Third-Order Differential Subordination Features of Meromorphic Functions: Erdelyi–Kober Model Integral Operator Application

Abstract

This study is concerned with the class of p-valent meromorphic functions, represented by the series $f\left(\zeta \right)={\zeta }^{-p}+{\sum }_{k=1-p}^{\infty }{d}_k{\zeta }^k$, with the domain characterized by $0<\left|\zeta \right|<1$. We apply an Erdelyi–Kober-type integral operator to derive two recurrence relations. From this, we draw specific conclusions on differential subordination and differential superordination. By looking into suitable classes of permitted functions, we obtain various outcomes, including results analogous to sandwich-type theorems. The operator used can provide generalizations of previous operators through specific parameter choices, thus providing more corollaries.

1. Introduction

Let $\mathcal{F}(\Delta )$ denote the collection of all analytic functions defined on the open unit disc $\Delta =\{\zeta :\zeta \in \mathbb{C}:|\zeta |<1\}$. For $n\in \mathbb{N}$ and $d\in \mathbb{C}$, let $\mathcal{F}[d,n]$ be the subclass of $\mathcal{F}(\Delta )$ consisting of functions of the form $f\left(\zeta \right)=d+d_n{\zeta }^n+d_{n+1}{\zeta }^{n+1}+...$. We note that ${\mathcal{F}}_1=\mathcal{F}[1,1]$. $f\left(\zeta \right)\prec g\left(\zeta \right)$ indicates that f is subordinate to g in $\Delta$ if there is an analytic function $l\left(\zeta \right)$ with $l\left(0\right)=0$, and $\left|l\left(\zeta \right)\right|<1$; $\left(\zeta \in \Delta \right),$ such that $f\left(\zeta \right)=g\left(l\right(\zeta \left)\right)$$\left(\zeta \in \Delta \right)$ [1,2,3]. Let ${\Sigma }_p$ be the family of all functions that have the following representation:

$$f\left(\zeta \right)={\zeta }^{-p}+{\displaystyle \sum _{k=1-p}^{\infty }}{d}_k{\zeta }^k(p\in \mathbb{N}),$$

(1)

which are p-valent and analytical in ${\Delta }^{\ast }=\Delta \setminus \left\{0\right\}=\{\zeta :\zeta \in \mathbb{C}:0<|\zeta |<1\}$. Considering El-Ashwah and Hassan’s current work in [4] (see also El-Ashwah [5]), for $f\left(\zeta \right)\in {\Sigma }_p$, $a,c\in \mathbb{C}$, and $\mu >0$, such that $Re\left(c\right)\ge Re\left(a\right)>\mu p$, the operator

$$J_{p,\mu }^{a,c}:{\Sigma }_p⟶{\Sigma }_p$$

is expressed by

$$J_{p,\mu }^{a,c}f\left(\zeta \right)=\left\{\begin{array}{ccc}{\displaystyle \frac{\Gamma (c-\mu p)}{\Gamma (a-\mu p)\Gamma (c-a)}}\underset{0}{\overset{1}{\int }}{t}^{a-1}{(1-t)}^{c-a-1}f\left(\zeta t^{\mu }\right)dt;& & Re(c-a)>0,\\ & & \\ f\left(\zeta \right);& & a=c.\end{array}\right.$$

(2)

From (2), it can easily be seen that series representation of the integral operator $J_{p,\mu }^{a,c}$ can be expressed as follows:

$$J_{p,\mu }^{a,c}f\left(\zeta \right)={\zeta }^{-p}+{\displaystyle \frac{\Gamma (c-\mu p)}{\Gamma (a-\mu p)}}{\displaystyle \sum _{k=1-p}^{\infty }}{\displaystyle \frac{\Gamma (a+\mu k)}{\Gamma (c+\mu k)}}{d}_k{\zeta }^k,$$

(3)

$$\left(a,c\in \mathbb{C},\mu >0,Re\left(c\right)\ge Re\left(a\right)>\mu p,p\in \mathbb{N}\right).$$

Remark 1.

By specifying values for a, c, p, and μ in (3), we obtain certain operators as special cases of the operator $J_{p,\mu }^{a,c}$; these operators were introduced by various authors as follows:

(i) 

$J_{1,A}^{a,c}f\left(\zeta \right)=I_A^{a,c}f\left(\zeta \right)$ (see El-Ashwah [5]);

(ii) 

$J_{p,1}^{a+p,c+p}f\left(\zeta \right)={\ell }_p(a,c)f\left(\zeta \right)$ (see Liu and Srivastava [6]);

(iii) 

$J_{1,1}^{\nu +1,n+2}f\left(\zeta \right)={\ell }_{n,\nu }f\left(\zeta \right)$ (see Yuan et al. [7]);

(iv) 

$J_{p,1}^{n+2p,p+1}f\left(\zeta \right)=D^{n+p-1}f\left(\zeta \right)$ (see Uralegaddi and Somanatha [8], see also Aouf [9] and Aouf and Srivastava [10]).

Thus, the equivalent conclusions for other well-known operators can be derived from the results of this work by selecting specific values for the parameters a, c, p, and μ, as previously described.

It is easily verified from definition (3) that:

$$\zeta \left(J_{p,\mu }^{a,c}f\left(\zeta \right)\right)\prime ={\displaystyle \frac{a-\mu p}{\mu }}{J}_{p,\mu }^{a+1,c}f\left(\zeta \right)-{\displaystyle \frac{a}{\mu }}{J}_{p,\mu }^{a,c}f\left(\zeta \right),$$

(4)

and

$$\zeta \left(J_{p,\mu }^{a,c+1}f\left(\zeta \right)\right)\prime ={\displaystyle \frac{c-\mu p}{\mu }}{J}_{p,\mu }^{a,c}f\left(\zeta \right)-{\displaystyle \frac{c}{\mu }}{J}_{p,\mu }^{a,c+1}f\left(\zeta \right).$$

(5)

The following lemmas and definitions are necessary to present the primary results:

Definition 1

([11]). Suppose $\psi :{\mathbb{C}}^4\times \Delta ⟶\mathbb{C}$. Let $h\left(\zeta \right)$ be a univariate function in Δ, and let $p\left(\zeta \right)$ be analytic in Δ. Additionally, if the following third-order differential subordination is satisfied:

$$\psi (p\left(\zeta \right),\zeta p\prime \left(\zeta \right),{\zeta }^{2}p″\left(\zeta \right),{\zeta }^{3}p‴\left(\zeta \right);\zeta )\prec h\left(\zeta \right),$$

(6)

then, $p\left(\zeta \right)$ is a solution of (6). A univalent function $q\left(\zeta \right)$ is considered a dominating solution if $p\left(\zeta \right)\prec q\left(\zeta \right)$ for every $p\left(\zeta \right)$ that satisfies (6). A dominant $\tilde{q}\left(\zeta \right)$ is considered the best if $\tilde{q}\left(\zeta \right)\prec q\left(\zeta \right)$ for all dominants $q\left(\zeta \right)$ of (6).

Definition 2

([12]). Assume that $\psi :{\mathbb{C}}^4\times \Delta ⟶\mathbb{C}$ and $h\left(\zeta \right)$ are analytic in Δ. If $p\left(\zeta \right)$ and $\psi (p\left(\zeta \right),\zeta p\prime \left(\zeta \right),{\zeta }^{2}p″\left(\zeta \right),{\zeta }^{3}p‴\left(\zeta \right);\zeta )$ are univalent in Δ, and fulfilling the following third-order superordination:

$$h\left(\zeta \right)\prec \psi (p\left(\zeta \right),\zeta p\prime \left(\zeta \right),{\zeta }^{2}p″\left(\zeta \right),{\zeta }^{3}p‴\left(\zeta \right);\zeta ),$$

(7)

then, $p\left(\zeta \right)$ is said to be a solution of the superordination. An analytic function $q\left(\zeta \right)$ is a subordinant of the differential superordination solutions, or simply a subordinant, if $q\left(\zeta \right)\prec p\left(\zeta \right)$ for every $p\left(\zeta \right)$ fulfills (7). A univalent subordinant $\tilde{q}\left(\zeta \right)$ has the property that $q\left(\zeta \right)\prec \tilde{q}\left(\zeta \right)$ for all subordinants. $q\left(\zeta \right)$ of (7) is known as the best subordinant.

Definition 3

([11]). Let Q denote the set of all functions q that are analytic and injective on $\overline{\Delta }\setminus E\left(q\right)$, where

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

(8)

such that $q\prime \left(\xi \right)\ne 0$ for $\xi \in \partial \Delta \setminus E\left(q\right)$. Moreover, let $Q\left(a\right)$ indicate the subclass of Q where $q\left(0\right)=a$, and $Q_1:=Q\left(1\right)=\left\{q\left(\zeta \right)\in Q:q\left(0\right)=1\right\}$.

Definition 4

([11]). Let Ω be a set in $\mathbb{C}$, with $q\in Q$ and $n\in \mathbb{N}\setminus \left\{1\right\}$. The admissible functions ${\Psi }_n[\Omega ,q]$ include functions $\psi :{\mathbb{C}}^4\times \Delta \to \mathbb{C}$ that meet the aforementioned admissibility circumstance:

$$\psi (r,s,t,u;\zeta )\notin \Omega ,$$

(9)

such that

$$r=q\left(\xi \right),s=k\xi q\prime \left(\xi \right),$$

$$Re\left\{{\displaystyle \frac{t}{s}}+1\right\}\ge kRe\left\{1+{\displaystyle \frac{\xi q″\left(\xi \right)}{q\prime \left(\xi \right)}}\right\},$$

(10)

$$Re\left\{{\displaystyle \frac{u}{s}}\right\}\ge k^{2}Re\left\{{\displaystyle \frac{{\xi }^{2}q‴\left(\xi \right)}{q\prime \left(\xi \right)}}\right\},$$

where $\zeta \in \Delta ,\xi \in \partial \Delta \setminus E\left(q\right)$, and $k\ge n$.

Definition 5

([12]). Assume that $\Omega \subseteq \mathbb{C}$, $q\in \mathcal{F}[a,n]$, such that $q\prime \left(\zeta \right)\ne 0$ and $n\in \{2,3,4,...\}$. The set of admissible functions ${\Psi }_n^{\prime }[\Omega ,q]$ are the functions $\psi :{\mathbb{C}}^4\times \overline{\Delta }\to \mathbb{C}$ that meet the following admissibility criteria

$$\psi (r,s,t,u;\xi )\in \Omega ,$$

(11)

whenever

$$r=q\left(\zeta \right),s={\displaystyle \frac{\zeta q\prime \left(\zeta \right)}{m}},$$

$$Re\left\{{\displaystyle \frac{t}{s}}+1\right\}\le {\displaystyle \frac{1}{m}}Re\left\{1+{\displaystyle \frac{\zeta q″\left(\zeta \right)}{q\prime \left(\zeta \right)}}\right\},$$

(12)

$$Re\left\{{\displaystyle \frac{u}{s}}\right\}\le {\displaystyle \frac{1}{m^2}}Re\left\{{\displaystyle \frac{{\zeta }^{2}q‴\left(\zeta \right)}{q\prime \left(\zeta \right)}}\right\},$$

where $\zeta \in \Delta$, $\xi \in \partial \Delta$ and $m\ge n.$

Lemma 1

([11], Theorem 1). Suppose that $q\in Q\left(a\right)$ and $p\in \mathcal{F}[a,n]$, such that $n\ge 2$. Additionally, the following criteria are fulfilled

$$Re\left\{{\displaystyle \frac{\xi q″\left(\xi \right)}{q\prime \left(\xi \right)}}\right\}\ge 0,\left|{\displaystyle \frac{\zeta p\prime \left(\zeta \right)}{q\prime \left(\xi \right)}}\right|\le k,$$

(13)

where $k\ge n$, $\xi \in \partial \Delta \setminus E\left(q\right)$, and $\zeta \in \Delta$. Assuming that $\Omega \subseteq \mathbb{C}$, $\psi \in {\Psi }_n[\Omega ,q]$ and

$$\psi (p\left(\zeta \right),\zeta p\prime \left(\zeta \right),{\zeta }^{2}p″\left(\zeta \right),{\zeta }^{3}p‴\left(\zeta \right);\zeta )\in \Omega ,$$

(14)

therefore,

$$p\left(\zeta \right)\prec q\left(\zeta \right)\left(\zeta \in \Delta \right).$$

(15)

Lemma 2

([12], Theorem 8). Assuming $\psi \in {\Psi }_n^{\prime }[\Omega ,q]$. If $p\in Q\left(a\right)$, $\psi (p\left(\zeta \right),\zeta p\prime \left(\zeta \right),{\zeta }^{2}p″\left(\zeta \right)$, ${\zeta }^{3}p‴\left(\zeta \right);\zeta )$ is univalent in Δ, and $q\in \mathcal{F}[a,n]$, meet the following circumstances:

$$Re\left\{{\displaystyle \frac{\xi q″\left(\xi \right)}{q\prime \left(\xi \right)}}\right\}\ge 0,\left|{\displaystyle \frac{\zeta p\prime \left(\zeta \right)}{q\prime \left(\xi \right)}}\right|\le m,$$

(16)

where $m\ge n\ge 2$, $\zeta \in \Delta$, and $\xi \in \partial \Delta$, then

$$\Omega \subset \left\{\psi (p\left(\zeta \right),\zeta p\prime \left(\zeta \right),{\zeta }^{2}p″\left(\zeta \right),{\zeta }^{3}p‴\left(\zeta \right);\zeta ):\zeta \in \Delta \right\},$$

(17)

leads to

$$q\left(\zeta \right)\prec p\left(\zeta \right)\left(\zeta \in \Delta \right).$$

(18)

Oros et al. [13] indicated new conclusions about the fundamental challenge of providing adequate criteria for identifying the best subordinant of a third-order differential superordination using the Gaussian hypergeometric function. Seoudy [14] demonstrated the use of third-order differential subordination for admissible functions in $\Delta$, described by the k-Ruscheweyh derivative operator. Oros et al. [15] proposed the dual notion of 3rd-order fuzzy differential superordination. The conclusions of this study provide essential and sufficient criteria for determining subordinants of a third-order fuzzy differential superordination, as well as selecting the best possible subordinant for such fuzzy differential superordination, where available. Shexoa et al. [16] suggested specific sets of admissible functions and discussed certain applications of third-order differential subordination for the normalized analytic functions related to Zeta–Riemann fractional differential operators. Soren and Cotîrlă [17] (see also [18]) studied fuzzy differential subordination and superordination findings for analytic functions, including Pascal distribution series and the Mittag–Leffler function. They specified the conditions for a function to act as fuzzy dominant and the fuzzy subordinant in fuzzy differential subordination and superordination We refer to [19,20,21,22] for further reading. We identify some appropriate classes of admissible functions and examine some third-order differential subordination and superordination properties of multivalent meromorphic functions involving the operator $J_{p,\mu }^{a,c}$ specified by (3), by using the third-order differential subordination established by Antonino and Miller [11] in $\Delta$ and the third-order differential superordination results in $\Delta$ obtained by Tang et al. [12] (see also [23]).

2. Subordination Results

Throughout this study, we will make the assumption that $\mu >0$, $a,c\in \mathbb{C}$, $Re\left(c\right)\ge Re\left(a\right)>\mu p$, and $p\in \mathbb{N}$, except if otherwise specified. We derive several third-order differential subordination insights. The following definition provides a class of admissible functions for this purpose.

Definition 6.

Suppose that $q\in Q_1\cap {\mathcal{F}}_1$ and $\Omega \subseteq \mathbb{C}$. ${\Phi }_J[\Omega ,q]$ is the class of admissible functionsm which includes the functions $\varphi :{\mathbb{C}}^4\times \Delta \to \mathbb{C}$ that meet the subsequent admissibility requirement:

$$\varphi \left(\alpha ,\beta ,\gamma ,\delta ;\zeta \right)\notin \Omega ,$$

(19)

as

$$\alpha =q\left(\xi \right),\beta ={\displaystyle \frac{n\xi q\prime \left(\xi \right)+\frac{c-\mu p}{\mu }q\left(\xi \right)}{\frac{c-\mu p}{\mu }}},$$

(20)

$$Re\left\{{\displaystyle \frac{c-\mu p-1}{\mu }}\left({\displaystyle \frac{\gamma -\alpha }{\beta -\alpha }}\right)-{\displaystyle \frac{2\left(c-\mu p\right)-1}{\mu }}\right\}\ge nRe\left\{{\displaystyle \frac{\xi q″\left(\xi \right)}{q\prime \left(\xi \right)}}+1\right\},$$

(21)

$$Re\left\{{\displaystyle \frac{\left(c-\mu p-1\right)\left[\left(c-\mu p-2\right)\left(\delta -\alpha \right)-3\left(c-(p-1)\mu -1\right)\left(\gamma -\alpha \right)\right]}{{\mu }^2\left(\beta -\alpha \right)}}\right.$$

$$\left.+{\displaystyle \frac{\left(2\mu -1\right)\left(3(c-\mu p)+\mu -1\right)+3{\left(c-\mu p\right)}^2}{{\mu }^2}}\right\}\ge n^{2}Re\left\{{\displaystyle \frac{{\xi }^{2}q‴\left(\xi \right)}{q\prime \left(\xi \right)}}\right\},$$

(22)

where $\zeta \in \Delta$, $n\in \left\{2,3,4,...\right\}$, and $\xi \in \partial \Delta \setminus E\left(q\right)$.

Theorem 1.

Suppose that $\varphi \in {\Phi }_J[\Omega ,q]$. Additionally, $q\in Q_1\cap {\mathcal{F}}_1$ and $f\in {\Sigma }_p$ meet the criteria

$$Re\left\{{\displaystyle \frac{\xi q″\left(\xi \right)}{q\prime \left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\right|\le \mu n\left|{\displaystyle \frac{q\prime \left(\xi \right)}{c-\mu p}}\right|.$$

(23)

If

$$\left\{\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right):\zeta \in \Delta \right\}\subset \Omega ,$$

(24)

$$\left(n\in \left\{2,3,4,...\right\},\xi \in \partial \Delta \setminus E\left(q\right)and\zeta \in \Delta \right),$$

therefore,

$${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\prec q\left(\zeta \right)(\zeta \in \Delta ).$$

(25)

Proof. 

Let g be defined by

$$g\left(\zeta \right)={\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right).$$

(26)

Making use of (5) and (26), we have

$${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)={\displaystyle \frac{\zeta g\prime \left(\zeta \right)+{\eta }_{1}g\left(\zeta \right)}{{\eta }_1}},$$

(27)

where ${\eta }_1=\left(c-\mu p\right)/\mu .$ Further computations show that

$${\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right)={\displaystyle \frac{{\zeta }^{2}g″\left(\zeta \right)+{\eta }_2\zeta g\prime \left(\zeta \right)+{\eta }_{3}g\left(\zeta \right)}{{\eta }_3}},$$

(28)

where ${\eta }_2=1+{\displaystyle \frac{2\left(c-\mu p\right)-1}{\mu }}$ and ${\eta }_3={\displaystyle \frac{\left(c-\mu p\right)\left(c-\mu p-1\right)}{{\mu }^2}},$ Also,

$${\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right)={\displaystyle \frac{{\zeta }^{3}g‴\left(\zeta \right)+{\eta }_4{\zeta }^{2}g″\left(\zeta \right)+{\eta }_5\zeta g\prime \left(\zeta \right)+{\eta }_{6}g\left(\zeta \right)}{{\eta }_6}},$$

(29)

where

$${\eta }_4=3\left(1+\frac{c-\mu p-1}{\mu }\right),$$

$${\eta }_5=\left(1+{\displaystyle \frac{2\left(c-\mu p\right)-1}{\mu }}+{\displaystyle \frac{\left(c-\mu p\right)\left(c-\mu p-1\right)}{{\mu }^2}}+{\displaystyle \frac{c-\mu p-2}{\mu }}\left(1+{\displaystyle \frac{2\left(c-\mu p\right)-1}{\mu }}\right)\right),$$

and

$${\eta }_6={\displaystyle \frac{\left(c-\mu p\right)\left(c-\mu p-1\right)\left(c-\mu p-2\right)}{\mu 3}}.$$

The transformation from ${\mathbb{C}}^4$ to $\mathbb{C}$ is defined by

$$\alpha (r,s,t,u)=r,\beta (r,s,t,u)={\displaystyle \frac{s+{\eta }_{1}r}{{\eta }_1}},$$

(30)

$$\gamma (r,s,t,u)={\displaystyle \frac{t+{\eta }_{2}s+{\eta }_{3}r}{{\eta }_3}},$$

(31)

and

$$\delta (r,s,t,u)={\displaystyle \frac{u+{\eta }_{4}t+{\eta }_{5}s+{\eta }_{6}r}{{\eta }_6}}.$$

(32)

Let

$$\begin{array}{cc}\hfill \psi \left(r,s,t,u;\zeta \right)& =\varphi \left(\alpha ,\beta ,\gamma ,\delta ;\zeta \right)\hfill \\ \hfill & =\varphi \left(r,{\displaystyle \frac{s+{\eta }_{1}r}{{\eta }_1}},{\displaystyle \frac{t+{\eta }_{2}s+{\eta }_{3}r}{{\eta }_3}},{\displaystyle \frac{u+{\eta }_{4}t+{\eta }_{5}s+{\eta }_{6}r}{{\eta }_6}};\zeta \right).\hfill \end{array}$$

(33)

Using Lemma 1, (26)–(29), and (30)–(33), we have

$$\psi \left(g\left(\zeta \right),\zeta g\prime \left(\zeta \right),{\zeta }^{2}g″\left(\zeta \right),{\zeta }^{3}g‴\left(\zeta \right);\zeta \right)$$

$$=\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right).$$

(34)

Hence, (24) shows that

$$\psi \left(g\left(\zeta \right),\zeta g\prime \left(\zeta \right),{\zeta }^{2}g″\left(\zeta \right),{\zeta }^{3}g‴\left(\zeta \right);\zeta \right)\in \Omega .$$

(35)

By using Equations (30)–(32), we obtain

$${\displaystyle \frac{t}{s}}+1={\displaystyle \frac{c-\mu p-1}{\mu }}\left({\displaystyle \frac{\gamma -\alpha }{\beta -\alpha }}\right)-{\displaystyle \frac{2\left(c-\mu p\right)-1}{\mu }},$$

(36)

$$\begin{array}{cc}\hfill {\displaystyle \frac{u}{s}}& ={\displaystyle \frac{\left(c-\mu p-1\right)\left[\left(c-\mu p-2\right)\left(\delta -\alpha \right)-3\left(c-(p-1)\mu -1\right)\left(\gamma -\alpha \right)\right]}{{\mu }^2\left(\beta -\alpha \right)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mu }^2}}\left[\left(2\mu -1\right)\left(3(c-\mu p)+\mu -1\right)+3{\left(c-\mu p\right)}^2\right].\hfill \end{array}$$

(37)

As $\varphi \in {\Phi }_J[\Omega ,q]$ in Definition 6 has the same admissibility condition as $\psi \in {\Psi }_n[\Omega ,q]$ in Definition 4, the two conditions are identical. Thus, $g\left(\zeta \right)\prec q\left(\zeta \right)$$(\zeta \in \Delta )$, or equivalently, ${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\prec q\left(\zeta \right)$$(\zeta \in \Delta )$, may be obtained by using (23) and Lemma 1. This accomplishes the proof of Theorem 1. □

The following result is a modification of Theorem 1 for the case when the behvior of q on $\partial \Delta$ is unknown, using the identical reasons as in ([1], Corollary 2.3b.1, p. 30).

Corollary 1.

Assuming that q is univalent in Δ, such that $q\left(0\right)=1$ and $\Omega \subset \mathbb{C}$. Additionally, $\varphi \in {\Phi }_J[\Omega ,q_{\rho }]$ for some $\rho \in (0,1)$ where $q_{\rho }\left(\zeta \right)=q\left(\rho \zeta \right)$. If the following criteria are met by functions $f\in {\Sigma }_p$ and $q_{\rho }$:

$$Re\left\{{\displaystyle \frac{\xi q_{\rho }^{″}\left(\xi \right)}{q_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\right|\le \mu n\left|{\displaystyle \frac{q_{\rho }^{\prime }\left(\xi \right)}{c-\mu p}}\right|,$$

(38)

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right)\in \Omega ,$$

(39)

then

$${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\prec q\left(\zeta \right),$$

$$(\xi \in \partial \Delta \setminus E\left(q\right),n\in \left\{2,3,4,...\right\}\mathit{and}\zeta \in \Delta ),$$

Proof. 

It follows from Theorem 1 that

$${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\prec q_{\rho }\left(\zeta \right).$$

(40)

The following subordination property may now be used to infer the following proof:

$$q_{\rho }\left(\zeta \right)\prec q\left(\zeta \right).$$

(41)

This accomplishes the proof of the corollary. □

Let $\Omega =h\left(\Delta \right)$ for some conformal mapping h of $\Delta$ onto $\Omega$, where $\Omega \ne \mathbb{C}$ is a simply connected domain. In this case, the class ${\Phi }_J[h(\Delta ),q]$ is denoted by ${\Phi }_J[h,q]$. Two direct consequences of Theorem 1 and Corollary 1 are as follows.

Theorem 2.

If $q\in Q_1$, $f\in {\Sigma }_p$ and $\varphi \in {\Phi }_J[h,q]$ fulfill the requirements listed below:

$$Re\left\{{\displaystyle \frac{\xi q″\left(\xi \right)}{q\prime \left(\xi \right)}}\right\}\ge 0and\left|{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\right|\le \mu n\left|{\displaystyle \frac{q\prime \left(\xi \right)}{c-\mu p}}\right|.$$

(42)

If

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right)\prec h\left(\zeta \right),$$

(43)

therefore

$${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\prec q\left(\zeta \right),$$

$$(n\in \left\{2,3,4,...\right\},\xi \in \partial \Delta \setminus E\left(q\right),\zeta \in \Delta ).$$

Corollary 2.

Let q, be univalent in Δ with $q\left(0\right)=1$, and let $\Omega \subset \mathbb{C}$. Assume $\varphi \in {\Phi }_J[h,q_{\rho }]$, where $q_{\rho }\left(\zeta \right)=q\left(\rho \zeta \right)$ for any $\rho \in (0,1)$. If the following criteria are satisfied by the functions $f\in {\Sigma }_p$ and $q_{\rho }$:

$$Re\left\{{\displaystyle \frac{\xi q_{\rho }^{″}\left(\xi \right)}{q_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\right|\le \mu n\left|{\displaystyle \frac{q_{\rho }^{\prime }\left(\xi \right)}{c-\mu p}}\right|,$$

(44)

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right)\prec h\left(\zeta \right),$$

(45)

thus

$${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\prec q\left(\zeta \right),$$

(46)

$$(n\in \left\{2,3,4,...\right\},\xi \in \partial \Delta \setminus E\left(q\right)\mathit{and}\zeta \in \Delta ).$$

The best dominant of the subordination in (24) or (39) is obtained using the following theorem.

Theorem 3.

Suppose that $\varphi :{\mathbb{C}}^4\times \Delta ⟶\mathbb{C}$, $h\left(\zeta \right)$ is univalent in Δ, and ψ is defined in (33). Moreover, consider the differential equation

$$\psi \left(q\left(\zeta \right),\zeta q\prime \left(\zeta \right),{\zeta }^{2}q″\left(\zeta \right),{\zeta }^{3}q‴\left(\zeta \right);\zeta \right)=h\left(\zeta \right),$$

(47)

has the solution $q\left(\zeta \right)\in Q_1\cap {\mathcal{F}}_1$, which fulfills the circumstances in (23). If function $f\in {\Sigma }_p$ satisfies the condition in (39) and

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right),$$

is analytic in Δ; therefore,

$${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\prec q\left(\zeta \right),$$

(48)

additionally, $q\left(\zeta \right)$ is the best dominant.

Proof. 

We determine that q is a dominant of (43) by using Theorem 1. q is a solution of (43) as it adheres to (47). As a result, all dominants will dominate q. Thus, the best dominant is q. □

Then, we present ${\tilde{\Phi }}_J[\Omega ,q]$, a new admissible class as follows:

Definition 7.

Assuming that $q\in Q_1\cap {\mathcal{F}}_1$ and $\Omega \subseteq \mathbb{C}$, then set of admissible functions ${\tilde{\Phi }}_J[\Omega ,q]$ is the set of functions $\varphi :{\mathbb{C}}^4\times \Delta \to \mathbb{C}$ that fulfill the following admissibility criteria:

$$\varphi \left(\alpha ,\beta ,\gamma ,\delta ;\zeta \right)\notin \Omega ,$$

(49)

as

$$\alpha =q\left(\xi \right),\beta ={\displaystyle \frac{n\xi q\prime \left(\xi \right)+\frac{a-\mu p}{\mu }q\left(\xi \right)}{\frac{a-\mu p}{\mu }}},$$

(50)

$$Re\left\{{\displaystyle \frac{a-\mu p+1}{\mu }}\left({\displaystyle \frac{\gamma -\alpha }{\beta -\alpha }}\right)-{\displaystyle \frac{2\left(a-\mu p\right)+1}{\mu }}\right\}\ge nRe\left\{{\displaystyle \frac{\xi q″\left(\xi \right)}{q\prime \left(\xi \right)}}+1\right\},$$

(51)

$$Re\left\{{\displaystyle \frac{\left(a-\mu p+1\right)\left[\left(a-\mu p+2\right)\left(\delta -\alpha \right)-3\left(a-(p-1)\mu +1\right)\left(\gamma -\alpha \right)\right]}{{\mu }^2\left(\beta -\alpha \right)}}\right.$$

$$\left.+{\displaystyle \frac{\left(2\mu +1\right)\left(3(a-\mu p)+\mu +1\right)+3{\left(a-\mu p\right)}^2}{{\mu }^2}}\right\}\ge n^{2}Re\left\{{\displaystyle \frac{{\xi }^{2}q‴\left(\xi \right)}{q\prime \left(\xi \right)}}\right\},$$

(52)

such that $\xi \in \partial \Delta \setminus E\left(q\right)$, $n\in \left\{2,3,4,...\right\}$, and $\zeta \in \Delta$.

Theorem 4.

Suppose that $f\in {\Sigma }_p$, $q\in Q_1\cap {\mathcal{F}}_1$, and $\varphi \in {\tilde{\Phi }}_J[\Omega ,q]$ fulfill the criteria:

$$Re\left\{{\displaystyle \frac{\xi q″\left(\xi \right)}{q\prime \left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\right|\le \mu n\left|{\displaystyle \frac{q\prime \left(\xi \right)}{a-\mu p}}\right|.$$

(53)

If

$$\left\{\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right):\zeta \in \Delta \right\}\subset \Omega ,$$

(54)

$$\left(n\in \left\{2,3,4,...\right\},\xi \in \partial \Delta \setminus E\left(q\right)\mathit{and}\zeta \in \Delta \right),$$

then

$${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\prec q\left(\zeta \right)(\zeta \in \Delta ).$$

(55)

Proof. 

Define g as follows

$$g\left(\zeta \right)={\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right).$$

(56)

which is an analytic function. Applying (4) and (56), we obtain

$${\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right)={\displaystyle \frac{\zeta g\prime \left(\zeta \right)+{\lambda }_{1}g\left(\zeta \right)}{{\lambda }_1}},$$

(57)

where ${\lambda }_1=\left(a-\mu p\right)/\mu .$ Further computations lead to

$${\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right)={\displaystyle \frac{{\zeta }^{2}g″\left(\zeta \right)+{\lambda }_2\zeta g\prime \left(\zeta \right)+{\lambda }_{3}g\left(\zeta \right)}{{\lambda }_3}},$$

(58)

where ${\lambda }_2=1+[\left(2\left(a-\mu p\right)+1\right)/\mu ],$ ${\lambda }_3=\left(a-\mu p\right)\left(a-\mu p+1\right)/\left({\mu }^2\right)$ and

$${\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right)={\displaystyle \frac{{\zeta }^{3}g‴\left(\zeta \right)+{\lambda }_4{\zeta }^{2}g″\left(\zeta \right)+{\lambda }_5\zeta g\prime \left(\zeta \right)+{\lambda }_{6}g\left(\zeta \right)}{{\lambda }_6}},$$

(59)

such that

$${\lambda }_4=3\left(1+\frac{a-\mu p+1}{\mu }\right),$$

$${\lambda }_5=1+\frac{2\left(a-\mu p\right)+1}{\mu }+\frac{\left(a-\mu p\right)\left(a-\mu p+1\right)}{{\mu }^2}+\frac{a-\mu p+2}{\mu }\left(1+\frac{2\left(a-\mu p\right)+1}{\mu }\right),$$

and

$${\lambda }_6={\displaystyle \frac{\left(a-\mu p\right)\left(a-\mu p+1\right)\left(a-\mu p+2\right)}{\mu 3}}.$$

We provide a transformation from ${\mathbb{C}}^4$ into $\mathbb{C}$, as follows:

$$\alpha (r,s,t,u)=r,\beta (r,s,t,u)={\displaystyle \frac{s+{\lambda }_{1}r}{{\lambda }_1}},$$

(60)

$$\gamma (r,s,t,u)={\displaystyle \frac{t+{\lambda }_{2}s+{\lambda }_{3}r}{{\lambda }_3}},$$

(61)

and

$$\delta (r,s,t,u)={\displaystyle \frac{u+{\lambda }_{4}t+{\lambda }_{5}s+{\lambda }_{6}r}{{\lambda }_6}}.$$

(62)

Let

$$\begin{array}{cc}\hfill \psi \left(r,s,t,u;\zeta \right)& =\varphi \left(\alpha ,\beta ,\gamma ,\delta ;\zeta \right)\hfill \\ \hfill & =\varphi \left(r,{\displaystyle \frac{s+{\lambda }_{1}r}{{\lambda }_1}},{\displaystyle \frac{t+{\lambda }_{2}s+{\lambda }_{3}r}{{\lambda }_3}},{\displaystyle \frac{u+{\lambda }_{4}t+{\lambda }_{5}s+{\lambda }_{6}r}{{\lambda }_6}};\zeta \right).\hfill \end{array}$$

(63)

Using Lemmas 1, (56)–(59), and (60)–(63), we have

$$\psi \left(g\left(\zeta \right),\zeta g\prime \left(\zeta \right),{\zeta }^{2}g″\left(\zeta \right),{\zeta }^{3}g‴\left(\zeta \right);\zeta \right)$$

$$=\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right).$$

(64)

Hence, (64) leads to

$$\psi \left(g\left(\zeta \right),\zeta g\prime \left(\zeta \right),{\zeta }^{2}g″\left(\zeta \right),{\zeta }^{3}g‴\left(\zeta \right);\zeta \right)\in \Omega .$$

(65)

Using (60)–(62), then we have

$${\displaystyle \frac{t}{s}}+1={\displaystyle \frac{a-\mu p+1}{\mu }}\left({\displaystyle \frac{\gamma -\alpha }{\beta -\alpha }}\right)-{\displaystyle \frac{2\left(a-\mu p\right)+1}{\mu }},$$

(66)

$$\begin{array}{cc}\hfill {\displaystyle \frac{u}{s}}& ={\displaystyle \frac{\left(a-\mu p+1\right)\left[\left(a-\mu p+2\right)\left(\delta -\alpha \right)-3\left(a-(p-1)\mu +1\right)\left(\gamma -\alpha \right)\right]}{{\mu }^2\left(\beta -\alpha \right)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{{\mu }^2}}\left[\left(2\mu +1\right)\left(3(a-\mu p)+\mu +1\right)+3{\left(a-\mu p\right)}^2\right].\hfill \end{array}$$

(67)

As $\varphi \in {\tilde{\Phi }}_J[\Omega ,q]$ in Definition 7 has the same admissibility condition as $\psi \in {\Psi }_n[\Omega ,q]$ in Definition 4, the two conditions are identical. Thus, $g\left(\zeta \right)\prec q\left(\zeta \right)$$(\zeta \in \Delta )$, or equivalently, ${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\prec q\left(\zeta \right)$$(\zeta \in \Delta )$, may be obtained by using (53) and Lemma 1. This completes the proof. □

Similarly, the following assertion is an extension of Theorem 4 for the case when the behavior of q on $\partial \Delta$ is unknown, using the same indications as in ([1], Corollary 2.3b.1, p. 30).

Corollary 3.

Suppose that q is univalent in Δ, such that $q\left(0\right)=1$, and let $\varphi \in {\tilde{\Phi }}_J[\Omega ,q_{\rho }]$ for some $\rho \in (0,1)$, such that $q_{\rho }\left(\zeta \right)=q\left(\rho \zeta \right)$. Also, assume that $f\in {\Sigma }_p$ and $q_{\rho }$ fulfill the following criteria:

$$Re\left\{{\displaystyle \frac{\xi q_{\rho }^{″}\left(\xi \right)}{q_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0,$$

(68)

and

$$\left|{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\right|\le \mu n\left|{\displaystyle \frac{q_{\rho }^{\prime }\left(\xi \right)}{a-\mu p}}\right|.$$

(69)

Additionally, if

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right)\in \Omega ,$$

(70)

then

$${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\prec q\left(\zeta \right),$$

$$(\xi \in \partial \Delta \setminus E\left(q\right),n\in \left\{2,3,4,...\right\}\mathit{and}\zeta \in \Delta ).$$

Proof. 

It follows from Theorem 4 that

$${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\prec q_{\rho }\left(\zeta \right).$$

(71)

The subsequent subordination characteristic can now be used to infer the corollary’s proof:

$$q_{\rho }\left(\zeta \right)\prec q\left(\zeta \right).$$

(72)

This finishes the proof. □

$\Omega =h\left(\Delta \right)$ for some conformal mapping $h\left(\zeta \right)$ of $\Delta$ onto $\Omega$ if $\Omega \ne \mathbb{C}$ is a simply connected domain. Here, ${\tilde{\Phi }}_J[h(\Delta ),q]$ is the written form of the class ${\tilde{\Phi }}_J[h,q]$. The following two conclusions are direct outcomes of the Corollary 3 and Theorem 4.

Theorem 5.

For $\varphi \in {\tilde{\Phi }}_J[h,q]$, and for $f\in {\Sigma }_p$ and $q\in Q_1$ fulfilling the following two criteria:

$$Re\left\{{\displaystyle \frac{\xi q″\left(\xi \right)}{q\prime \left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\right|\le \mu n\left|{\displaystyle \frac{q\prime \left(\xi \right)}{a-\mu p}}\right|.$$

(73)

If

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right)\prec h\left(\zeta \right),$$

(74)

therefore

$${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\prec q\left(\zeta \right),$$

$$(\xi \in \partial \Delta \setminus E\left(q\right),n\in \left\{2,3,4,...\right\}\mathit{and}\zeta \in \Delta ).$$

Corollary 4.

Suppose that $\Omega \subset \mathbb{C}$ and q is univalent in Δ, such that $q\left(0\right)=1$. Also, let $\varphi \in {\tilde{\Phi }}_J[h,q_{\rho }]$, where $\rho \in (0,1)$ and $q_{\rho }\left(\zeta \right)=q\left(\rho \zeta \right)$. Additionally, assume that $f\in {\Sigma }_p$ and $q_{\rho }$ satisfy the following criteria:

$$Re\left\{{\displaystyle \frac{\xi q_{\rho }^{″}\left(\xi \right)}{q_{\rho }^{\prime }\left(\xi \right)}}\right\}\ge 0\mathit{and}\left|{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\right|\le \mu n\left|{\displaystyle \frac{q_{\rho }^{\prime }\left(\xi \right)}{a-\mu p}}\right|.$$

(75)

Now, if

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right)\prec h\left(\zeta \right),$$

(76)

then

$${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\prec q\left(\zeta \right),$$

(77)

$$(\xi \in \partial \Delta \setminus E\left(q\right),n\in \left\{2,3,4,...\right\},\mathit{and}\zeta \in \Delta ).$$

The best dominant of the subordination in (54) or (70) is obtained using the following theorem.

Theorem 6.

Assuming that h is univalent in Δ, and that $\varphi :{\mathbb{C}}^4\times \Delta ⟶\mathbb{C}$ and ψ is given in (63). Additionally, suppose that the differential equation

$$\psi \left(q\left(\zeta \right),\zeta q\prime \left(\zeta \right),{\zeta }^{2}q″\left(\zeta \right),{\zeta }^{3}q‴\left(\zeta \right);\zeta \right)=h\left(\zeta \right),$$

(78)

has a solution $q\left(\zeta \right)\in Q_1\cap {\mathcal{F}}_1$ that satisfies (53). Now, if $f\in {\Sigma }_p$ complies with condition in (70), and

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right),$$

is an analytic function in Δ, then

$${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\prec q\left(\zeta \right),$$

(79)

moreover, $q\left(\zeta \right)$ is the best dominant.

Proof. 

We determined that q is a dominant of (74) by using Theorem 4. As q satisfies (74), it is also a solution of (78). As a result, all dominants will dominate q. The best dominant is thus q. □

3. Superordination Results

We derive several third-order differential superordination insights in this section. The class of admissible functions for this purpose is defined as follows:

Definition 8.

Suppose that $q\in {\mathcal{F}}_1$ with $q\prime \left(\zeta \right)\ne 0$, and $\Omega \subseteq \mathbb{C}$. The set of admissible functions ${\Phi }_J^{\prime }[\Omega ,q]$ includes the following functions: $\psi :{\mathbb{C}}^4\times \overline{\Delta }\to \mathbb{C}$, which meet the subsequent requirements for admission:

$$\varphi \left(\alpha ,\beta ,\gamma ,\delta ;\xi \right)\in \Omega ,$$

(80)

such that

$$\alpha =q\left(\zeta \right),\beta ={\displaystyle \frac{\zeta q\prime \left(\zeta \right)+m\frac{c-\mu p}{\mu }q\left(\zeta \right)}{m\frac{c-\mu p}{\mu }}},$$

(81)

$$Re\left\{{\displaystyle \frac{c-\mu p-1}{\mu }}\left({\displaystyle \frac{\gamma -\alpha }{\beta -\alpha }}\right)-{\displaystyle \frac{2\left(c-\mu p\right)-1}{\mu }}\right\}\le {\displaystyle \frac{1}{m}}Re\left\{1+{\displaystyle \frac{\zeta q″\left(\zeta \right)}{q\prime \left(\zeta \right)}}\right\},$$

(82)

$$Re\left\{{\displaystyle \frac{\left(c-\mu p-1\right)\left[\left(c-\mu p-2\right)\left(\delta -\alpha \right)-3\left(c-(p-1)\mu -1\right)\left(\gamma -\alpha \right)\right]}{{\mu }^2\left(\beta -\alpha \right)}}\right.$$

$$\left.+{\displaystyle \frac{\left[\left(2\mu -1\right)\left(3(c-\mu p)+\mu -1\right)+3{\left(c-\mu p\right)}^2\right]}{{\mu }^2}}\right\}\le {\displaystyle \frac{1}{m^2}}Re\left\{{\displaystyle \frac{{\zeta }^{2}q‴\left(\zeta \right)}{q\prime \left(\zeta \right)}}\right\},$$

(83)

where $m\in \{2,3,4,...\}$, $\xi \in \partial \Delta$ and $\zeta \in \Delta$.

Theorem 7.

Assume that $\varphi \in {\Phi }_J^{\prime }[\Omega ,q]$, and that $f\in {\Sigma }_p$ and ${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\in Q_1$ satisfy the following conditions:

$$Re\left\{{\displaystyle \frac{\zeta q″\left(\zeta \right)}{q\prime \left(\zeta \right)}}\right\}\ge 0\mathit{and}\left|{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\right|\le \mu m\left|{\displaystyle \frac{q\prime \left(\zeta \right)}{c-\mu p}}\right|,$$

(84)

and $\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right)$ be univalent in Δ. Therefore

$$\Omega \subset \left\{\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right):\zeta \in \Delta \right\},$$

(85)

implies

$$q\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)(\zeta \in \Delta ).$$

(86)

Proof. 

Taking into account $g\left(\zeta \right)$, which is defined by (26), and $\psi$ expressed in (33). For $\varphi \in {\Phi }_J^{\prime }[\Omega ,q],$ then relations in (34) and (85) yield the following:

$$\Omega \subset \left\{\psi \left(g\left(\zeta \right),\zeta g\prime \left(\zeta \right),{\zeta }^{2}g″\left(\zeta \right),{\zeta }^{3}g‴\left(\zeta \right);\zeta \right):\zeta \in \Delta \right\}.$$

(87)

The admissible condition for $\varphi \in {\Phi }_J^{\prime }[\Omega ,q]$ in Definition 8 is similar to the admissible condition for $\psi$ as stated in Definition 5, as we can infer from (33). Therefore, by applying Lemma 2 and the requirements in (84), we obtain

$$q\left(\zeta \right)\prec g\left(\zeta \right),$$

(88)

or, similarly,

$$q\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)(\zeta \in \Delta ),$$

(89)

this completes the proof. □

The class ${\Phi }_J^{\prime }[h(\Delta ),q]$ can be simplified to ${\Phi }_J^{\prime }[h,q]$ if $\Omega \ne \mathbb{C}$ is a simply connected domain and $\Omega =h(\Delta )$ for some conformal mapping $h\left(\zeta \right)$ of $\Delta$ onto $\Omega$. Following the same steps as in the previous section, Theorem 7 immediately leads to the following consequence.

Theorem 8.

Assume that h is analytic in Δ, and $\varphi \in {\Phi }_J^{\prime }[h,q]$. If $f\in {\Sigma }_p$ and ${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\in Q_1$ satisfy (84), and

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right),$$

is univalent in Δ. Therefore,

$$h\left(\zeta \right)\prec \varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right),$$

(90)

implies that

$$q\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)(\zeta \in \Delta ).$$

(91)

For a sufficiently selected $\varphi$, the following theorem establishes the availability of the best subordinant of (90).

Theorem 9.

Suppose that $\varphi :{\mathbb{C}}^4\times \overline{\Delta }⟶\mathbb{C}$, and let h be analytic in Δ and ψ is expressed in (33). Moreover, suppose that the differential equation

$$\psi \left(q\left(\zeta \right),\zeta q\prime \left(\zeta \right),{\zeta }^{2}q″\left(\zeta \right),{\zeta }^{3}q‴\left(\zeta \right);\zeta \right)=h\left(\zeta \right),$$

has a solution $q\left(\zeta \right)\in Q_1$. Additionally, if $f\in {\Sigma }_p$ and ${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\in Q_1$ satisfy condition (84) and

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right),$$

is univalent in Δ; therefore,

$$h\left(\zeta \right)\prec \varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right),$$

(92)

implies

$$q\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)(\zeta \in \Delta ).$$

(93)

and q is the best subordinant.

Then, in this order, we present an alternative admissible class ${\tilde{\Phi }}_J^{\prime }[\Omega ,q]$:

Definition 9.

Suppose that $\Omega \subseteq \mathbb{C}$, and let $q\in {\mathcal{F}}_1$ be such that $q\prime \left(\zeta \right)\ne 0$. The set of admissible functions ${\tilde{\Phi }}_J^{\prime }[\Omega ,q]$ consists of functions $\psi :{\mathbb{C}}^4\times \overline{\Delta }\to \mathbb{C}$ that fulfill the admissibility criteria:

$$\varphi \left(\alpha ,\beta ,\gamma ,\delta ;\xi \right)\in \Omega ,$$

(94)

such that

$$\alpha =q\left(\zeta \right),\beta ={\displaystyle \frac{\zeta q\prime \left(\zeta \right)+m\frac{a-\mu p}{\mu }q\left(\zeta \right)}{m\frac{a-\mu p}{\mu }}},$$

(95)

$$Re\left\{{\displaystyle \frac{a-\mu p+1}{\mu }}\left({\displaystyle \frac{\gamma -\alpha }{\beta -\alpha }}\right)-{\displaystyle \frac{2\left(a-\mu p\right)+1}{\mu }}\right\}\le {\displaystyle \frac{1}{m}}Re\left\{1+{\displaystyle \frac{\zeta q″\left(\zeta \right)}{q\prime \left(\zeta \right)}}\right\},$$

(96)

$$Re\left\{{\displaystyle \frac{\left(a-\mu p+1\right)\left[\left(a-\mu p+2\right)\left(\delta -\alpha \right)-3\left(a-(p-1)\mu +1\right)\left(\gamma -\alpha \right)\right]}{{\mu }^2\left(\beta -\alpha \right)}}\right.$$

$$\left.+{\displaystyle \frac{\left(2\mu +1\right)\left(3(a-\mu p)+\mu +1\right)+3{\left(a-\mu p\right)}^2}{{\mu }^2}}\right\}\le {\displaystyle \frac{1}{m^2}}Re\left\{{\displaystyle \frac{{\zeta }^{2}q‴\left(\zeta \right)}{q\prime \left(\zeta \right)}}\right\},$$

(97)

where $m\in \{2,3,4,...\}$, $\xi \in \partial \Delta$ and $\zeta \in \Delta .$

Theorem 10.

Suppose that $\varphi \in {\tilde{\Phi }}_J^{\prime }[\Omega ,q]$, and let ${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\in Q_1$ and $f\in {\Sigma }_p$ fulfill the following criteria:

$$Re\left\{{\displaystyle \frac{\zeta q″\left(\zeta \right)}{q\prime \left(\zeta \right)}}\right\}\ge 0\mathit{and}\left|{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right)-{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\right|\le \mu m\left|{\displaystyle \frac{q\prime \left(\zeta \right)}{a-\mu p}}\right|,$$

(98)

and

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right),$$

is univalent in Δ. Then

$$\Omega \subset \left\{\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right):\zeta \in \Delta \right\},$$

(99)

implies

$$q\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)(\zeta \in \Delta ).$$

(100)

Proof. 

Recall that g and $\psi$ were defined by (56) and (63), respectively. Also, as $\varphi \in {\tilde{\Phi }}_J^{\prime }[\Omega ,q]$, then (64) and (99) yield

$$\Omega \subset \left\{\psi \left(g\left(\zeta \right),\zeta g\prime \left(\zeta \right),{\zeta }^{2}g″\left(\zeta \right),{\zeta }^3{g}^{‴}\left(\zeta \right);\zeta \right):\zeta \in \Delta \right\}.$$

(101)

From (63), we conclude that, as stated in Definition 9, the admissible condition for $\varphi \in {\tilde{\Phi }}_J^{\prime }[\Omega ,q]$ is equal to the acceptable condition for $\psi$, as stated in Definition 5. Therefore, applying Lemma 2 and the circumstances in (98), we obtain

$$q\left(\zeta \right)\prec g\left(\zeta \right),$$

(102)

or, similarly,

$$q\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)(\zeta \in \Delta ),$$

(103)

This concludes the proof. □

Given a simply connected domain $\Omega \ne \mathbb{C}$ and a conformal mapping $h\left(\zeta \right)$ of $\Delta$ onto $\Omega$ such that $\Omega =h(\Delta )$, the class ${\tilde{\Phi }}_J^{\prime }[h(\Delta ),q]$ can be expressed directly as ${\tilde{\Phi }}_J^{\prime }[h,q]$. The subsequent conclusion is a direct result of Theorem 10, with procedures identical to those in the previous section.

Theorem 11.

Suppose that $\varphi \in {\tilde{\Phi }}_J^{\prime }[h,q]$, and h is analytic in Δ. Moreover, if $f\in {\Sigma }_p$ and ${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\in Q_1$ satisfies the conditions in (98) and

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right),$$

is univalent in Δ. Therefore,

$$h\left(\zeta \right)\prec \varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right),$$

(104)

implies

$$q\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)(\zeta \in \Delta ).$$

(105)

For an adequately selected $\varphi$, the preceding theorem establishes the availability of the best subordinant of (104).

Theorem 12.

Assuming that $\varphi :{\mathbb{C}}^4\times \overline{\Delta }⟶\mathbb{C}$, and h is analytic in Δ, then ψ is also expressed in (63). Additionally, the differential equation

$$\psi \left(q\left(\zeta \right),\zeta q\prime \left(\zeta \right),{\zeta }^{2}q″\left(\zeta \right),{\zeta }^{3}q‴\left(\zeta \right);\zeta \right)=h\left(\zeta \right),$$

has a solution $q\left(\zeta \right)\in Q_1$. If $f\in {\Sigma }_p$ and ${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\in Q_1$ satisfy the criteria in (98), also

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right),$$

be univalent in Δ. Then,

$$h\left(\zeta \right)\prec \varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right),$$

(106)

implies

$$q\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)(\zeta \in \Delta ).$$

(107)

where the best subordinant is q.

4. Sandwich Results

Two sandwich-type consequences are shown in this section by combining Theorems 2 and 8, as follows:

Theorem 13.

Suppose that $h_1$ and $q_1$ are analytic in Δ, and let $h_2$ be univalent in Δ, with $q_2\in Q_1$$q_1\left(0\right)=q_2\left(0\right)=1$, and $\varphi \in {\Phi }_J[h_2,q_2]\cap {\Phi }_J^{\prime }[h_1,q_1]$. Moreover, $f\in {\Sigma }_p$, ${\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\in Q_1\cap {\mathcal{F}}_1$, and

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right),$$

is univalent in Δ; additionally, criteria (23) and (84) are fulfilled. Therefore

$$h_1\left(\zeta \right)\prec \varphi \left({\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-1}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a,c-2}f\left(\zeta \right);\zeta \right)\prec h_2\left(\zeta \right),$$

(108)

implies that

$$h_1\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c+1}f\left(\zeta \right)\prec h_2\left(\zeta \right).$$

(109)

Likewise, we obtain additional sandwich-type consequences through the combination of Theorems 5 and 11, which are given as follows:

Theorem 14.

Assume that ${\tilde{h}}_1$ and ${\tilde{q}}_1$ are analytic in Δ, also, let ${\tilde{h}}_2$ be univalent in Δ, ${\tilde{q}}_2\in Q_1$ with ${\tilde{q}}_1\left(0\right)={\tilde{q}}_2\left(0\right)=1$ and $\varphi \in {\tilde{\Phi }}_J[{\tilde{h}}_2,{\tilde{q}}_2]\cap {\tilde{\Phi }}_J^{\prime }[{\tilde{h}}_1,{\tilde{q}}_1]$. Moreover, let $f\in {\Sigma }_p$, ${\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\in Q_1\cap {\mathcal{F}}_1$, and

$$\varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right),$$

is univalent in Δ; additionally, criteria (50) and (98) are fulfilled. Therefore

$${\tilde{h}}_1\left(\zeta \right)\prec \varphi \left({\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+1,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+2,c}f\left(\zeta \right),{\zeta }^p{J}_{p,\mu }^{a+3,c}f\left(\zeta \right);\zeta \right)\prec {\tilde{h}}_2\left(\zeta \right),$$

(110)

implies that

$${\tilde{h}}_1\left(\zeta \right)\prec {\zeta }^p{J}_{p,\mu }^{a,c}f\left(\zeta \right)\prec {\tilde{h}}_2\left(\zeta \right).$$

(111)

Remark 2.

We may derive the appropriate results of the operators described in Remark 1 as special instances by specifying the parameters in all of the theorems provided in this study.

5. Conclusions

We focused on the set of meromorphic functions with a pole of order p defined on the open punctured unit disc $0<\left|\zeta \right|<1$. Two recurrence relations (concerning the two parameters a and c, Equations (4) and (5)) were obtained by using an integral operator of the Erdelyi–Kober type. We concluded some results about differential subordination and superordination for these meromorphic functions. Evaluating suitable classes of admissible functions yielded the results. Furthermore, sandwich-like consequences were also achieved. By making certain parameter selections, the operator can provide generalizations for other earlier operators, resulting in more corollaries. For future works, we suggest further investigation using the quantum analog of the Erdelyi–Kober operator for both the analytic and meromorphic function classes.