New Results on Higher-Order Differential Subordination and Superordination for Univalent Analytic Functions Using a New Operator

Abstract

We present several new results for higher-order (fourth-order) differential subordination and superordination in this paper by using the new operator H_{α,β,δ,ξ,γ,n}f(v) and offer numerous new findings for fourth-order differential subordination and superordination. The innovative discoveries presented here are connected to those mentioned in previous articles. The differential subordination theory’s characteristics and outcomes are symmetric to the properties gained. Sandwich-type theorems are created by merging differential superordination theory with sandwich-type theorems.

1. Introduction

The discussion conducted in this paper uses the well-known concepts of differential subordination and differential superordination. The concept of differential subordination, introduced by Miller and Mocanu, was presented in the monograph published in 2000 [1] and the concept of differential superordination was introduced by the same authors as a dual concept to subordination in 2003 [2]. Third-order differential inequalities in the complex plane were considered in 1992 [3], and the concept of third-order differential subordination was introduced in 2011 by Antonino and Miller [4]. Further investigations were conducted on third-order differential subordination results for univalent analytic functions involving an operator [5,6,7,8] and, continuing the idea, the concept of fourth-order differential subordination was introduced and studied in 2020 [9,10]. Further results were published in 2021 and 2022 [11,12,13,14] regarding the new concepts of higher-order differential subordinations. The present paper continues this study.

$K\left(U\right)$ denotes the family of analytic functions in $U$ that have the form:

$$K\left[a,n\right]=\left\{f\in K\left(U\right):f\left(\nu \right)=a+a_n{\nu }^n+a_{n+1}{\nu }^{n+1}+\cdots \right\},a\in C,n\in N=\left\{1,2,\dots \right\},$$

and let $Y_n$ to be the collection of the form:

$$Y_n=\left\{f\in K\left(U\right):f\left(\nu \right)=\nu +a_{n+1}{\nu }^{n+1}+\cdots \right\},$$

where $Y_1=Y$, is the subclass of normalized analytic functions in $U.$ Further, indicated by $M$, the subfamily of $K\left(U\right)$ takes the form:

$$f\left(\nu \right)=\nu +\sum _{n=2}^{\infty }{a}_n{\nu }^n,\nu \in U,$$

(1)

which are univalent in $U$. For analytic functions $f\mathrm{and}F$, the function $f$ is said to be subordinate to $F$, if

$$f\left(\nu \right)=F\left(\Theta \left(\nu \right)\right),\left(\nu \in U\right),$$

where $\Theta \left(\nu \right)$ is analytic and $\Theta \left(0\right)=0,\left|\Theta \left(\nu \right)\right|<1$. This subordination is indicated by $f\left(\nu \right)\prec F\left(\nu \right)$.

If $f,g\in M$, where $f$ given by (1) and $g$ is defined by

$$g\left(\nu \right)=\nu +\sum _{n=2}^{\infty }{b}_n{\nu }^n,\nu \in U,$$

then

$$\left(f\ast g\right)\left(\nu \right)=\nu +\sum _{n=2}^{\infty }{a}_n{b}_n{\nu }^n=\left(g\ast f\right)\left(\nu \right).$$

Definition 1.

Suppose$f\in M,$where$\left|\nu \right|<1$. We create a new operator$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right):M\to M$, where

$$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)=\nu +{\displaystyle \sum _{n=2}^{\infty }}{\left(\frac{\delta \gamma +\xi +n}{\delta \gamma +\xi +1}\right)}^{\alpha +\beta +1}{a}_n{\nu }^n,$$

(2)

where

$$H_{0,-1,\delta ,\xi ,\gamma ,n}f\left(\nu \right)=f\left(\nu \right)$$

$$H_{1,-1,1,0,0,n}f\left(\nu \right)=\nu f^{\prime }\left(\nu \right)$$

$$\begin{gathered} H_{1,-1,\delta ,\xi ,\gamma ,n}f\left(\nu \right)=\frac{\left(\delta \gamma +\xi +1\right)f\left(\nu \right)-{\nu }^{3}f\left(\nu \right)+\left(n^3+3n\left(1-n\right)-1\right){\sum }_{n=2}^{\infty }{a}_n{\nu }^n}{\delta \gamma +\xi +1} \\ =\nu +{\displaystyle \sum _{n=2}^{\infty }}{\left(\frac{\delta \gamma +\xi +n}{\delta \gamma +\xi +1}\right)}^1{a}_n{\nu }^n \end{gathered}$$

$$⋮$$

$$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)=\nu +\sum _{n=2}^{\infty }{\left(\frac{\delta \gamma +\xi +n}{\delta \gamma +\xi +1}\right)}^{\alpha +\beta +1}{a}_n{\nu }^n,$$

where$\alpha ,\beta \in Z,\delta ,\xi ,\gamma \in C\backslash Z_0^{-}.$

After that, we obtain the relation:

$$\nu {\left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\right)}^{\prime }=\left(\delta \gamma +\xi +1\right)H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)-\left(\delta \gamma +\xi \right)H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right).$$

(3)

Many scholars have discussed and dealt with second-order differential subordination and superordination, see [11,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]). Several authors have recently written about superordination and the principle of third-order differential subordination. For examples of asymmetrical subordination and superordination on a third-order case, see ([3,4,30,31,32,33,34,35,36,37,38,39]). Antonino and Miller [4] presented basic concepts and expanded Miller and Mocanu’s [1] principle of second-order differential subordination in the open unit disk to the third-order case. The third-order case was expanded to fourth-order differential subordination by Atshan et al. [9,10,12,13,14]. The third-order case was extended to fourth-order differential subordination and discovered characteristics of functions meeting the fourth-order differential subordination:

$$\varphi \left(f\left(\nu \right),\nu f^{\prime }\left(\nu \right),{\nu }^2{f}^{″}\left(\nu \right),{\nu }^3{f}^{‴}\left(\nu \right),{\nu }^4{f}^{⁗}\left(\nu \right);\nu \right)\prec h\left(\nu \right),$$

where h is an analytic univalent function in U, f is an analytic function, and $\varphi :C^5\times U⟶C$. Subsequently, we then extended the third-order case to fourth-order differential superordination and uncovered properties of the functions $f$ that satisfy the fourth-order differential superordination with numerous applications:

$$h\left(\nu \right)\prec \varphi \left(f\left(\nu \right),\nu f^{\prime }\left(\nu \right),{\nu }^2{f}^{″}\left(\nu \right),{\nu }^3{f}^{‴}\left(\nu \right),{\nu }^4{f}^{⁗}\left(\nu \right);\nu \right),$$

where h is an analytic univalent function in U and f is a form analytic function $\varphi :C^5\times U⟶C.$

2. Problem Formulation

To demonstrate our primary findings, we need to understand the fundamental principles of fourth-order theory.

Definition 2

([4]). Indicate by $\left(\vartheta \right)$ the collection of all analytic functions $q$ on $\mathsf{Ū}\backslash E\left(q\right)$, where $E\left(q\right)=\{J:J\in \partial U$ and $li{m}_{\nu \to J}q\left(\nu \right)=\infty \}$ are such that $min\left|q^{\prime }\left(J\right)\right|=\gamma >0$ for $J\in \partial U\backslash E\left(q\right)$.

Furthermore, indicate by$\vartheta \left(a\right)$the subclass of functions q for which$q\left(0\right)=a$. Note that${\vartheta }_1=\vartheta \left(1\right)=\left\{q\left(\nu \right)\in \vartheta :q\left(0\right)=1\right\}$.

Definition 3

([9,10]). Assume that $k$ is univalent in $U$ and $\psi :C^5\times U\to C$. If the analytic functions $𝒷$ satisfy the fourth-order differential subordination condition,

$$\psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{⁗}\left(\nu \right);\nu )\prec k\left(\nu \right),$$

(4)

then the function$𝒷$ is named a solution of the differential subordination (4). A univalent function q is named a dominant of the solutions of the differential subordination if $𝒷\prec q$ for all $p$ satisfying (4). A dominant $\tilde{q}\left(\nu \right)$ that satisfies $\tilde{q}\prec q$ for all dominants q of (4) is named the best dominant.

Definition 4

([9,10]). Suppose that $\Omega$ is a set in $C$ and $q\in \vartheta$ . The admissible functions class ${\Phi }_n\left[\Omega ,q\right],\left(n\in N\backslash \left\{2\right\}\right)$ consists of those functions $\psi :C^5\times U\to C$ that fulfill the following admissibility condition: $\psi \left(r,s,t,u,b;\nu \right)\notin \Omega ,$

wherever

$$\begin{gathered} r=q\left(\tau \right),s=m\tau q^{\prime }\left(\tau \right),Re\left(\frac{t}{s}+1\right)\ge mRe\left(1+\frac{\tau q^{″}\left(\tau \right)}{q^{\prime }\left(\tau \right)}\right),Re\left(\frac{u}{s}\right)\ge m^{2}Re\left(\frac{{\tau }^2{q}^{‴}\left(\tau \right)}{q^{\prime }\left(\tau \right)}\right),Re\left(\frac{b}{s}\right) \\ \ge m^{3}Re\left(\frac{{\tau }^3{q}^{‴}\left(\tau \right)}{q^{\prime }\left(\tau \right)}\right),\left(\nu \in U,\tau \in \partial U\backslash E\left(q\right)andm\ge n\right). \end{gathered}$$

Theorem 1

([10]). Let $𝒷\in K\left[a,n\right],\left(n\in N\backslash \left\{2\right\}\right)$. In addition, let $q\in \vartheta$ and fulfill the conditions:

$$Re\left(\frac{{\tau }^2{q}^{‴}\left(\tau \right)}{q^{\prime }\left(\tau \right)}\right)\ge 0,\left|\left(\frac{{\nu }^2{𝒷}^{″}\left(\tau \right)}{q^{\prime }\left(\tau \right)}\right)\right|\le m^2,$$

(5)

where $\nu \in U,\tau \in \partial U\backslash E\left(q\right)$ and $m\ge n$. If $\psi \in {\Phi }_n\left[\Omega ,q\right],\Omega$ is a set in $C$ and $\psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{⁗}\left(\nu \right);\nu )\in \Omega$, then

$$𝒷\left(\nu \right)\prec q\left(\nu \right),\hspace{1em}\left(\nu \in U\right).$$

Definition 5

([9,10]). Suppose that $\psi :C^5\times U\to C$ and $k$ is an analytic function in $U$. If $𝒷\left(\nu \right)$ and

$$\psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{⁗}\left(\nu \right);\nu )$$

are univalent in U and satisfy the fourth-order differential superordination

$$k\left(\nu \right)\prec \psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{⁗}\left(\nu \right);\nu ),$$

(6)

then $𝒷\left(\nu \right)$ is called a solution of the differential superordination. An analytic function q($\nu$) is denoted a subordinate of the solutions of the differential superordination, or, more simply, a subordinate is an analytic function $q\left(\nu \right)$ if $q\left(\nu \right)\prec 𝒷\left(\nu \right)$ for all $𝒷\left(\nu \right)$ satisfying (6). A univalent subordinate $\tilde{q}\left(\nu \right)$ that satisfies the condition $q\left(\nu \right)\prec \tilde{q}\left(\nu \right)$ for all subordinates $q\left(\nu \right)$ of (6) is referred to as the best subordinate. We note that the best subordinate is unique up to a rotation of U.

Definition 6

([9,10]). Assume $q\left(\nu \right)\in K\left[a,n\right],q^{\prime }\left(\nu \right)\ne 0$ and $\Omega$ is a set in $C$. The class of admissible functions ${\Phi }_n^{\prime }\left[\Omega ,q\right]$ consists of those functions:

$$\psi :C^5\times \mathsf{Ū}\to C$$

that satisfy the following admissibility condition:

$$\psi \left(\tau ,s,t,u,b;\nu \right)\notin \Omega ,$$

wherever

$$\tau =q\left(\nu \right),\hspace{1em}s=\frac{1}{\kappa }\nu \tilde{q}\left(\tau \right),\hspace{1em}Re\left(\frac{t}{s}+1\right)\ge \frac{1}{\kappa }Re\left(\frac{\nu q^{″}\left(\nu \right)}{q^{\prime }\left(\nu \right)}+1\right)$$

and

$$Re\left(\frac{u}{s}\right)\ge \frac{1}{{\kappa }^2}Re\left(\frac{{\tau }^2{q}^{‴}\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right)u,\hspace{1em}Re\left(\frac{t}{s}\right)\ge \frac{1}{{\kappa }^3}Re\left(\frac{{\tau }^3{q}^{‴}\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right),$$

where $\tau \in \partial U,\nu \in U$ and $\kappa \ge n\ge 3$.

Theorem 2

([9,10]). Let $\psi \in {\Phi }_n^{\prime }\left[\Omega ,q\right]$ and $q\left(\nu \right)\in K\left[a,n\right]$ . If $\psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{⁗}\left(\nu \right);\nu )$ is univalent in U and $𝒷\left(\nu \right)\in \vartheta \left(a\right)$ satisfy the conditions:

$$Re\left(\frac{{\nu }^2{q}^{‴}\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right)\ge 0,\left|\left(\frac{{\nu }^2{𝒷}^{″}\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right)\right|\le \frac{1}{{\kappa }^2},$$

$\tau \in \partial U,\nu \in U$ and $\kappa \ge n\ge 3$, then

$$\Omega \subset \{(\psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{⁗}\left(\nu \right);\nu ),\nu \in U)\},$$

thus, $q\left(\nu \right)\prec 𝒷\left(\nu \right),\nu \in U.$

Using those known definitions and results, in the next two sections we prove new fourth-order differential subordination and superordination results involving the operator introduced in Definition 1. Further, in the last section of the paper, we combine the results for obtaining a sandwich-type theorem.

3. Fourth-Order Differential Subordination Results Using the New Operator H_{α,β,δ,ξ,γ,n}f(ν)

We give the class of admissible functions, which is required in proving differential subordination theorems using the new operator $H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)$ given by (2).

Definition 7.

Assume$q\in {\vartheta }_1\cap K_1$and$\Omega$is a set in$C$. Let${\theta }_H\left[\Omega ,q\right]$denote the class of admissible functions, which consists of functions$\gamma :C^5\times U\to C$that satisfy the following admission criteria:

$$\gamma \left(r,s,x,y,g;\nu \right)\notin \Omega ,$$

wherever

$$r=q\left(J\right),s=\frac{mJ{P}^{\prime }\left(\tau J\right)+\left(\delta \gamma +\xi \right)q\left(\nu \right)}{\delta \gamma +\xi +1},Re\left\{\frac{{(\delta \gamma +\xi +1)}^{2}x-{\left(\delta \gamma +\xi \right)}^{2}r}{\left(\delta \gamma +\xi +1\right)s-\left(\delta \gamma +\xi \right)r}-2\mu \right\}\ge mRe\left\{\frac{J{q}^{″}\left(J\right)}{q^{\prime }\left(J\right)}+1\right\},$$

$$\begin{gathered} Re\left\{\frac{{(\delta \gamma +\xi +1)}^2\left[\left(\delta \gamma +\xi +1\right)y-\left(3+3\left(\delta \gamma +\xi \right)\right)x\right]+(3{\left(\delta \gamma +\xi \right)}^2+2{\left(\delta \gamma +\xi \right)}^3)r}{\left(\delta \gamma +\xi +1\right)s-\left(\delta \gamma +\xi \right)r} \\ +(2+6\left(\delta \gamma +\xi \right)+3{\left(\delta \gamma +\xi \right)}^2)\right\}\ge m^{2}Re\left\{\frac{J^2{q}^{″}\left(J\right)}{q^{\prime }\left(J\right)}\right\} \end{gathered}$$

and

$$\begin{gathered} Re\{\frac{\left(\delta \gamma +\xi +1\right)[{(\delta \gamma +\xi +1)}^{3}g-{(\delta \gamma +\xi +1)}^2\left(6+4\left(\delta \gamma +\xi \right)\right)y+}{\left(\delta \gamma +\xi +1\right)s+\left(\delta \gamma +\xi \right)r} \\ \frac{\left(\delta \gamma +\xi +1\right)(11+18\left(\delta \gamma +\xi \right)+8{\left(\delta \gamma +\xi \right)}^2)x-(6s+22\left(\delta \gamma +\xi \right)+18{\left(\delta \gamma +\xi \right)}^2+8{\left(\delta \gamma +\xi \right)}^3)]}{\left(\delta \gamma +\xi +1\right)s+\left(\delta \gamma +\xi \right)r} \\ \frac{+(6\left(\delta \gamma +\xi \right)+11{\left(\delta \gamma +\xi \right)}^2+6{\left(\delta \gamma +\xi \right)}^3+3{\left(\delta \gamma +\xi \right)}^4)s}{\left(\delta \gamma +\xi +1\right)s+\left(\delta \gamma +\xi \right)r}\}\ge m^3\left\{\frac{J^3{q}^{‴}\left(J\right)}{q^{\prime }\left(J\right)}\right\}, \end{gathered}$$

where$\nu \in U,\mu \in \partial U\backslash E\left(q\right),\delta \gamma +\xi >-1$and$m\ge 3$.

Theorem 3.

Assume that$\gamma \in {\theta }_{\Gamma }\left[\Omega ,q\right]$. If$f\in M$and$q\in Q_1$satisfy the following conditions:

$$\left(\frac{J^2{q}^{″}\left(J\right)}{q^{\prime }\left(J\right)}\right)\ge 0,\left|\left(\frac{H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)}{q^{\prime }\left(J\right)}\right)\right|\le m^2,$$

(7)

and

$$\left\{\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\right):\nu \in U\right\}\subset \Omega ,$$

(8)

then$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\prec q\left(\nu \right),\left(\nu \in U\right)$.

Proof. 

Put

$$𝒷\left(\nu \right)=H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right).$$

(9)

Now, by separating (9) with regard to z and applying (3), we obtain:

$$H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)=\frac{\nu {𝒷}^{\prime }\left(\nu \right)+\left(\delta \gamma +\xi \right)𝒷\left(\nu \right)}{\left(\delta \gamma +\xi +1\right)}.$$

(10)

□

Additional calculations reveal that

$$H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)=\frac{{\nu }^2{𝒷}^{″}\left({\nu }^{\prime }\right)+\left(12\left(\delta \gamma +\xi \right)+1\right)\nu {𝒷}^{\prime }\left(\nu \right)+{\left(\delta \gamma +\xi \right)}^2𝒷\left(\nu \right)}{{(1+\mu )}^2},$$

(11)

$$\begin{array}{c}{H}_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\hfill \\ =\frac{{\left(\delta \gamma +\xi \right)}^3{\nu }^3{𝒷}^{‴}\left(\nu \right)+\left(3+3\left(\delta \gamma +\xi \right)\right){\nu }^2{𝒷}^{″}\left(\nu \right)+(1+3\left(\delta \gamma +\xi \right)+3{\left(\delta \gamma +\xi \right)}^2)\nu {𝒷}^{\prime }\left(\nu \right)+{\left(\delta \gamma +\xi \right)}^3𝒷\left(\nu \right)}{{(\delta \gamma +\xi +1)}^4}\hfill \end{array}$$

(12)

and

$$\begin{gathered} H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)=\frac{{\nu }^4{𝒷}^{⁗}\left(\nu \right)+\left(6+4\left(\delta \gamma +\xi \right)\right){\nu }^3{𝒷}^{‴}\left(\nu \right)+(7+12\left(\delta \gamma +\xi \right)+4{\left(\delta \gamma +\xi \right)}^2){\nu }^2{𝒷}^{″}\left(\nu \right)}{{(\delta \gamma +\xi +1)}^4} \\ \frac{+(1+4\left(\delta \gamma +\xi \right)+4{\left(\delta \gamma +\xi \right)}^2)+4{\left(\delta \gamma +\xi \right)}^3)\nu {𝒷}^{\prime }\left(\nu \right)+{\left(\delta \gamma +\xi \right)}^4𝒷\left(\nu \right)}{{(\delta \gamma +\xi +1)}^4}. \end{gathered}$$

(13)

We now show the change from $C^5$ to $C$ by

$$\begin{array}{cc}\hfill u\left(r,s,t,w,b\right)=& r,\hfill \\ & c\left(t,s,t,w,b\right)=\frac{s+\left(\delta \gamma +\xi \right)r}{\delta \gamma +\xi +1},x\left(r,s,t,w,b\right)\hfill \\ & =\frac{t+\left(1+2\left(\delta \gamma +\xi \right)\right)s+{\left(\delta \gamma +\xi \right)}^{2}r}{{(\delta \gamma +\xi +1)}^2},y\left(r,s,t,w,b\right)\hfill \\ & =\frac{w+\left(3+3\left(\delta \gamma +\xi \right)\right)t+\left(1+3\left(\delta \gamma +\xi \right)+3{\left(\delta \gamma +\xi \right)}^2\right)s+{\left(\delta \gamma +\xi \right)}^{3}r}{{(\delta \gamma +\xi +1)}^3},\hfill \end{array}$$

and

$$\begin{gathered} g\left(r,s,t,w,b;\nu \right)=\frac{b+\left(4\left(\delta \gamma +\xi \right)+6\right)w+(4{\left(\delta \gamma +\xi \right)}^2+12\left(\delta \gamma +\xi \right)+7)t}{{(\delta \gamma +\xi +1)}^4} \\ \frac{+(4{\left(\delta \gamma +\xi \right)}^3+4{\left(\delta \gamma +\xi \right)}^2+4\left(\delta \gamma +\xi \right)+1)s+{\left(\delta \gamma +\xi \right)}^{4}r}{{(\delta \gamma +\xi +1)}^4}. \end{gathered}$$

(14)

Assume

$$\begin{gathered} \psi \left(r,s,t,w,b;\nu \right)=\gamma \left(u,v,x,y,g;\nu \right), \\ =\gamma (r,\frac{s+\left(\delta \gamma +\xi \right)r}{\delta \gamma +\xi +1},\frac{t+\left(1+2\left(\delta \gamma +\xi \right)\right)s+{\left(\delta \gamma +\xi \right)}^{2}r}{{(\delta \gamma +\xi +1)}^2}, \\ \frac{w+\left(3+3\left(\delta \gamma +\xi \right)\right)t+(1+3\left(\delta \gamma +\xi \right)+3{\left(\delta \gamma +\xi \right)}^2)s+{\left(\delta \gamma +\xi \right)}^{3}r}{{(\delta \gamma +\xi +1)}^3}, \\ \frac{b+\left(6+4\left(\delta \gamma +\xi \right)\right)w+\left(7+12\left(\delta \gamma +\xi \right)+4{\mu }^2\right)t}{{(\left(\delta \gamma +\xi \right))}^4} \\ \frac{+(1+4\left(\delta \gamma +\xi \right)+4{\left(\delta \gamma +\xi \right)}^2+4{\left(\delta \gamma +\xi \right)}^3)s+{\left(\delta \gamma +\xi \right)}^{4}r}{{(\left(\delta \gamma +\xi \right))}^4};\nu ) \end{gathered}$$

(15)

We complete the proof with Theorem 1, and, using Equation (9) in (13), we can derive from (15) that

$$\begin{gathered} (\psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{⁗}\left(\nu \right);\nu )= \\ \gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \right). \end{gathered}$$

(16)

Therefore, (8) transforms into

$$\psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{⁗}\left(\nu \right);\nu )\in \Omega ,$$

and we observe that

$$\frac{t}{s}+1=\frac{{(\delta \gamma +\xi +1)}^{2}x-{\left(\delta \gamma +\xi \right)}^{2}u}{\left(\delta \gamma +\xi +1\right)v-\left(\delta \gamma +\xi \right)u}-2\mu ,$$

$$\begin{gathered} \frac{w}{s}=\frac{{(\delta \gamma +\xi +1)}^2\left[\left(\delta \gamma +\xi +1\right)y-\left(3+3\left(\delta \gamma +\xi \right)\right)x\right]+(3{\left(\delta \gamma +\xi \right)}^2+2{\left(\delta \gamma +\xi \right)}^3)u}{\left(\delta \gamma +\xi +1\right)v-\left(\delta \gamma +\xi \right)u} \\ +(2+6\left(\delta \gamma +\xi \right)+3{\left(\delta \gamma +\xi \right)}^2) \end{gathered}$$

and

$$\begin{gathered} \frac{b}{s}=\frac{\left(\delta \gamma +\xi +1\right)[{(\delta \gamma +\xi +1)}^{3}g-{(\delta \gamma +\xi +1)}^2\left(6+4\left(\delta \gamma +\xi \right)\right)y}{\left(\delta \gamma +\xi +1\right)v-\left(\delta \gamma +\xi \right)u} \\ +\left(\delta \gamma +\xi +1\right)(11+18\left(\delta \gamma +\xi \right)+8{\left(\delta \gamma +\xi \right)}^2)x-(6+22\left(\delta \gamma +\xi \right)+18{\mu }^2+8{\mu }^3)v \\ +(6\left(\delta \gamma +\xi \right)+11{\left(\delta \gamma +\xi \right)}^2+6{\left(\delta \gamma +\xi \right)}^3+3{\left(\delta \gamma +\xi \right)}^4)u. \end{gathered}$$

As a result, we obtain the equivalent of Definition 7, admissibility condition for ${\theta }_{\Gamma }\left[\Omega ,q\right]$ with Definition 4, and n = 3 admissibility condition for $\psi \in {\Phi }_n\left[\Omega ,q\right]$. Using Theorem 1 and Equation (7), we obtain $𝒷\left(\nu \right)=H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\prec q\left(\nu \right)$.

Corollary 1.

Assume the function $q\left(\nu \right)$ is univalent in$U$with$q\left(0\right)=1$and$\Omega \subset C$. Assume$\gamma \in {\theta }_H\left[\Omega ,q_{\Gamma }\right]$for some$\Gamma \in \left(0,1\right)$such that$q_{\Gamma }\left(\nu \right)=q\left(\Gamma \nu \right)$. If the function$f\left(\nu \right)\in Y$and$q_{\Gamma }\left(\nu \right)$satisfy the following conditions:

$$Re\left(\frac{J^2{q}^{‴}\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right)\ge 0,\left|\left(\frac{H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right)\right|\le m^2,\left(\nu \in U,J\in \partial U\backslash E\left(q_{\Gamma }\right)\right)$$

(17)

and

$$\left\{\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \right)\in U\right\}\subset \Omega ,$$

then

$$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\prec q\left(\nu \right),\nu \in U.$$

Proof. 

Using the preceding theorem, we obtain $H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\prec q_{\Gamma }\left(\nu \right)$. We then have the result from $q_{\Gamma }\left(\nu \right)\prec q\left(\nu \right),\nu \in U$. This concludes the Corollary proof (1).

If $\Omega \ne C$ is a simply linked domain, then $\Omega =k\left(U\right)$, taking a conformal mapping $k\left(\nu \right)$ of $U$ onto into account. The class ${\theta }_H\left[K\left(U\right),q\right]$ may be expressed as ${\theta }_H\left[k,q\right]$ in this situation.

We may now derive the following two findings from the preceding theorem and corollary. □

Theorem 4.

Let$\gamma \in {\theta }_H\left[k,q\right]$. If$q\in Q_1$and$f\in M$fulfills condition (7) and

$$\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \right)\prec k\left(\nu \right),$$

(18)

then$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\prec q\left(\nu \right),\nu \in U.$

Corollary 2.

Let$q\left(\nu \right)$be univalent functions in$U,q\left(0\right)=1$and$\Omega \subset C$. Assume$\gamma \in {\theta }_H\left[k,q_{\Gamma }\right]$for several$\Gamma \in \left(0,1\right)$such that$q_{\Gamma }\left(\nu \right)=q\left(\Gamma \nu \right)$. If the function$f\in M$and$q_{\Gamma }$satisfies conditions (17) and

$$\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \right)\prec k\left(\nu \right),$$

(19)

then

$$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\prec q\left(\nu \right),\nu \in U.$$

The following theorem determines the optimum dominant of the differential subordination (18).

Theorem 5.

Assume$\gamma :C^5\times U\to C$. Moreover, suppose that the function$k$is univalent in$U$and that the next differential equation:

$$\begin{gathered} \gamma (q\left(\nu \right),\frac{\nu q^{\prime }\left(\nu \right)+\left(\delta \gamma +\xi \right)q\left(\nu \right)}{\delta \gamma +\xi +1},\frac{{\nu }^2{q}^{″}\left(\nu \right)+\left(1+2\left(\delta \gamma +\xi \right)\right)\nu q^{\prime }\left(\nu \right)+{\left(\delta \gamma +\xi \right)}^{2}q\left(\nu \right)}{{(\delta \gamma +\xi +1)}^2}, \\ \frac{{\nu }^3{q}^{‴}\left(\nu \right)+\left(3+3\left(\delta \gamma +\xi \right)\right){\nu }^2{q}^{″}\left(\nu \right)+(1+3\left(\delta \gamma +\xi \right)+3{\left(\delta \gamma +\xi \right)}^2)\nu q^{\prime }\left(\nu \right)+{\left(\delta \gamma +\xi \right)}^{3}q\left(\nu \right)}{{(\delta \gamma +\xi +1)}^3}, \\ \frac{{\nu }^4{q}^{⁗}\left(\nu \right)+\left(6+4\left(\delta \gamma +\xi \right)\right){\nu }^3{q}^{‴}\left(\nu \right)+(7+12\left(\delta \gamma +\xi \right)+4{\left(\delta \gamma +\xi \right)}^2){\nu }^2{q}^{″}\left(\nu \right)}{{(\delta \gamma +\xi +1)}^4} \\ \frac{+(1+4\left(\delta \gamma +\xi \right)+4{\left(\delta \gamma +\xi \right)}^2+4{\left(\delta \gamma +\xi \right)}^3)\nu q^{\prime }\left(\nu \right)+{\left(\delta \gamma +\xi \right)}^{4}q\left(\nu \right);\nu }{{(\delta \gamma +\xi +1)}^4})=k\left(\nu \right), \end{gathered}$$

(20)

has a solution of$q\left(\nu \right)$with$q\left(0\right)=1$, which satisfies condition (7). If the function$f\in M$satisfies condition (18) and if$\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \right)$is analytic in$U$, then$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\prec q\left(\nu \right)$and$q\left(\nu \right)$is the best dominant.

Proof. 

Using Theorem 3, it is possible to demonstrate that $q\left(\nu \right)$ is a dominant of Equation (18), since $q\left(\nu \right)$ satisfies (20), implying that $q\left(\nu \right)$ is a solution of (18). Thus, $q\left(\nu \right)$ will be more dominant than other dominants. As a result, $q\left(\nu \right)$ is the best dominant.

It is possible to establish that $q\left(\nu \right)$ is a dominant of Equation (18) using Theorem 3, since $q\left(\nu \right)$ satisfies (20), meaning that $q\left(\nu \right)$ is a solution of (18) and $q\left(\nu \right)$ is thus more dominant than other dominants. As a consequence, the best dominant function is q(z).

Now, we put $q\left(\nu \right)=M\nu ,M>0$, and using Definition (7), the class of admissible functions ${\theta }_H\left[\Omega ,q\right]$, denoted by ${\theta }_H\left[\Omega ,M\right]$, is given below. □

Definition 8.

Let$M>0$and consider$\Omega$is a set in$C$. The class${\theta }_H\left[\Omega ,M\right]$of admissible functions consists of those functions$\gamma :C^5\times U\to C$that satisfy the admissibility condition:

$$\begin{gathered} \gamma (M{e}^{i\theta },\frac{k+\left(\delta \gamma +\xi \right)}{\delta \gamma +\xi +1}M{e}^{i\theta },\frac{L+[\left(2\left(\delta \gamma +\xi \right)+1\right)k+{\left(\delta \gamma +\xi \right)}^2]M{e}^{i\theta }}{{(\delta \gamma +\xi +1)}^2},\frac{N+\left(3\left(\delta \gamma +\xi \right)+3\right)L+(3{\left(\delta \gamma +\xi \right)}^2+3\left(\delta \gamma +\xi \right)+1)k+{\left(\delta \gamma +\xi \right)}^3}{{(\delta \gamma +\xi +1)}^3}, \\ \frac{A+\left(4\left(\delta \gamma +\xi \right)+6\right)N+(4{\left(\delta \gamma +\xi \right)}^2+12\left(\delta \gamma +\xi \right)+7)L+[(4{\left(\delta \gamma +\xi \right)}^3+4{\left(\delta \gamma +\xi \right)}^2+4\left(\delta \gamma +\xi \right)+1)k+{\left(\delta \gamma +\xi \right)}^4]M{e}^{i\theta }}{{(\delta \gamma +\xi +1)}^4};\nu )\notin \Omega , \end{gathered}$$

(21)

such that$1>-\left(\delta \gamma +\xi \right),\nu \in U,Re\left(L{e}^{-i\theta }\right)\ge \left(k-1\right)kM,Re\left(N{e}^{-i\theta }\right)\ge 0$and$Re\left(A{e}^{-i\theta }\right)\ge 0$for all$\theta \in R$and$k\ge 3$.

Theorem 6.

Assume that$\gamma \in {\theta }_H\left[\Omega ,M\right]$. If$f\in M$fulfills the conditions:$\left|H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\right|\le$$k^{2}M,k\ge 3,M>0$, and$\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \right)\in \Omega ,$then

$$\left|H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\right|<M.$$

Now, talking$\Omega =q\left(U\right)=\{w:\left|w\right|<M\}$, the class${\theta }_H\left[\Omega ,M\right]$is simply denoted by${\theta }_H\left[M\right]$.

Theorem 7.

Assume$k\ge 3$,$M>0,\left(\delta \gamma +\xi \right)>-1$. If$f\in M$satisfies the conditions$\left|H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\right|\le$$k^{2}M$and$\left|{(\delta \gamma +\xi +1)}^4{H}_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)-\left(\delta \gamma +\xi \right){(\delta \gamma +\xi +1)}^3{H}_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\right|<(|1+3\left(\delta \gamma +\xi \right)+\lambda {\left(\delta \gamma +\xi \right)}^2+{\left(\delta \gamma +\xi \right)}^3\mid +2\mid 7+9\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^2\mid )3M$, then

$$\left|H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\right|<M.$$

Proof. 

Suppose that $\gamma \left(r,s,x,y,g,\nu \right)={(\delta \gamma +\xi +1)}^{4}g-\left(\delta \gamma +\xi \right){(\delta \gamma +\xi +1)}^{3}y,\Omega =k\left(U\right)$ such that

$$k\left(\nu \right)=(|1+3\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^2+{\left(\delta \gamma +\xi \right)}^3\mid +2|7+9\left(\delta \gamma +\xi \right)+\delta {\left(\delta \gamma +\xi \right)}^2\mid )3M\nu ,M>0.$$

Now, by applying Theorem 6, we show that $\gamma \in {\theta }_{H,1}\left[\Omega ,M\right]$. Because

$$\begin{gathered} |\gamma (M{e}^{i\theta },\frac{k+\left(\delta \gamma +\xi \right)}{\delta \gamma +\xi +1}M{e}^{i\theta },\frac{L+[\left(2\left(\delta \gamma +\xi \right)+1\right)k+{\left(\delta \gamma +\xi \right)}^2]M{e}^{i\theta }}{{(\delta \gamma +\xi +1)}^2},\frac{N+\left(3\left(\delta \gamma +\xi \right)+3\right)L+(3{\left(\delta \gamma +\xi \right)}^2+3\left(\delta \gamma +\xi \right)+1)k+{\left(\delta \gamma +\xi \right)}^3}{{(\delta \gamma +\xi +1)}^3}, \\ \frac{A+\left(4\left(\delta \gamma +\xi \right)+6\right)N+(4{\left(\delta \gamma +\xi \right)}^2+12\left(\delta \gamma +\xi \right)+7)L+[(4{\left(\delta \gamma +\xi \right)}^3+4{\left(\delta \gamma +\xi \right)}^2+4\left(\delta \gamma +\xi \right)+1)k+{\left(\delta \gamma +\xi \right)}^4]M{e}^{i\theta }}{{(\delta \gamma +\xi +1)}^4};\nu )\mid \\ =|A+\left(3\left(\delta \gamma +\xi \right)+6\right)N+({\left(\delta \gamma +\xi \right)}^2+9\left(\delta \gamma +\xi \right)+7)L+[({\left(\delta \gamma +\xi \right)}^3+{\left(\delta \gamma +\xi \right)}^2+3\left(\delta \gamma +\xi \right)+1)k+{\left(\delta \gamma +\xi \right)}^4]kM{e}^{i\theta }\mid \\ =|A{e}^{-i\theta }+\left(6+3\left(\delta \gamma +\xi \right)\right)N{e}^{-i\theta }+(7+9\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^2)L{e}^{-i\theta }+(1+3\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^3)kM\mid \\ \ge Re\left(A{e}^{-i\theta }\right)+\left|\left(6+3\left(\delta \gamma +\xi \right)\right)\right|Re\left(N{e}^{-i\theta }\right)+|(7+9\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^2)\mid L{e}^{-i\theta }+|(1+3\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^2+{\left(\delta \gamma +\xi \right)}^3)\mid kM \\ \ge |(1+3\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^2+{\left(\delta \gamma +\xi \right)}^3)\mid kM+2|(7+9\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^2)\mid k\left(k-1\right)M \\ \ge (|(1+3\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^2+{\left(\delta \gamma +\xi \right)}^3)\mid +2|(7+9\left(\delta \gamma +\xi \right)+{\left(\delta \gamma +\xi \right)}^2)\mid )3M, \end{gathered}$$

such that

$$Re\left(A{e}^{-i\theta }\right)\ge 0,Re\left(N{e}^{-i\theta }\right)\ge 0andRe\left(L{e}^{-i\theta }\right)\ge \left(k-1\right)kMaforall\theta \in R,\nu \in Uandk\ge 3.$$

The proof is complete. □

4. Results Using the Operator for Fourth-Order Differential Superordination H_{α,β,δ,ξ,γ,n}f(ν)

We use $H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)$ to introduce fourth-order differential superordination. The class of admissible functions for this major goal is defined as follows:

Definition 9.

Let$q^{\prime }\left(\nu \right)\ne 0,q\in K_1$and let$\Omega$be a set in$C$. Those functions$\gamma :C^5\times U\to C$that meet the admissibility criterion make up the admissible class${\theta }_H^{\prime }[\Omega ,q]$

$$\gamma \left(r,s,x,y,g;J\right)\in \Omega ,$$

where

$$r=q\left(\nu \right),s=\frac{\nu J{q}^{\prime }\left(\nu \right)+mq\left(\nu \right)}{\left(\delta \gamma +\xi +1\right)m},Re\left\{\frac{{(\delta \gamma +\xi +1)}^{2}x-{\left(\delta \gamma +\xi \right)}^{2}r}{\left(\delta \gamma +\xi +1\right)s-\left(\delta \gamma +\xi \right)r}-2\left(\delta \gamma +\xi \right)\right\}\ge \frac{1}{m}Re\left\{\frac{\nu q^{″}\left(\nu \right)}{q^{\prime }\left(\nu \right)}+1\right\},$$

$$\begin{array}{c}Re\{\frac{{(\delta \gamma +\xi +1)}^2\left[\left(\delta \gamma +\xi +1\right)y-\left(3+3\left(\delta \gamma +\xi \right)\right)x\right]+(3{\left(\delta \gamma +\xi \right)}^2+2{\left(\delta \gamma +\xi \right)}^3)r}{\left(\delta \gamma +\xi +1\right)s-\left(\delta \gamma +\xi \right)r}\\ +(2+6\left(\delta \gamma +\xi \right)+3{\left(\delta \gamma +\xi \right)}^2)\}\ge \frac{1}{m^2}Re\left\{\frac{{\nu }^2{q}^{‴}\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right\},\end{array}$$

and

$$\begin{gathered} Re\{\frac{\left(\delta \gamma +\xi +1\right)[{(\delta \gamma +\xi +1)}^{3}g-{(\delta \gamma +\xi +1)}^2\left(6+4\left(\delta \gamma +\xi \right)\right)y+\left(\delta \gamma +\xi +1\right)(11+18\left(\delta \gamma +\xi \right)+8{\left(\delta \gamma +\xi \right)}^2)x}{\left(\delta \gamma +\xi +1\right)s+\left(\delta \gamma +\xi \right)r} \\ \frac{-(6+22\left(\delta \gamma +\xi \right)+18(\delta \gamma +\xi {)}^2+8(\delta \gamma +\xi {)}^3)s]+(6\left(\delta \gamma +\xi \right)+11{(\delta \gamma +\xi )}^2+{(\delta \gamma +\xi )}^3+3(\delta \gamma +\xi {)}^4)r}{\left(\delta \gamma +\xi +1\right)s+\left(\delta \gamma +\xi \right)r}\} \\ \ge \frac{1}{m^3}Re\left\{\frac{{\nu }^3{q}^{⁗}\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right\}, \end{gathered}$$

where$\nu \in U,J\in \partial U,\left(\delta \gamma +\xi \right)\in C\backslash Z_0^{-},Z_0^{-}=\left\{0,-1,-2,\dots \right\}$and$m\ge 3.$

Theorem 8.

Assume that$\gamma \in {\theta }_H^{\prime }[\Omega ,q]$. If$f\in M$and$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\in Q_1$satisfy the conditions

$$Re\left(\frac{{\nu }^2{q}^{‴}\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right)\ge 0,\left|\left(\frac{H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)}{q^{\prime }\left(\nu \right)}\right)\right|\le \frac{1}{m^2}$$

(22)

and

$$\gamma \left(H_{\alpha :\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \in U\right)$$

is univalent in$U$with

$$\Omega \subset \left\{\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \right),\nu \in U\right\},$$

(23)

then$q\left(\nu \right)\prec H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)$.

Proof. 

By (9) and (15), respectively, define the functions $𝒷\left(\nu \right)$. We have $\gamma \in {\theta }_H^{\prime }\left[\Omega ,q\right]$. Therefore, from (16) and (23), we obtain

$$\Omega \subset \{\psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{⁗}\left(\nu \right);\nu \in U)\}.$$

Take note of the fact that the admission requirement for $\gamma \in {\theta }_H^{\prime }\left[\Omega ,q\right]$ in Definition 9 is the same as the admissibility condition for $\psi$ in Definition 6 with n = 3. As a result of using (7) and Theorem 2 and knowing $\psi \in {\theta }_H^{\prime }[\Omega ,q]$, we obtain $q\left(\nu \right)\prec 𝒷\left(\nu \right)=H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)$. The theorem has been fully proved.

If $k\left(\nu \right)$ of U onto $\Omega$ is equal to k(U), and $\Omega \ne C$ is a simply connected domain, the class ${\theta }_H^{\prime }[k\left(U\right),q]$ is represented as ${\theta }_H^{\prime }\left[k,q\right].$

The theorem below follows directly from the previous theorem. □

Theorem 9.

Considering the analytic functions$k\left(\nu \right)$in$U$and$\gamma \in {\theta }_H^{\prime }[k\left(U\right),q]$. If$f\in M,H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\in Q_1$and$q\in K_1$satisfies condition (22),

$$\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \in U\right)$$

is univalent in$U$, and

$$\Omega \subset \left\{\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \in U\right)\right\},$$

(24)

then$q\left(\nu \right)\prec H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)$.

Proof. 

The proof of the theorem is identical to that of Theorem 8; hence it will not be included here. □

Theorem 10.

Let$\gamma :C^5\times U\to C$,$k\left(\nu \right)$be analytic functions in$U$, and$\psi$be defined by (15). Suppose that the differential equation

$$\psi (𝒷\left(\nu \right),\nu {𝒷}^{\prime }\left(\nu \right),{\nu }^2{𝒷}^{″}\left(\nu \right),{\nu }^3{𝒷}^{‴}\left(\nu \right),{\nu }^4{𝒷}^{‴}\left(\nu \right);\nu \in U)=k\left(\nu \right),$$

(25)

has a solution$q\left(\nu \right)\in Q_1.$If$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\in Q_1,q\in K_1,q^{\prime }\left(\nu \right)\ne 0$and$f\in M$satisfy conditions (7) and (22),

$$\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \in U\right)$$

is univalent in$U$, and

$$\Omega \subset \left\{\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \in U\right)\right\},$$

then$q\left(\nu \right)\prec H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)$and$q\left(\nu \right)$is the best subordinate of (24).

Proof. 

The theorem’s proof is similar to that of Theorem 5; hence it will not be provided here. □

5. Sandwich-Type Results

We now have the sandwich-type result using Theorems 5 and 9.

Theorem 11.

Consider$k_1\left(\nu \right)$and$q_1\left(\nu \right)$to be two analytic functions in U, as well as$q_2\left(\nu \right)\in Q_1$where$q_1\left(0\right)=q_2\left(0\right)=1$. Let$k_1\left(\nu \right)$also be univalent in U and$\gamma \in$${\theta }_H[k_2,q_2]\cap {\theta }_H^{\prime }[k_1,q_1]$. If$H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\in Q_1\cap K,f\in M$,

$$\gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \in U\right)$$

is univalent in$U$, and the two conditions (7) and (22) are satisfied as$k_1\left(\nu \right)\prec \gamma \left(H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +1,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +2,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +3,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right),H_{\alpha +4,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right);\nu \in U\right)\prec k_2\left(\nu \right)$, then

$$q_1\left(\nu \right)\prec H_{\alpha ,\beta ,\delta ,\xi ,\gamma ,n}f\left(\nu \right)\prec q_2\left(\nu \right).$$

6. Conclusions

In Definition 1 it is introduced a new operator and regarding it following the fourth-order differential subordination and superordination method are defined classes of admissible functions. It is obtained several properties for the defined classes related fourth-order subordination in Section 3 and fourth-order superordination in Section 4. Combining the results from Section 3 and Section 4 it is obtained sandwich-type theorem in Section 5.

The method used in the paper can be applied to other operators to obtain fourth-order differential subordinations and superordinations. New classes of univalent functions could be defined by using the operator introduced in Definition 1 and studied using the fourth-order differential subordination and superordination method using the admissible condition from Definitions 7–9.