes the class of all analytic functions in the open unit disk $U=\left[w:\left|w\right|<1 anq w\in ₵\right]$ and $\mathsf{\Omega }\left[a, k\right]$ as the subclass of $\mathsf{\Omega }\left(U\right)$, which consists of the form functions

$$f\left(w\right)=a+a_k{w}^k+a_{k+1}{w}^{k+1}+\dots , \left(a\in ₵, w\in U, k\in N\right) .$$

(1)

With ${\mathcal{A}}_p$ as the class of all multivalent functions in open unit disk $U$ of the form

$$f\left(w\right)=w^p+{\displaystyle \sum }_{k=1+p}^{\infty }{a}_k{w}^k, w\in U,p\in N.$$

(2)

Additionally, we use $\mathcal{A}={\mathcal{A}}_1$ to denote the class of analytic functions in the open unit disk $U$ and normalize them with $f$(0) = 0, $f^{\prime }$(0) = 1.

Additionally, consider $\mathcal{S}$ as the class of the univalent function in $U$,

Let ${\mathcal{S}}^{*}\left(\varrho \right), \mathcal{C}\left(\varrho \right)$ and $\mathcal{K}$ be the subclasses of $\mathcal{A}$ such that:

$\left\{f\in {\mathcal{S}}^{*}:Re\left\{\frac{w{f}^{\prime }\left(w\right)}{ f\left(w\right)}\right\}>\varrho \right\},w\in U, \left(0<\varrho <1\right),$ then $f$ is a starlike function;

$\left\{f\in \mathcal{C}:Re\left\{1+\frac{w{f}^{″}\left(w\right)}{ f^{\prime }\left(w\right)}\right\}>\varrho \right\},w\in U, \left(0<\varrho <1\right)$, then $f$ is a convex function;

$\left\{f\in \mathcal{K}:Re\left\{\frac{{f_1}^{\prime }\left(w\right)}{g^{\prime }\left(w\right)}\right\}>0:g\in \mathcal{C}\right\},w\in U$, then $f$ is a close-to-convex function.

If the functions $f$ and $g$ are analytic in $U$, then we say $f$ is subordinate to $g$ or f is said to be superordinate to f in $U$, written as $f\prec g$ or $f\left(w\right)\prec g\left(w\right)$ if there is a Schwarz function $\upsilon \left(w\right)$ analytic in $U$, with $\left|\upsilon \left(w\right)\right|<1$, so that $f\left(w\right)=g\left(\upsilon \left(w\right)\right)$ and $w\in U$. In particular, if the function g is univalent in $U$, then the subordination $f\prec g$ is equivalent to $f\left(0\right)=g\left(0\right)$ and $f\left(U\right)\subset g\left(U\right)$, (see [1,2,3,4,5,6,7,8]).

If $f,g\in {\mathcal{A}}_p$, where $f\left(w\right)$ is provided by (1) and $g\left(w\right)$ is defined by

$$g\left(w\right)=w^p+{\displaystyle \sum }_{k=1+p}^{\infty }{a}_k{w}^k,w\in U,$$

the Hadamard product (or convolution) of the function $f \mathrm{and} g$ is defined by

$$f\left(w\right)\times g\left(w\right)=w^p+{\displaystyle \sum }_{k=1+p}^{\infty }{a}_k{b}_k{w}^k,\left(w\in U\right)=\left(f\times g\right)\left(w\right).$$

(3)

Let $\delta >0, a,c\in ₵$ such that $Re\left(c-a\right)\ge 0$ and $Rea\ge -\delta p, p\in N, n\in \mathbb{Z}, \theta \ge 0$$\mathrm{and} \lambda >-p.$

El-Ashwah and Drbuk [5] introduced the linear operator ${\mathcal{B}}_{p,n}^{\theta , \lambda }\left(a,c,\delta \right):{\mathcal{A}}_p\to {\mathcal{A}}_p$ defined by

$$\begin{array}{l}{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\\ =w^p+{\displaystyle \frac{\mathsf{\Gamma }\left(c+\delta p\right)}{\mathsf{\Gamma }\left(c+\delta p\right)}}{\displaystyle {\displaystyle \sum }_{k=1+p}^{\infty }}{\left({\displaystyle \frac{p+\lambda +\theta \left(k-p\right)}{p+\lambda }}\right)}^n{\displaystyle \frac{\mathsf{\Gamma }\left(c+\delta p\right)}{\mathsf{\Gamma }\left(c+\delta p\right)}}{a}_k{w}^k. \end{array}$$

(4)

It is readily verified from (4) that

$$\begin{array}{l}{\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)=\left(1-{\displaystyle \frac{p\theta }{p+\lambda }}\right){\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\\ +{\displaystyle \frac{\theta }{p+\lambda }}w{\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}^{\prime }.\end{array}$$

(5)

Putting $a=c$ in (4), we obtain the Prajapat operator $J_p^n\left(\theta ,\lambda \right)$, see [9].

Additionally, when $n=0$, we obtain the Erdelyi-Kober integral operator $I_{p,\delta }^{a,c}$, see [10].

Definition 1.

Let${\rm Y}$:${₵}^3\times U\to ₵$and$h$ ($w)$ be univalent in $U$. If $p$ ($w$) is analytic in $U$, that fulfils the second-order differential subordination [11]:

$${\rm Y}\left(p\left(w\right),w{p}^{\prime }\left(w\right), w^2{p}^{″}\left(w\right);w\right)\prec h(w),$$

(6) 

then $p\left(w\right)$ is the differential subordination solution of (6).

Definition 2.

Let${{\rm Y}}_1$:${₵}^3\times U\to ₵$and$h$($w)$be univalent in$U$. If$p$($w$) and${{\rm Y}}_1\left(p\left(w\right),w{p}^{\prime }\left(w\right), w^2{p}^{″}\left(w\right);w\right)$are univalent in$U$and$p\left(w\right)$fulfill the second-order differential superordination [11]:

$$h\left(w\right)\prec {{\rm Y}}_1\left(p\left(w\right),w{p}^{\prime }\left(w\right), w^2{p}^{″}\left(w\right);w\right),$$

(7) 

then$p\left(w\right)$is the differential superordination solution of (7).

Definition 3.

Let$Q$be the collections of functions$f$that are analytic and injective on$\overline{U}\setminus E\left(f\right)$, when $E\left(f\right)=\left\{\varsigma \in \partial U :\underset{w\to \varsigma }{\mathit{lim}} f\left(w\right)=\infty \right\}$and$f^{\prime }\left(w\right)\ne 0$for$\varsigma \in \partial U\setminus E\left(f\right)$[11].

Lemma 1.

Let$p_1\left(w\right)$be the univalent function in$U$and let$\Sigma$and$\vartheta$be holomorphic in a domain$p_1$($U)\subset D,$with$\vartheta \left(\omega \right)\ne 0$, when$\omega \in p_1$ ($U$). Set $\mathcal{O}\left(w\right)=w {p_1}^{\prime }\left(w\right)\vartheta \left(p_1\left(w\right)\right)$ and $\hslash \left(w\right)=\Sigma (p_1\left(w\right)+\mathcal{O}\left(w\right)$. Suppose that [12]

$\left(i\right)$$\mathcal{O}$is starlike in$U$.

$\left(ii\right) Re\left(\frac{w{\hslash }^{\prime }\left(w\right)}{\mathcal{O}\left(w\right)}\right)>0$,$w\in U$.

If$p_2\left(w\right)$is holomorphic in$U$with$p_2\left(0\right)=p_1\left(0\right)$,$p_2$($U)\subset D$, and$\Sigma \left(p_2\left(w\right)\right)+w{p_2}^{\prime }\left(w\right)\vartheta \left(p_2\left(w\right)\right)\prec \Sigma \left(p_1\left(w\right)\right)+w{p_1}^{\prime }\left(w\right)\vartheta \left(p_1\left(w\right)\right)$, then$p_2\left(w\right)\prec p_1\left(w\right)$.

Lemma 2.

Let$p_1\left(w\right)$be convex in$U$and${\beta }_1\in ₵,{\beta }_2\in {₵}^{*}$with$Re\left(1+\frac{{p_1}^{″}\left(w\right)}{{p_1}^{\prime }\left(w\right)}\right)>\mathit{max}\left\{0,-Re\frac{{\beta }_1}{{\beta }_2}\right\}$. If$p_2\left(w\right)$is holomorphic in$U$and${\beta }_1{p}_2\left(w\right)+{\beta }_{2}w{p_2}^{\prime }\left(w\right)\prec {\beta }_1{p}_1\left(w\right)+{\beta }_{2}w{p_1}^{\prime }\left(w\right),$then$p_2\left(w\right)\prec p_1\left(w\right)$[11].

Lemma 3.

Let$p_1\left(w\right)$be convex univalent in$U$and let$\Sigma$and$\vartheta$be holomorphic in a domain$D$,$p_1$ ($U)\subset D$. Suppose that [12]

$\left(i\right)$$w{p_1}^{\prime }\left(w\right)\vartheta (p_1\left(w\right)$is starlike univalent in$U$.

$\left(ii\right) Re\left(\frac{ \Sigma \prime \left(p_1\left(w\right)\right)}{\vartheta \left(p_1\left(w\right)\right)}\right)>0$,$w\in U$.

If$p_2\left(w\right)\in \mathcal{A}\left[p_1\left(0\right),1\right]\bigcap Q$, with$p_2$ ($U)\subset D$, $\Sigma \left(p_2\left(w\right)+w{p_2}^{\prime }\left(w\right)\vartheta (p_2\left(w\right)\right)$ is univalent in $U$ and $\Sigma \left(p_1\left(w\right)\right)+w{p_1}^{\prime }\left(w\right)\vartheta \left(p_1\left(w\right)\right)\prec \Sigma \left(p_2\left(w\right)\right)+w{p_2}^{\prime }\left(w\right)\vartheta \left(p_2\left(w\right)\right)$, then $p_1\left(w\right)\prec p_2\left(w\right)$.

Lemma 4.

Let$p_1\left(w\right)$be convex in$U$and$Re\left(\beta \right)>0$.

If$p_2\left(w\right)\in \mathcal{A}\left[p_1\left(0\right),1\right]\bigcap Q$,$p_2\left(w\right)+\beta w{p_2}^{\prime }\left(w\right)$is univalent in$U$and$p_1\left(w\right)+\beta w{p_1}^{\prime }\left(w\right)\prec p_2\left(w\right)+\beta w{p_2}^{\prime }\left(w\right)$, then$p_1\left(w\right)\prec p_2\left(w\right)$[12].

2. Subordination Results

Theorem 1.

Let$b\left(w\right)$be convex univalent in$U$, with$b\left(0\right)=1$, $a_1>0,0\ne a_2\in ₵$and suppose

$$Re\left(1+\frac{b^{″}\left(w\right)}{b^{\prime }\left(w\right)}\right)>\max \left\{0,-Re\frac{a_1}{a_2}\right\}.$$

If$f\in \mathcal{A}$, it satisfies the subordination:

$$\left(1-\frac{a_2\left(p+\lambda \right)}{\theta }\right){\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}+\frac{a_2\left(p+\lambda \right)}{\theta }{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)}{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}\right)}^{a_1}{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}\prec b\left(w\right)+\frac{a_2}{a_1}w{b}^{\prime }\left(w\right),$$

then

$${\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}\prec b\left(w\right).$$

Proof. 

Consider

$$q\left(w\right)={\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}.$$

Then

$$\begin{array}{l}{q}^{\prime }\left(w\right)=a_1{\left({\displaystyle \frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}}\right)}^{a_1}\left({\displaystyle \frac{w^p}{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}}\right)\\ \left({\displaystyle \frac{w^p{\left[{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right]}^{\prime }-p{w}^{p-1}\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{{\left(w^p\right)}^2}}\right)\\ =a_1{\left({\displaystyle \frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}}\right)}^{a_1}\left({\displaystyle \frac{{\left[{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right]}^{\prime }}{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}}-{\displaystyle \frac{p}{w}}\right)\end{array}$$

We have

$$\frac{w{q}^{\prime }\left(w\right)}{q\left(w\right)}=a_1\left(\frac{w{\left[{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right]}^{\prime }}{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}-p\right).$$

By using (5) we obtain

$$\begin{array}{l}{\displaystyle \frac{w{q}^{\prime }\left(w\right)}{q\left(w\right)}}\\ =a_1\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+p\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}}-p\right)\\ =a_1{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\displaystyle \frac{\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)}{\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}}-1\right),\end{array}$$

and

$$\begin{gathered} \frac{a_{2}w{q}^{\prime }\left(w\right)}{a_1}=\frac{a_2\left(p+\lambda \right)}{\theta }\left(\frac{\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)}{\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}\right){\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1} \\ -\frac{a_2\left(p+\lambda \right)}{\theta }{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}. \end{gathered}$$

By using the hypothesis, we obtain $q\left(w\right)+\frac{a_2}{a_1}w{q}^{\prime }\left(w\right)\prec b\left(w\right)+\frac{a_2}{a_1}w{b}^{\prime }\left(w\right)$.

Additionally, apply Lemma 2, when ${\beta }_1=1$ and ${\beta }_2=\frac{a_2}{a_1}$, then

$${\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}\prec b\left(w\right).$$

□

Corollary 1.

Let$b\left(w\right)$be convex univalent in$U$, with$b\left(0\right)=1$, $a_1>0,0\ne a_2\in ₵$and suppose

$$Re\left(1+\frac{b^{″}\left(w\right)}{b^{\prime }\left(w\right)}\right)>\mathit{max}\left\{0,-Re\frac{a_1}{a_2}\right\}.$$

If$f\in \mathcal{A}$, it satisfies the subordination:

$$\left(1-\frac{a_2\left(p+\lambda \right)}{\theta }\right){\left(\frac{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}{w^p}\right)}^{a_1}+\frac{a_2\left(p+\lambda \right)}{\theta }{\left(\frac{J_p^{n+1}\left(\theta ,\lambda \right)f\left(w\right)}{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}\right)}^{a_1}{\left(\frac{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}{w^p}\right)}^{a_1}\prec b\left(w\right)+\frac{a_2}{a_1}w{b}^{\prime }\left(w\right),$$

then

$${\left(\frac{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}{w^p}\right)}^{a_1}\prec b\left(w\right).$$

Theorem 2.

Let$b$be convex univalent in,$b\left(0\right)=1$, and$b\left(w\right)\ne 0$for all$w\in U$, and suppose that$b$satisfies:

$$Re\left\{p+\frac{wt\sigma }{w^p{a}_2}+\frac{w\epsilon \left(\sigma +1\right)}{w^p{a}_2}\left(w\right)+\left(\sigma -1\right)\frac{w{b}^{\prime }\left(w\right)}{b\left(w\right)}+\frac{w{b}^{″}\left(w\right)}{b^{\prime }\left(w\right)}\right\}>0,$$

(8)

where$\sigma$, $\epsilon ,t\in ₵,a_1>0,0\ne a_2\in ₵$and$w\in U$. Suppose that$w^p{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$is a starlike univalent in$U$.

If$f\in \mathcal{A}$satisfies the subordination:

$$\mathcal{M}\left(p,n,\lambda ,\theta ,\epsilon ,a_1,a_2;w\right)\prec \left(t+\epsilon b\left(w\right)\right){\left(b\left(w\right)\right)}^{\sigma }+a_2{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$$

where

$$\begin{array}{l}\mathcal{M}\left(p,n,\lambda ,\theta ,\epsilon ,a_1,a_2;w\right)\\ =t{\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1\sigma }\\ +\epsilon {\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1\left(\sigma +1\right)}\\ +a_2{a}_1{\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1\sigma }\end{array}$$

$$\left(\frac{w{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}{\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)}^{\prime }+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right){\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}^{\prime }}{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}-p\right),$$

(9) 

then

$$\begin{array}{l}{\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1}\\ \prec b\left(w\right).\end{array}$$

Proof. 

Let $H\left(\beta \right)=\left(t+\epsilon \beta \right){\beta }^{\sigma }$ and $L\left(\beta \right)=a_2{\left(\beta \right)}^{\sigma -1}$, $0\ne \beta \in ₵$, when $H\left(\beta \right)$ and $L\left(\beta \right)$ are analytic in $₵$. □

Then, we obtain $G\left(w\right)=w{b}^{\prime }\left(w\right)L\left(b\left(w\right)\right)=a_2{w}^p{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$ and $y\left(w\right)=H\left(b\left(w\right)\right)+G\left(w\right)=\left(t+\epsilon \left(b\left(w\right)\right)\right){\left(b\left(w\right)\right)}^{\sigma }+a_2{w}^p{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$.

Since $w^p{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$ is starlike, then $G\left(w\right)$ is starlike in $U,$ and

$$\begin{array}{l}Re\left({\displaystyle \frac{y^{\prime }\left(w\right)}{G\left(w\right)}}\right)=Re\left\{p+{\displaystyle \frac{wt\sigma }{w^p{a}_2}}+{\displaystyle \frac{w\epsilon \left(\sigma +1\right)}{w^p{a}_2}}\left(w\right)+\left(\sigma -1\right){\displaystyle \frac{w{b}^{\prime }\left(w\right)}{b\left(w\right)}}+{\displaystyle \frac{w{b}^{″}\left(w\right)}{b^{\prime }\left(w\right)}}\right\}\\ >0\end{array}$$

.

Additionally, consider

$$\begin{array}{l}q\left(w\right)\\ ={\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1}.\end{array}$$

Then,

$$\begin{array}{l}{q}^{\prime }\left(w\right)\\ =a_1{\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1}\end{array}$$

$$\left[{\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}{\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)}^{\prime }+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right){\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}^{\prime }}{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}}-{\displaystyle \frac{1}{w}}\right].$$

We obtain

$$\begin{array}{l}t{\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1\sigma }\\ +\epsilon {\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1\left(\sigma +1\right)}\\ =t{\left(q\left(w\right)\right)}^{\sigma }+\epsilon \left[{\left(q\left(w\right)\right)}^{\sigma }q\left(w\right)\right]=\left(t+\epsilon q\left(w\right)\right){\left(q\left(w\right)\right)}^{\sigma }.\end{array}$$

Since

$$a_1\left(\frac{w{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}{\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)}^{\prime }+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right){\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}^{\prime }}{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}-p\right)=\frac{w{q}^{\prime }\left(w\right)}{q\left(w\right)},$$

That

$$\begin{array}{c}{a}_2{a}_1{\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1\sigma }\\ \left({\displaystyle \frac{w{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}{\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)}^{\prime }+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right){\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}^{\prime }}{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}}-p\right)\\ =a_2 w{\left(q\left(w\right)\right)}^{\sigma -1}{q}^{\prime }\left(w\right).\end{array}$$

From (8) we obtain $\left(t+\epsilon q\left(w\right)\right){\left(q\left(w\right)\right)}^{\sigma }+a_2 w{\left(q\left(w\right)\right)}^{\sigma -1}{q}^{\prime }\left(w\right)\prec \left(t+\epsilon b\left(w\right)\right){\left(b\left(w\right)\right)}^{\sigma }+a_2{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$ and using Lemma 1 we obtain $q\left(w\right)\prec b\left(w\right).$

Corollary 2.

Let$b$be convex univalent in,$b\left(0\right)=1$, and$b\left(w\right)\ne 0$for all$w\in U$, and suppose that$b$satisfies:

$$Re\left\{p+\frac{wt\sigma }{w^p{a}_2}+\frac{w\epsilon \left(\sigma +1\right)}{w^p{a}_2}\left(w\right)+\left(\sigma -1\right)\frac{w{b}^{\prime }\left(w\right)}{b\left(w\right)}+\frac{w{b}^{″}\left(w\right)}{b^{\prime }\left(w\right)}\right\}>0,$$

where$\sigma ,$$\epsilon ,t\in ₵,a_1>0,0\ne a_2\in ₵$and$w\in U$.

Suppose that$w^p{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$is a starlike univalent in$U$.

If$f\in \mathcal{A}$, it satisfies the subordination:

$$\mathcal{M}\left(\sigma ,t,\epsilon ,h_{\mu },\mu ,a_1,a_2;w\right)\prec \left(t+\epsilon b\left(w\right)\right){\left(b\left(w\right)\right)}^{\sigma }+a_2{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right),$$

then

$${\left(\frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left(J_p^{n+1}\left(\theta ,\lambda \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left(J_p^n\left(\theta ,\lambda \right)f\left(w\right)\right)}{w^p}\right)}^{a_1}\prec b\left(w\right).$$

3. Superordination Results

Theorem 3.

Let$b\left(w\right)$be convex in$U$, with$b\left(0\right)=1$, $a_1>0$, $Re{a}_2>0$, if$f\in \mathcal{A}$,

$${\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}\in \mathsf{\Omega }\left[q\left(0\right),1\right]\cap Q$$

and

$$\left(1-\frac{a_2\left(p+\lambda \right)}{\theta }\right){\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}+\frac{a_2\left(p+\lambda \right)}{\theta }{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)}{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}\right)}^{a_1}{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}$$

is univalent in$U$and satisfies the superordination.

$$b\left(w\right)+\frac{a_2}{a_1}w{b}^{\prime }\left(w\right)\prec \left(1-\frac{a_2\left(p+\lambda \right)}{\theta }\right){\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}+\frac{a_2\left(p+\lambda \right)}{\theta }{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)}{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}\right)}^{a_1}{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}$$

then

$$b\left(w\right)\prec {\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}.$$

Proof. 

Consider

$$q\left(w\right)={\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1},$$

then

$$q^{\prime }\left(w\right)=a_1{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1-1}\left(\frac{w^p{\left[{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right]}^{\prime }}{{\left(w^p\right)}^2}-\frac{p{w}^{p-1}\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)\left(w\right)}{{\left(w^p\right)}^2}\right).$$

We have

$$\frac{q^{\prime }\left(w\right)}{q\left(w\right)}=a_1\left(\frac{{\left[{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right]}^{\prime }}{\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}-\frac{1}{w}\right),$$

with the same steps of Theorem 1 and using the hypothesis, we obtain

$$b\left(w\right)+\frac{a_2}{a_1}w{b}^{\prime }\left(w\right)\prec b\left(w\right)+\frac{a_2}{a_1}w{b}^{\prime }\left(w\right).$$

Apply Lemma 4 we obtain

$$b\left(w\right)\prec {\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}.$$

□

Corollary 3.

Let$b\left(w\right)$be convex in$U$, with$b\left(0\right)=1$, $a_1>0$, $Re{a}_2>0$, if$f\in \mathcal{A}$,

$${\left(\frac{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}{w^p}\right)}^{a_1}\in \Omega \left[q\left(0\right),1\right]\cap Q$$

and

$$\left(1-\frac{a_2\left(p+\lambda \right)}{\theta }\right){\left(\frac{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}{w^p}\right)}^{a_1}+\frac{a_2\left(p+\lambda \right)}{\theta }{\left(\frac{J_p^{n+1}\left(\theta ,\lambda \right)f\left(w\right)}{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}\right)}^{a_1}{\left(\frac{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}{w^p}\right)}^{a_1}$$

is univalent in$U$and satisfies the superordination

$$\begin{array}{r}b\left(w\right)+{\displaystyle \frac{a_2}{a_1}}w{b}^{\prime }\left(w\right)\prec \left(1-{\displaystyle \frac{a_2\left(p+\lambda \right)}{\theta }}\right){\left({\displaystyle \frac{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}{w^p}}\right)}^{a_1}\\ +\frac{a_2\left(p+\lambda \right)}{\theta }{\left(\frac{J_p^{n+1}\left(\theta ,\lambda \right)f\left(w\right)}{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}\right)}^{a_1}{\left(\frac{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}{w^p}\right)}^{a_1},\end{array}$$

then

$$b\left(w\right)\prec {\left(\frac{J_p^n\left(\theta ,\lambda \right)f\left(w\right)}{w^p}\right)}^{a_1}.$$

Theorem 4.

Let$b$be convex univalent in$b\left(0\right)=1$and$b\left(w\right)\ne 0$for all$w\in U$and suppose that$b$satisfies:

$$Re\left\{\frac{t\sigma }{a_2}{b}^{\prime }\left(w\right)+\frac{\epsilon \left(\sigma +1\right)}{a_2}b\left(w\right)b^{\prime }\left(w\right)\right\}>0,$$

(10)

where,$\epsilon ,t\in ₵,0\ne a_2\in {₵}^{*}$, $w\in U$, and$w{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$are all starlike univalent in$U$.

If$f\in \mathcal{A}$, satisfies the condition:

$${\left(\frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}\right)}^{a_1}\in \Omega \left[b\left(0\right),1\right]\bigcap Q,$$

and$\mathcal{M}\left(p,n,\lambda ,\theta ,\epsilon ,a_1,a_2;w\right)$is univalent in$U$.

If$\left(t+\epsilon b\left(w\right)\right){\left(b\left(w\right)\right)}^{\sigma }+a_2{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)\prec \mathcal{M}\left(\sigma ,t,\epsilon ,h_{\mu },\mu ,a_1,a_2;w\right)$, then

$$b\left(w\right)\prec {\left(\frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}\right)}^{a_1}.$$

Proof. 

Let $H\left(\beta \right)=\left(t+\epsilon \beta \right){\beta }^{\sigma }$ and $L\left(\beta \right)=a_2{\left(\beta \right)}^{\sigma -1}$, $0\ne \beta \in ₵$, when $H\left(\beta \right)$ is analytic in $₵$ and $L\left(\beta \right)\ne 0$ is analytic in $₵∕0$. Then, we obtain $G\left(w\right)=w^p{b}^{\prime }\left(w\right)L\left(b\left(w\right)\right)=a_2{w}^p{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$. □

Since $w^p{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$ is starlike, then $G\left(w\right)$ is starlike in $U,$ and

$$\begin{gathered} Re\left(\frac{H^{\prime }\left(b\left(w\right)\right)}{L\left(b\left(w\right)\right)}\right)=Re\left(\frac{{\left[\left(t+\epsilon \left(b\left(w\right)\right)\right){\left(b\left(w\right)\right)}^{\sigma }\right]}^{\prime }}{a_2{\left(b\left(w\right)\right)}^{\sigma -1}}\right)=Re\left\{\frac{t\sigma }{a_2}{b}^{\prime }\left(w\right)+\frac{\epsilon \left(\sigma +1\right)}{a_2}b\left(w\right)b^{\prime }\left(w\right)\right\}> \\ 0; \end{gathered}$$

Now, let

$$\begin{array}{l}q\left(w\right)\\ ={\left({\displaystyle \frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}}\right)}^{a_1}.\end{array}$$

From (8) we obtain

$$\begin{gathered} \left(t+\epsilon b\left(w\right)\right){\left(b\left(w\right)\right)}^{\sigma }+a_2{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right) \\ \prec \left(t+\epsilon q\left(w\right)\right){\left(q\left(w\right)\right)}^{\sigma }+a_2 w{\left(q\left(w\right)\right)}^{\sigma -1}{q}^{\prime }\left(w\right). \end{gathered}$$

Using Lemma 3 we obtain $b\left(w\right)\prec q\left(w\right).$

Corollary 4.

Let$b$be convex univalent in U,$b\left(0\right)=1$, and$b\left(w\right)\ne 0$for all$w\in U,$and suppose that$b$satisfies:

$$\begin{gathered} Re\left\{\frac{t\sigma }{a_2}{b}^{\prime }\left(w\right)+\frac{\epsilon \left(\sigma +1\right)}{a_2}b\left(w\right)b^{\prime }\left(w\right)\right\}, \\ >0, \end{gathered}$$

(11) 

where,$\epsilon ,t\in ₵,0\ne a_2\in {₵}^{*}$, $w\in U$, and$w{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)$are starlike univalent in$U$.

Let$f\in \mathcal{A}$, satisfies the condition:

$${\left(\frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left(J_p^{n+1}\left(\theta ,\lambda \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left(J_p^n\left(\theta ,\lambda \right)f\left(w\right)\right)}{w^p}\right)}^{a_1}\in \Omega \left[b\left(0\right),1\right]\bigcap Q,$$

and$\mathcal{M}\left(p,n,\lambda ,\theta ,\epsilon ,a_1,a_2;w\right)$is univalent in$U$.

If$\left(t+\epsilon b\left(w\right)\right){\left(b\left(w\right)\right)}^{\sigma }+a_2{\left(b\left(w\right)\right)}^{\sigma -1}{b}^{\prime }\left(w\right)\prec \mathcal{M}\left(\sigma ,t,\epsilon ,h_{\mu },\mu ,a_1,a_2;w\right)$, then

$$b\left(w\right)\prec {\left(\frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left(J_p^{n+1}\left(\theta ,\lambda \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left(J_p^n\left(\theta ,\lambda \right)f\left(w\right)\right)}{w^p}\right)}^{a_1}.$$

4. Sandwich Results

By combining the above theories, we obtain the following two sandwich theories.

Theorem 5.

Let$b_1,b_2$be convex univalent in$U$, with$b_1\left(0\right)=b_2\left(0\right)=1$$Re{a}_2>0$and

$$Re\left(1+\frac{q^{″}\left(w\right)}{q^{\prime }\left(w\right)}\right)>\max \left\{0,-Re\frac{a_1}{a_2}\right\},$$

where$a_1>0,0\ne a_2\in ₵$.

If$f\in \mathcal{A}$and

$${\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}\in \Omega \left[1,1\right]\bigcap Q,$$

and

$$\begin{array}{l}\left(1-{\displaystyle \frac{a_2\left(p+\lambda \right)}{\theta }}\right){\left({\displaystyle \frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}}\right)}^{a_1}\\ +{\displaystyle \frac{a_2\left(p+\lambda \right)}{\theta }}{\left({\displaystyle \frac{{\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)}{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}}\right)}^{a_1}{\left({\displaystyle \frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}}\right)}^{a_1}\end{array}$$

is univalent in$U$, it satisfies:

$$\begin{gathered} b_1\left(w\right)+\frac{a_2}{a_1}w{b^{\prime }}_1\left(w\right)\prec \left(1-\frac{a_2\left(p+\lambda \right)}{\theta }\right){\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1} \\ +\frac{a_2\left(p+\lambda \right)}{\theta }{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)}{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}\right)}^{a_1}{\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}\prec b_2\left(w\right)+\frac{a_2}{a_1}w{b^{\prime }}_2\left(w\right), \end{gathered}$$

then$b_1\left(w\right)\prec {\left(\frac{{\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)}{w^p}\right)}^{a_1}\prec b_2\left(w\right)$.

Theorem 6.

Let$b_1,b_2$be convex univalent in$U$, with$b_1\left(0\right)=b_2\left(0\right)=1$, and let$f\in \mathcal{A}$satisfy the condition:

$${\left(\frac{{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-{\displaystyle \frac{\left(p+\lambda \right)}{\theta }}\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}\right)}^{a_1}\in \Omega \left[1,1\right]\bigcap Q,$$

and$\mathcal{M}\left(p,n,\lambda ,\theta ,\epsilon ,a_1,a_2;w\right)$is univalent in$U$.

If

$$\begin{gathered} \left(t+\epsilon b_1\left(w\right)\right){\left(b_1\left(w\right)\right)}^{\sigma }+a_2{\left(b_1\left(w\right)\right)}^{\sigma -1}{b_1}^{\prime }\left(w\right)\prec \mathcal{M}\left(p,n,\lambda ,\theta ,\epsilon ,a_1,a_2;w\right) \\ \prec \left(t+\epsilon b_2\left(w\right)\right){\left(b_2\left(w\right)\right)}^{\sigma }+a_2{\left(b_2\left(w\right)\right)}^{\sigma -1}{b_2}^{\prime }\left(w\right), \end{gathered}$$

then

$$b_1\left(w\right)\prec {\left(\frac{\frac{\left(p+\lambda \right)}{\theta }\left({\mathcal{B}}_{\theta ,\lambda }^{n+1, p}\left(a,c,\delta \right)f\left(w\right)\right)+\left(1-\frac{\left(p+\lambda \right)}{\theta }\right)\left({\mathcal{B}}_{\theta ,\lambda }^{n, p}\left(a,c,\delta \right)f\left(w\right)\right)}{w^p}\right)}^{a_1}\prec b_2\left(w\right).$$

5. Conclusions

In this paper, using the convolution (or Hadamard product) we defined the El-Ashwah and Drbuk linear operator, which is a multivalent function in the unit disk U and satisfied its specific relationship to derive the subordination, superordination, and some sandwich results for this operator using the properties of subordination and superordination concepts. The interesting results can be obtained for other operators using the same techniques of subordinations and superordinations.