Third Order Differential Subordination Results Associated with Tremblay Fractional Derivative Operator for p-Valent Analytic Functions

Abstract

This paper studies a class of p-valent analytic functions in the open unit disk using a modified Tremblay operator based on the Riemann–Liouville fractional integral and derivative. We derive a series representation of the operator, which serves as a useful tool to explore its analytic and geometric properties. New third-order differential subordination results are obtained by defining suitable classes of admissible functions. We provide sufficient conditions to ensure subordination relations with a given dominant function, leading to inclusion results in general complex domains. In addition, several applications are given for specific choices of the target domain, showing how the framework is flexible and able to produce unified results in the theory of differential subordination for multivalent analytic functions.

1. Introduction and Preliminaries

The methods related to the concept of differential subordination, introduced by S.S. Miller and P.T. Mocanu in [1,2], have facilitated the proof of previously established results and inspired numerous new studies focused on the specific techniques of this theory. The main features of the theory of differential subordination are presented in the book by S.S. Miller and P.T. Mocanu, published in 2000 [3].

In 2011, J.A. Antonino and S.S. Miller [4] extended several results on second-order differential subordinations, laying the groundwork for the investigation of third-order differential subordinations. Based on the results in [4], many studies have explored third-order differential subordinations using various operators. Notably, the Liu–Srivastava operator and meromorphic multivalent functions were employed to obtain novel results concerning third-order differential subordinations [5]. Additionally, generalized Mittag–Leffler functions were utilized in [6,7] to derive third-order differential subordinations, while linear operators defined via the class of meromorphic multivalent functions were applied to generate further new outcomes in this context [8].

Let $\mathsf{\Lambda }\left(\mathsf{p}\right)$ be the class of normalized p-valent functions f analytic in the unit disk $\mathsf{\Theta }=\{\varsigma \in \mathbb{C}:|\varsigma |<1\}$, with series representation

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

(1)

Note that $\mathsf{\Lambda }\left(1\right)=\mathsf{\Lambda }$, the standard class of univalent functions.

A function $f\in \mathsf{\Lambda }\left(\mathsf{p}\right)$ is said to be $\mathsf{p}$-valent starlike of order $\varkappa$$(0\le \varkappa <p)$ if

$$\Re \left\{{\displaystyle \frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}}\right\}>\varkappa ,\varsigma \in \mathsf{\Theta },$$

and we denote the class by ${\mathcal{S}}_{\mathsf{p}}\left(\varkappa \right)$. Similarly, f is $\mathsf{p}$-valent convex of order $\varkappa$ if

$$\Re \left\{1+{\displaystyle \frac{\varsigma f^{″}\left(\varsigma \right)}{f^{\prime }\left(\varsigma \right)}}\right\}>\varkappa ,\varsigma \in \mathsf{\Theta },$$

denoted by ${\mathcal{C}}_{\mathsf{p}}\left(\varkappa \right)$, with

$$f\in {\mathcal{C}}_{\mathsf{p}}\left(\varkappa \right)⟺\varsigma f^{\prime }\in {\mathcal{S}}_{\mathsf{p}}\left(\varkappa \right),\varsigma \in \mathsf{\Theta }.$$

Further, f is a $\mathsf{p}$-valent close-to-convex or quasi-convex function of order $\beta$ and type $\varkappa$ if there exists $g\in {\mathcal{S}}_{\mathsf{p}}\left(\varkappa \right)$ or $g\in {\mathcal{C}}_{\mathsf{p}}\left(\varkappa \right)$ such that

$$\Re \left\{{\displaystyle \frac{\varsigma g^{\prime }\left(\varsigma \right)}{g\left(\varsigma \right)}}\right\}>\beta \mathrm{or}\Re \left\{{\displaystyle \frac{\varsigma g^{\prime }\left(\varsigma \right)}{g^{\prime }\left(\varsigma \right)}}\right\}>\beta ,\varsigma \in \mathsf{\Theta }.$$

Both classes are denoted by ${\mathcal{K}}_{\mathsf{p}}(\beta ,\varkappa )$, satisfying

$$g\in {\mathcal{K}}_{\mathsf{p}}(\beta ,\varkappa )⟺\varsigma g^{\prime }\in {\mathcal{K}}_{\mathsf{p}}(\beta ,\varkappa ),\varsigma \in \mathsf{\Theta }.$$

For details on these classes, see [9,10,11].

Let $\mathcal{R}\left(\mathsf{\Theta }\right)$ represent the family of analytic functions defined on the open unit disk $\mathsf{\Theta }=\{\varsigma \in \mathbb{C}:|\varsigma |<1\}$. For a fixed $a\in \mathbb{C}$ and $j\in \mathbb{N}=\{1,2,3,\dots \}$, we denote by $\mathcal{R}[a,j]$ the subclass of $\mathcal{R}\left(\mathsf{\Theta }\right)$ formed by functions of the form

$$\mathcal{R}[a,j]=\left\{f\in \mathcal{R}\left(\mathsf{\Theta }\right):f\left(\varsigma \right)=a+\sum _{n=j}^{\infty }{a}_n{\varsigma }^n\right\}.$$

In particular, we write $\mathcal{R}[1,j]={\mathcal{R}}_j$.

Let $f,g\in \mathcal{R}\left(\mathsf{\Theta }\right)$. We say that $f\left(\varsigma \right)$ is subordinate to $\mathcal{I}\left(\varsigma \right)$, denoted by $f\left(\varsigma \right)\prec \mathcal{I}\left(\varsigma \right)$, if there exists an analytic Schwarz function $\mathcal{W}\left(\varsigma \right)$ in $\mathsf{\Theta }$ satisfying $\mathcal{W}\left(0\right)=0$ and $\left|\mathcal{W}\right(\varsigma \left)\right|<1$ for all $\varsigma \in \mathsf{\Theta }$, such that

$$f\left(\varsigma \right)=\mathcal{I}\left(\mathcal{W}\left(\varsigma \right)\right),\varsigma \in \mathsf{\Theta }.$$

When $\mathcal{I}$ is univalent in $\mathsf{\Theta }$, the subordination $f\left(\varsigma \right)\prec \mathcal{I}\left(\varsigma \right)$ holds if and only if $f\left(0\right)=\mathcal{I}\left(0\right)$ and $f\left(\mathsf{\Theta }\right)\subset \mathcal{I}\left(\mathsf{\Theta }\right)$ (see [3,12,13]).

Example 1. 

Let

$$\mathcal{I}\left(\varsigma \right)={\displaystyle \frac{1+\varsigma }{1-\varsigma }},f\left(\varsigma \right)={\displaystyle \frac{1+{\varsigma }^2}{1-{\varsigma }^2}}.$$

We claim that $f\left(\varsigma \right)\prec \mathcal{I}\left(\varsigma \right)$. Indeed, define the Schwarz function

$$\mathcal{W}\left(\varsigma \right)={\varsigma }^2.$$

It is analytic in Θ, satisfies $\mathcal{W}\left(0\right)=0$, and $\left|\mathcal{W}\right(\varsigma \left)\right|<1$ for all $\varsigma \in \mathsf{\Theta }$. Then we observe that

$$\mathcal{I}\left(\mathcal{W}\left(\varsigma \right)\right)={\displaystyle \frac{1+{\varsigma }^2}{1-{\varsigma }^2}}=f\left(\varsigma \right),$$

proving the subordination.

Example 2. 

Let

$$\mathcal{I}\left(\varsigma \right)=\varsigma ,f\left(\varsigma \right)=\varsigma +1.$$

Suppose there exists a Schwarz function $\mathcal{W}\left(\varsigma \right)$ such that $f\left(\varsigma \right)=\mathcal{I}\left(\mathcal{W}\right(\varsigma \left)\right)=\mathcal{W}\left(\varsigma \right)$. This would require $\mathcal{W}\left(\varsigma \right)=\varsigma +1$, but $\left|\mathcal{W}\right(\varsigma \left)\right|<1$ fails for any $\varsigma \in \mathsf{\Theta }$, hence no such Schwarz function exists. Therefore,

$$f\left(\varsigma \right)\nprec \mathcal{I}\left(\varsigma \right).$$

Let $\varphi ({\mathsf{s}}_1,{\mathsf{s}}_2,{\mathsf{s}}_3,{\mathsf{s}}_4;\varsigma ):{\mathbb{C}}^4\times \mathsf{\Theta }\to \mathbb{C}$ and let $\ell \left(\varsigma \right)$ be univalent in $\mathsf{\Theta }$. Suppose that $\mathcal{I}\left(\varsigma \right)$ is analytic in $\mathsf{\Theta }$ and satisfies

$$\varphi \left(\mathcal{I}\left(\varsigma \right),\varsigma {\mathcal{I}}^{\prime }\left(\varsigma \right),{\varsigma }^2{\mathcal{I}}^{″}\left(\varsigma \right),{\varsigma }^3{\mathcal{I}}^{‴}\left(\varsigma \right);\varsigma \right)\prec \ell \left(\varsigma \right).$$

(2)

In this context, $\mathcal{I}\left(\varsigma \right)$ is referred to as a solution of the differential subordination (2). A univalent function $\beth \left(\varsigma \right)$ is called a dominant of (2) if

$$\mathcal{I}\left(\varsigma \right)\prec \beth \left(\varsigma \right)$$

for every solution $\mathcal{I}\left(\varsigma \right)$ of (2). Furthermore, a univalent dominant $\tilde{\mathcal{I}}$ is termed the best dominant whenever $\tilde{\mathcal{I}}\prec \beth$ for all dominants ℶ associated with (2) (see [4]).

Recent studies have concentrated on third-order differential subordination involving fractional operators, since such operators play an important role in extending many classical results in the theory of analytic and univalent functions. Oros et al. studied third-order differential subordination problems using the fractional integral of the Gaussian hypergeometric function and showed that fractional methods lead to broader classes of analytic functions [14]. Moreover, Abdulnabi et al. obtained several third-order differential subordination and superordination results by means of a general fractional differential operator, with applications to different subclasses of analytic functions [15]. In a related direction, Maktoof et al. derived third-order differential subordination results associated with a new integral operator, demonstrating its effectiveness in generating wider function classes and sharper inclusion relations [16]. Additionally, Farzana et al. investigated third-order differential subordination for analytic functions defined via a functional derivative operator, further highlighting the role of non-classical operators in this setting [17].

Despite these contributions, the existing literature on third-order differential subordination involving fractional operators does not include results based on the Tremblay fractional derivative operator. The present work addresses this point by studying third-order differential subordination problems associated with the Tremblay operator and by showing that this operator can be used to obtain new and meaningful results within this framework.

Recent studies have demonstrated the growing importance of fractional calculus in both theoretical and applied contexts, particularly through modern computational methods and applications on complex geometries such as fractal curves [18,19]. Despite these advances, classical fractional operators continue to play a fundamental role in analytical investigations. Accordingly, in this work we adopt the Riemann–Liouville fractional operators as the main analytical tool (see [20,21,22]).

Definition 1. 

Let $f\left(\varsigma \right)$ be analytic in a simply connected domain that includes the origin. The fractional integral of order ∝ $>0$ is defined as

$${\mathbb{D}}_{\varsigma }^{\propto }f\left(\varsigma \right)={\displaystyle \frac{1}{\mathsf{\Gamma }(\propto )}}{\int }_0^{\varsigma }{\displaystyle \frac{f\left(\varrho \right)}{{(\varsigma -\varrho )}^{1-\propto }}}d\varrho ,$$

where the branch of ${(\varsigma -\varrho )}^{\propto -1}$ is chosen so that $log(\varsigma -\varrho )$ is real for $\varsigma -\varrho >0$.

Definition 2. 

For $0\le \propto <1$, the fractional derivative of order ∝ of f is given by

$${\mathbb{D}}_{\varsigma }^{\propto }f\left(\varsigma \right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\propto )}}{\displaystyle \frac{d}{d\varsigma }}{\int }_0^{\varsigma }{\displaystyle \frac{f\left(\varrho \right)}{{(\varsigma -\varrho )}^{\propto }}}d\varrho ,$$

where the branch of ${(\varsigma -\varrho )}^{\propto }$ is fixed in the same manner as above.

For $n\in {\mathbb{N}}_0$, the derivative of order $n+\propto$ is defined by

$${\mathbb{D}}_{\varsigma }^{n+\propto }f\left(\varsigma \right)={\displaystyle \frac{d^n}{d{\varsigma }^n}}\left({\mathbb{D}}_{\varsigma }^{\propto }f\left(\varsigma \right)\right).$$

Tremblay [23] introduced a fractional operator constructed via the Riemann–Liouville derivative in his doctoral thesis. This construction was subsequently generalized to the class of univalent functions by Ibrahim and Jahangiri [24]. In the present work, we deal with its extension to the multivalent setting.

Definition 3 

([24,25]). For $\varsigma \in \mathsf{\Theta }$, the Tremblay fractional derivative operator $T_{\eta ,\lambda }^{\mathsf{p}}$ is defined by

$$\left(T_{\eta ,\lambda }^{\mathsf{p}}f\right)\left(\varsigma \right)={\displaystyle \frac{\mathsf{\Gamma }\left(\lambda \right)}{\mathsf{\Gamma }\left(\eta \right)}}{\varsigma }^{1-\lambda }{\mathbb{D}}_{\varsigma }^{\eta -\lambda }\left({\varsigma }^{\eta -1}f\left(\varsigma \right)\right),$$

(3)

where $0<\lambda \le 1$, $0<\eta \le 1$, $0\le \eta -\lambda <1$, and $\eta >\lambda$.

It is immediate that the choice $\eta =\lambda =1$ yields $\left(T_{1,1}^{\mathsf{p}}f\right)\left(\varsigma \right)=f\left(\varsigma \right)$ (see [26]).

Definition 4 

([27]). Let $f\in \mathsf{\Lambda }\left(\mathsf{p}\right)$. The modified $\mathsf{p}$-valent Tremblay operator ${\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }:\mathsf{\Lambda }\left(\mathsf{p}\right)\to \mathsf{\Lambda }\left(\mathsf{p}\right)$ is defined by

$${\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)={\displaystyle \frac{\mathsf{\Gamma }\left(\eta \right)\mathsf{\Gamma }(\lambda +\mathsf{p})}{\mathsf{\Gamma }\left(\lambda \right)\mathsf{\Gamma }(\eta +\mathsf{p})}}\left(T_{\eta ,\lambda }^{z}f\right)\left(\varsigma \right)={\displaystyle \frac{\mathsf{\Gamma }(\lambda +\mathsf{p})}{\mathsf{\Gamma }(\eta +\mathsf{p})}}{\varsigma }^{1-\lambda }{\mathbb{D}}_{\varsigma }^{\eta -\lambda }\left({\varsigma }^{\eta -1}f\left(\varsigma \right)\right).$$

(4)

The series expansion of ${\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)$ takes the form

$${\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)={\varsigma }^{\mathsf{p}}+\sum _{k=\mathsf{p}+1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }(\lambda +\mathsf{p})\mathsf{\Gamma }(\eta +k)}{\mathsf{\Gamma }(\eta +\mathsf{p})\mathsf{\Gamma }(\lambda +k)}}{a}_k{\varsigma }^k={\varsigma }^{\mathsf{p}}+\sum _{k=\mathsf{p}+1}^{\infty }{\displaystyle \frac{{(\eta +\mathsf{p})}_{k-\mathsf{p}}}{{(\lambda +\mathsf{p})}_{k-\mathsf{p}}}}{a}_k{\varsigma }^k,$$

(5)

where ${\left(j\right)}_n=\mathsf{\Gamma }(j+n)/\mathsf{\Gamma }\left(j\right)$ denotes the Pochhammer symbol, given by

$${\left(j\right)}_n=\left\{\begin{array}{cc}j(j+1)\cdots (j+n-1),\hfill & n\in \mathbb{N},j\in \mathbb{C},\hfill \\ 1,\hfill & n=0,j\in \mathbb{C}.\hfill \end{array}\right.$$

Remark 1 

([27]). For $p=1$, we have ${\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)=T_{\eta ,\lambda }^{z}f\left(\varsigma \right),{\mathbb{T}}_{\lambda }^{\mathsf{p},\lambda +1}f\left(\varsigma \right)={\displaystyle \frac{1}{\lambda +\mathsf{p}}}(\lambda f\left(\varsigma \right)+\varsigma f^{\prime }\left(\varsigma \right)),$ which reduces to the modified Tremblay operator defined by Esa et al. [28]. From (1), the following recurrence relations hold and will be used later:

$$z{\left({\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }=(\eta +\mathsf{p}){\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right)-\eta {\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right),$$

(6)

$$z{\left({\mathbb{T}}_{\lambda +1}^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }=(\lambda +\mathsf{p}){\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)-\lambda {\mathbb{T}}_{\lambda +1}^{\mathsf{p},\eta }f\left(\varsigma \right).$$

(7)

Definition 5 

([4], p. 441). Let $\mathbb{Q}$ denote the class of all functions Υ which are analytic and univalent in $\mathsf{\Theta }\setminus E\left(\mathrm{{\rm Y}}\right)$, where

$$E\left(\mathrm{{\rm Y}}\right)=\left\{\varrho \in \partial \mathsf{\Theta }:\underset{z\to \varrho }{lim}\mathrm{{\rm Y}}\left(\varsigma \right)=\infty \right\},$$

and satisfy

$$\underset{\varrho \in \partial \mathsf{\Theta }\setminus E\left(\mathrm{{\rm Y}}\right)}{min}\left|{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)\right|=\epsilon >0.$$

Furthermore, let $\mathbb{Q}\left(a\right)$ be the subclass of $\mathbb{Q}$ consisting of functions Υ such that $\mathrm{{\rm Y}}\left(0\right)=a$, and note that $\mathbb{Q}\left(1\right)\equiv {\mathbb{Q}}_1$.

Definition 6. 

Let $\aleph \subset \mathbb{C}$, $\mathrm{{\rm Y}}\in \mathbb{Q}$, and $j\ge 2$. We denote by ${\beth }_j[\aleph ,\mathrm{{\rm Y}}]$ the class of admissible functions $\beth :{\mathbb{C}}^4\times \mathsf{\Theta }\to \mathbb{C}$ satisfying the condition

$$\beth ({\mathsf{s}}_1,{\mathsf{s}}_2,{\mathsf{s}}_3,{\mathsf{s}}_4;z)\notin \aleph ,$$

whenever

$${\mathsf{s}}_1=\mathrm{{\rm Y}}\left(\varrho \right),{\mathsf{s}}_2=m\varrho {\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right),$$

and

$$\Re \left\{1+{\displaystyle \frac{{\mathsf{s}}_3}{{\mathsf{s}}_2}}\right\}\ge m\Re \left\{1+{\displaystyle \frac{\varrho {\mathrm{{\rm Y}}}^{″}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\right\},$$

together with

$$\Re \left\{{\displaystyle \frac{{\mathsf{s}}_4}{{\mathsf{s}}_2}}\right\}\ge m^2\Re \left\{{\displaystyle \frac{{\varrho }^2{\mathrm{{\rm Y}}}^{‴}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\right\},$$

for $\varsigma \in \mathsf{\Theta }$, $\varrho \in \partial \mathsf{\Theta }\setminus E\left(\mathrm{{\rm Y}}\right)$, and $m\ge j$.

Lemma 1 

([4], Theorem 1, p. 449). Let $\mathcal{I}\in \mathcal{R}[a,j]$ with $j\ge 2$. Suppose further that $\mathrm{{\rm Y}}\in \mathbb{Q}\left(a\right)$ satisfies

$$\Re \left\{{\displaystyle \frac{\varrho {\mathrm{{\rm Y}}}^{″}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\right\}\ge 0and\left|z{\mathcal{I}}^{\prime }\left(\varsigma \right){\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)\right|\le m,$$

where $\varsigma \in \mathsf{\Theta }$, $\varrho \in \partial \mathsf{\Theta }\setminus E\left(\mathrm{{\rm Y}}\right)$, and $m\ge j$.

If ℵ is a subset of $\mathbb{C}$, $\beth \in {\beth }_j[\aleph ,\mathrm{{\rm Y}}]$, and

$$\beth \left(\mathcal{I}\left(\varsigma \right),z{\mathcal{I}}^{\prime }\left(\varsigma \right),{\varsigma }^2{\mathcal{I}}^{″}\left(\varsigma \right),{\varsigma }^3{\mathcal{I}}^{‴}\left(\varsigma \right):\varsigma \right)\in \varsigma ,$$

then

$$\mathcal{I}\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right).$$

A variety of results concerning differential subordination and superordination have been obtained through the use of different operators (see, for instance, [29,30,31]). In this paper, we make use of third-order differential subordination techniques developed by Antonino and Miller [4], together with some more recent developments due to Tang et al. [5,32,33], to establish sufficient conditions for certain appropriate classes of admissible functions such that

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right)$$

and

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right),$$

Here, $\mathrm{{\rm Y}}\left(\varsigma \right)$ denotes a univalent function in $\mathsf{\Theta }$ with $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1\cap {\mathcal{R}}_1$. Additionally, various particular cases of these admissible function classes are derived. The findings of this work, along with related studies from recent years, are anticipated to encourage further research in the theory of third-order differential subordinations.

The paper is organized as follows.

• Section 1 presents the introduction and preliminaries, including some basic definitions, a brief review of main results related to differential subordination, and the definition of the Tremblay fractional derivative operator.
• Section 2 is devoted to third-order differential subordination results involving the fractional operator ${\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)$, together with several corollaries.
• Section 3 extends the third-order differential subordination results obtained in the previous section by further applications of the operator ${\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)$, along with related corollaries.
• Section 4 contains applications and illustrative examples, including special cases and consequences of the main results.

2. Third-Order Subordination Results Using Fractional Operator ${\mathbb{T}}_{\mathbf{\lambda }}^{\mathsf{p},\mathbf{\eta }}\mathbf{f}\left(\mathbf{\varsigma }\right)$

Throughout this section, unless explicitly stated otherwise, we consider functions $f\in \mathsf{\Lambda }\left(\mathsf{p}\right)$. We assume that the parameters satisfy $0<\lambda \le 1$, $0<\eta \le 1$, and $0\le \eta -\lambda <1$ with $\eta >\lambda$. Moreover, let $\varrho \in \partial \mathsf{\Theta }\setminus \mathbb{E}\left(\mathrm{{\rm Y}}\right)$, $\theta \in [0,2\pi )$, and $\varsigma \in \mathsf{\Theta }$. These assumptions will be used consistently throughout this section without further mention.

Definition 7. 

Let $\aleph \subset \mathbb{C}$ and $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1\cap {\mathcal{R}}_1$. Define ${\psi }_1[\varsigma ,\mathrm{{\rm Y}}]$ as the class of admissible functions $\varphi :{\mathbb{C}}^4\times \mathsf{\Theta }\to \mathbb{C}$ satisfying the admissibility condition:

$$\varphi ({\mathsf{c}}_1,{\mathsf{c}}_2,{\mathsf{c}}_3,{\mathsf{c}}_4:\varsigma )\notin \varsigma ,$$

whenever

$${\mathsf{c}}_1=\mathrm{{\rm Y}}\left(\varrho \right),{\mathsf{c}}_2=\mathrm{{\rm Y}}\left(\varrho \right)+{\displaystyle \frac{m\varrho {\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}{\eta +\mathsf{p}}},$$

$$\Re \{1-{\displaystyle \frac{(2{\mathsf{c}}_2-{\mathsf{c}}_3-{\mathsf{c}}_1)(1+\mathsf{p}+\eta )}{{\mathsf{c}}_2-{\mathsf{c}}_1}}\}\ge m\Re \{1+{\displaystyle \frac{\varrho {\mathrm{{\rm Y}}}^{″}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\},$$

and

$$\Re \left\{{\displaystyle \frac{(3{\mathsf{c}}_2-3{\mathsf{c}}_3+{\mathsf{c}}_4-{\mathsf{c}}_1)(1+\mathsf{p}+\eta )(2+\mathsf{p}+\eta )}{{\mathsf{c}}_2-{\mathsf{c}}_1}}\right\}\ge m^2\Re \{{\displaystyle \frac{{\varrho }^2{\mathrm{{\rm Y}}}^{‴}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\},$$

for $\varsigma \in \mathsf{\Theta }$, $\varrho \in \partial \mathsf{\Theta }\setminus \mathbb{E}\left(\mathrm{{\rm Y}}\right)$, and $m\ge 1$.

Theorem 1. 

Let $\aleph \subset \mathbb{C}$ and $\varphi \in {\psi }_1[\varsigma ,\mathrm{{\rm Y}}]$. If $f\in \mathsf{\Lambda }\left(\mathsf{p}\right)$ and $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1$ satisfy

$$\Re \{{\displaystyle \frac{\varrho {\mathrm{{\rm Y}}}^{″}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\}\ge 0and|\varsigma {\left(z{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }|\le m|{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)|,$$

(8)

then

$$\{\varphi ({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +2}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +3}f\left(\varsigma \right):\varsigma ):\varsigma \in \mathsf{\Theta }\}\subset \varsigma ,$$

(9)

which implies

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right).$$

Proof. 

We introduce the function $\mathcal{I}\left(\varsigma \right)$ in the unit disk $\mathsf{\Theta }$ by setting

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)=\mathcal{I}\left(\varsigma \right),\varsigma \in \mathsf{\Theta }.$$

(10)

Differentiating (10) with respect to z and using the recurrence relation (7), we obtain

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right)=\mathcal{I}\left(\varsigma \right)+{\displaystyle \frac{z{\mathcal{I}}^{\prime }\left(\varsigma \right)}{\eta +\mathsf{p}}}.$$

(11)

Differentiating (11) and applying the recurrence relation (7) again gives

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +2}f\left(\varsigma \right)=\mathcal{I}\left(\varsigma \right)+{\displaystyle \frac{2z{\mathcal{I}}^{\prime }\left(\varsigma \right)}{\eta +\mathsf{p}}}+{\displaystyle \frac{{\varsigma }^2{\mathcal{I}}^{″}\left(\varsigma \right)}{(\eta +\mathsf{p})(\eta +\mathsf{p}+1)}}.$$

(12)

Further computations yield

$$\begin{array}{cc}\hfill {\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +3}f\left(\varsigma \right)& =\mathcal{I}\left(\varsigma \right)+{\displaystyle \frac{3z{\mathcal{I}}^{\prime }\left(\varsigma \right)}{\eta +\mathsf{p}}}+{\displaystyle \frac{3{\varsigma }^2{\mathcal{I}}^{″}\left(\varsigma \right)}{(\eta +\mathsf{p})(\eta +\mathsf{p}+1)}}+{\displaystyle \frac{{\varsigma }^3{\mathcal{I}}^{‴}\left(\varsigma \right)}{(\eta +\mathsf{p})(\eta +\mathsf{p}+1)(\eta +\mathsf{p}+2)}}.\hfill \end{array}$$

(13)

Let

$${\mathsf{s}}_1=\mathcal{I}\left(\varsigma \right),{\mathsf{s}}_2=\varsigma {\mathcal{I}}^{\prime }\left(\varsigma \right),{\mathsf{s}}_3={\varsigma }^2{\mathcal{I}}^{″}\left(\varsigma \right),{\mathsf{s}}_4={\varsigma }^3{\mathcal{I}}^{‴}\left(\varsigma \right).$$

Using (10)–(13), the corresponding transformation $\beth ({\mathsf{s}}_1,{\mathsf{s}}_2,{\mathsf{s}}_3,{\mathsf{s}}_4;\varsigma )$ is determined by defining

$$\begin{array}{cc}\hfill {\mathsf{c}}_1& ={\mathsf{s}}_1,\hfill \\ \hfill {\mathsf{c}}_2& ={\mathsf{s}}_1+{\displaystyle \frac{{\mathsf{s}}_2}{\eta +\mathsf{p}}},\hfill \\ \hfill {\mathsf{c}}_3& ={\mathsf{s}}_1+{\displaystyle \frac{2{\mathsf{s}}_2}{\eta +\mathsf{p}}}+{\displaystyle \frac{{\mathsf{s}}_3}{(\eta +\mathsf{p})(\eta +\mathsf{p}+1)}},\hfill \\ \hfill {\mathsf{c}}_4& ={\mathsf{s}}_1+{\displaystyle \frac{3{\mathsf{s}}_2}{\eta +\mathsf{p}}}+{\displaystyle \frac{3{\mathsf{s}}_3}{(\eta +\mathsf{p})(\eta +\mathsf{p}+1)}}+{\displaystyle \frac{{\mathsf{s}}_4}{(\eta +\mathsf{p})(\eta +\mathsf{p}+1)(\eta +\mathsf{p}+2)}}.\hfill \end{array}$$

(14)

Next, we introduce the mapping $\beth ({\mathsf{s}}_1,{\mathsf{s}}_2,{\mathsf{s}}_3,{\mathsf{s}}_4;z):{\mathbb{C}}^4\times \mathsf{\Theta }\to \mathbb{C}$ defined by

$$\begin{array}{cc}\hfill \beth ({\mathsf{s}}_1,{\mathsf{s}}_2,{\mathsf{s}}_3,{\mathsf{s}}_4:\varsigma )& =\varphi ({\mathsf{c}}_1,{\mathsf{c}}_2,{\mathsf{c}}_3,{\mathsf{c}}_4:\varsigma )\hfill \\ \hfill & =\varphi ({\mathsf{s}}_1,{\mathsf{s}}_1+{\displaystyle \frac{{\mathsf{s}}_2}{\eta +\mathsf{p}}},{\mathsf{s}}_1+{\displaystyle \frac{2{\mathsf{s}}_2}{\eta +\mathsf{p}}}+{\displaystyle \frac{{\mathsf{s}}_3}{(\eta +\mathsf{p})(\eta +\mathsf{p}+1)}},\hfill \\ \hfill & {\mathsf{s}}_1+{\displaystyle \frac{3{\mathsf{s}}_2}{\eta +\mathsf{p}}}+{\displaystyle \frac{3{\mathsf{s}}_3}{(\eta +\mathsf{p})(\eta +\mathsf{p}+1)}}+{\displaystyle \frac{{\mathsf{s}}_4}{(\eta +\mathsf{p})(\eta +\mathsf{p}+1)(\eta +\mathsf{p}+2)}}:\varsigma ).\hfill \end{array}$$

(15)

Then, using relations (10)–(13), we have

$$\begin{array}{c}\hfill \beth \left(\mathcal{I}\left(\varsigma \right),z{\mathcal{I}}^{\prime }\left(\varsigma \right),{\varsigma }^2{\mathcal{I}}^{″}\left(\varsigma \right),{\varsigma }^3{\mathcal{I}}^{‴}\left(\varsigma \right):\varsigma \right)\\ \hfill =\varphi \left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +2}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +3}f\left(\varsigma \right):\varsigma \right).\end{array}$$

(16)

Note that

$$1+{\displaystyle \frac{{\mathsf{s}}_3}{{\mathsf{s}}_2}}=1-{\displaystyle \frac{(2{\mathsf{c}}_2-{\mathsf{c}}_3-{\mathsf{c}}_1)(1+\mathsf{p}+\eta )}{{\mathsf{c}}_2-{\mathsf{c}}_1}}$$

$${\displaystyle \frac{{\mathsf{s}}_4}{{\mathsf{s}}_2}}={\displaystyle \frac{(3{\mathsf{c}}_2-3{\mathsf{c}}_3+{\mathsf{c}}_4-{\mathsf{c}}_1)(1+\mathsf{p}+\eta )(2+\mathsf{p}+\eta )}{{\mathsf{c}}_2-{\mathsf{c}}_1}}$$

Furthermore, the admissibility requirement for $\varphi \in {\psi }_1[\varsigma ,\epsilon ]$ stated in Definition (7) coincides with the admissibility condition for $\beth \in {\beth }_n[\varsigma ,\epsilon ]$ described in Definition (6). Consequently, the conclusion of Theorem (1) is an immediate consequence of Lemma 1. □

The next statement generalizes Theorem (1) to situations where the boundary behavior of the function $\mathrm{{\rm Y}}\left(\varsigma \right)$ on $\partial \mathsf{\Theta }$ is not specified.

Corollary 1. 

Let $\aleph \subset \mathbb{C}$ and suppose that Υ is univalent in Θ with $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1$. Assume that $\varphi \in {\psi }_1[\varsigma ,{\mathrm{{\rm Y}}}_{\epsilon }]$ for some $\epsilon \in (0,1)$, where ${\mathrm{{\rm Y}}}_{\epsilon }\left(\varsigma \right)=\mathrm{{\rm Y}}\left(\epsilon \varsigma \right)$. If $f\in \Sigma$ and ${\mathrm{{\rm Y}}}_{\epsilon }$ satisfy

$$\Re \left\{{\displaystyle \frac{\varrho {\mathrm{{\rm Y}}}_{\epsilon }^{″}\left(\varrho \right)}{{\mathrm{{\rm Y}}}_{\epsilon }^{\prime }\left(\varrho \right)}}\right\}\ge 0and\left|z{\left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }\right|\le m\left|{\mathrm{{\rm Y}}}_{\epsilon }^{\prime }\left(\varrho \right)\right|,$$

(17)

then

$$\left\{\varphi \left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +2}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +3}f\left(\varsigma \right);z\right):\varsigma \in \mathsf{\Theta }\right\}\subset \aleph ,$$

and consequently,

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right).$$

Proof. 

By Theorem (1), it follows that ${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec {\mathrm{{\rm Y}}}_{\epsilon }\left(\varsigma \right)$. Since ${\mathrm{{\rm Y}}}_{\epsilon }\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right)$, the desired subordination ${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right)$ is obtained. □

If $\aleph \ne \mathbb{C}$ is a simply connected domain, there exists a conformal mapping ℓ from $\mathsf{\Theta }$ onto ℵ such that $\ell \left(\mathsf{\Theta }\right)=\aleph$. Denote the class ${\psi }_1[\ell \left(\mathsf{\Theta }\right),\mathrm{{\rm Y}}]$ by ${\psi }_1[h,\mathrm{{\rm Y}}]$. The next two corollaries follow directly from Theorem 1 and Corollary (1).

Corollary 2. 

If ℓ is univalent in Θ and $\varphi \in {\psi }_1[\ell ,\mathrm{{\rm Y}}]$, with $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1$ satisfying (8), then

$$\varphi ({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +2}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +3}f\left(\varsigma \right):\varsigma )\prec \ell \left(z\right),$$

(18)

which implies

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \epsilon \left(\varsigma \right).$$

Corollary 3. 

Let Υ be univalent in Θ with $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1$ and assume that $\varphi \in {\psi }_1[h,{\mathrm{{\rm Y}}}_{\epsilon }]$ for some $\epsilon \in (0,1)$, where ${\mathrm{{\rm Y}}}_{\epsilon }\left(\varsigma \right)=\mathrm{{\rm Y}}\left(\epsilon \varsigma \right)$. If ${\mathrm{{\rm Y}}}_{\epsilon }$ fulfills the conditions given in (17), then the subordination relation in (18) yields

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right).$$

The corollary below clarifies how the best dominant associated with a third-order differential subordination is linked to the solution of the corresponding third-order differential equation.

Corollary 4. 

If ℓ is univalent in Θ and ℶ is given by (16) with $\varphi \in {\psi }_1[\ell ,\mathrm{{\rm Y}}]$, and the differential equation

$$\beth (\mathrm{{\rm Y}}\left(\varsigma \right),z{\mathrm{{\rm Y}}}^{\prime }\left(\varsigma \right),{\varsigma }^2{\mathrm{{\rm Y}}}^{″}\left(\varsigma \right),{\varsigma }^3{\mathrm{{\rm Y}}}^{‴}\left(\varsigma \right):\varsigma )=\ell \left(z\right)$$

has a solution ε with $\epsilon \in {\mathbb{Q}}_1$ satisfying (8), then the subordination (18) implies

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right),$$

and ε is the best dominant of (18).

Proof. 

Since

$$\begin{array}{cc}\hfill & \varphi \left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +2}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +3}f\left(\varsigma \right):\varsigma \right)\hfill \\ \hfill & =\beth \left(\mathcal{I}\left(\varsigma \right),\varsigma {\mathcal{I}}^{\prime }\left(\varsigma \right),{\varsigma }^2{\mathcal{I}}^{″}\left(\varsigma \right),{\varsigma }^3{\mathcal{I}}^{‴}\left(\varsigma \right):\varsigma \right)\prec \ell \left(\varsigma \right),\hfill \end{array}$$

(19)

it follows that $\mathcal{I}\left(\varsigma \right)$ is a solution of (19). From Corollary (2), we obtain

$$\mathcal{I}\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right),$$

meaning that $\mathrm{{\rm Y}}$ is a dominant of (19). Moreover, we have

$$\begin{array}{cc}\hfill & \varphi \left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +2}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +3}f\left(\varsigma \right):\varsigma \right)\hfill \\ \hfill & =\beth \left(\mathcal{I}\left(\varsigma \right),\varsigma {\mathcal{I}}^{\prime }\left(\varsigma \right),{\varsigma }^2{\mathcal{I}}^{″}\left(\varsigma \right),{\varsigma }^3{\mathcal{I}}^{‴}\left(\varsigma \right):\varsigma \right)\hfill \\ \hfill & \prec \ell \left(\varsigma \right)=\beth \left(\mathrm{{\rm Y}}\left(\varsigma \right),\varsigma {\mathrm{{\rm Y}}}^{\prime }\left(\varsigma \right),{\varsigma }^2{\mathrm{{\rm Y}}}^{″}\left(\varsigma \right),{\varsigma }^3{\mathrm{{\rm Y}}}^{‴}\left(\varsigma \right):\varsigma \right),\hfill \end{array}$$

which shows that $\mathrm{{\rm Y}}$ is the best dominant of (19). □

Theorem (1) remains valid for the choice $\epsilon \left(\varsigma \right)=1+\mu \varsigma$ with $\mu >0$. In this case, according to Definition (7), the associated class of admissible functions ${\psi }_1[\varsigma ,\epsilon ]$ will be written as ${\psi }_1[\aleph ,\mu ]$ and is described below.

Definition 8. 

Let $\aleph \subset \mathbb{C}$ and $\mu >0$. The class ${\psi }_1[\aleph ,\mu ]$ consists of functions $\varphi :{\mathbb{C}}^4\times \mathsf{\Theta }\to \mathbb{C}$ such that

$$\begin{array}{cc}\hfill \varphi (& 1+\mu e^{i\theta },1+\mu e^{i\theta }+{\displaystyle \frac{m}{\mathsf{p}+\eta }},1+\mu e^{i\theta }+{\displaystyle \frac{L+2m}{\mathsf{p}+\eta }}-{\displaystyle \frac{L}{1+\mathsf{p}+\eta }},\hfill \\ & 1+\mu e^{i\theta }+{\displaystyle \frac{3m}{\mathsf{p}+\eta }}+{\displaystyle \frac{N+3L(2+\mathsf{p}+\eta )}{(\mathsf{p}+\eta )(1+\mathsf{p}+\eta )(2+\mathsf{p}+\eta )}};\varsigma )\notin \aleph ,\hfill \end{array}$$

whenever $\varsigma \in \mathsf{\Theta }$, $\Re \left\{L{e}^{-i\theta }\right\}\ge m(m-1)\mu$, $\Re \left\{N{e}^{-i\theta }\right\}\ge 0$ for all $\theta \in [0,2\pi ]$, and $m\ge 2$.

By applying this definition together with Theorem (1), we obtain the following result.

Corollary 5. 

Let $\aleph \subset \mathbb{C}$ and assume that $\varphi \in {\psi }_1[\aleph ,\mu ]$. If

$$\left|\varsigma {\left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }\right|\le m\mu ,$$

(20)

then

$$\varphi \left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +2}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +3}f\left(\varsigma \right);z\right)\in \aleph ,\varsigma \in \mathsf{\Theta },$$

and hence

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec 1+\mu \varsigma .$$

3. Extended Differential Subordination Results Using ${\mathbb{T}}_{\mathbf{\lambda }}^{\mathsf{p},\mathbf{\eta }}\mathbf{f}\left(\mathbf{\varsigma }\right)$

In this section, we make use of the recurrence relation (9) to derive additional differential subordination results involving ${\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)$. The arguments required are analogous to those used earlier, and for this reason the detailed proofs are not included.

Definition 9. 

Let $\aleph \subset \mathbb{C}$ and $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1\cap {\mathcal{R}}_j$. The class ${\psi }_2[\aleph ,\mathrm{{\rm Y}}]$ consists of admissible functions $\varphi :{\mathbb{C}}^4\times \mathsf{\Theta }\to \mathbb{C}$ such that

$$\varphi ({\mathsf{c}}_1,{\mathsf{c}}_2,{\mathsf{c}}_3,{\mathsf{c}}_4;z)\notin \aleph$$

whenever

$${\mathsf{c}}_1=\mathrm{{\rm Y}}\left(\varrho \right),{\mathsf{c}}_2=\mathrm{{\rm Y}}\left(\varrho \right)+{\displaystyle \frac{m\varrho {\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}{\lambda +\mathsf{p}+2}},$$

$$\Re \left\{{\displaystyle \frac{{\mathsf{c}}_3(1+\mathsf{p}+\lambda )+{\mathsf{c}}_1(2+\mathsf{p}+\lambda )-{\mathsf{c}}_2(3+2\mathsf{p}+2\lambda )}{{\mathsf{c}}_2-{\mathsf{c}}_1}}\right\}\ge m\Re \left\{1+{\displaystyle \frac{\varrho {\mathrm{{\rm Y}}}^{″}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\right\},$$

$$\begin{array}{cc}\hfill \Re \{& {\displaystyle \frac{-12{\mathsf{c}}_1-3{\mathsf{c}}_3(1+\mathsf{p}+\lambda )(2+\mathsf{p}+\lambda )+3{\mathsf{c}}_2(2+\mathsf{p}+\lambda )(3+\mathsf{p}+\lambda )}{{\mathsf{c}}_2-{\mathsf{c}}_1}}\hfill \\ \hfill & +{\displaystyle \frac{(\mathsf{p}+\lambda )\left({\mathsf{c}}_4(1+\mathsf{p}+\lambda )-{\mathsf{c}}_1(7+\mathsf{p}+\lambda )\right)}{{\mathsf{c}}_2-{\mathsf{c}}_1}}\}\ge m^2\Re \left\{{\displaystyle \frac{{\varrho }^2{\mathrm{{\rm Y}}}^{‴}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\right\},\hfill \end{array}$$

for all $\varsigma \in \mathsf{\Theta }$, $\varrho \in \partial \mathsf{\Theta }\setminus E\left(\epsilon \right)$, and $m\ge 2$.

Theorem 2. 

Let $\aleph \subset \mathbb{C}$ and let $\varphi \in {\psi }_2[\aleph ,\mathrm{{\rm Y}}]$. Suppose that $f\in \mathsf{\Lambda }\left(\mathsf{p}\right)$ and $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1$ satisfy

$$\Re \left\{{\displaystyle \frac{\varrho {\mathrm{{\rm Y}}}^{″}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\right\}\ge 0and\left|z{\left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }\right|\le m\left|{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)\right|,$$

(21)

then

$$\left\{\varphi \left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +2}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +1}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right);\varsigma \right):\varsigma \in \mathsf{\Theta }\right\}\subset \aleph ,$$

(22)

which implies

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right).$$

The following Corollary extends Theorem (2) in the case when the boundary behavior of the function $\mathrm{{\rm Y}}\left(\varsigma \right)$ on $\partial \mathsf{\Theta }$ is not known.

Corollary 6. 

Let $\aleph \subset \mathbb{C}$ and let Υ be univalent in Θ with $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1$. Assume that $\varphi \in {\psi }_2[\varsigma ,{\mathrm{{\rm Y}}}_{\epsilon }]$ for some $\epsilon \in (0,1)$, where ${\mathrm{{\rm Y}}}_{\epsilon }\left(\varsigma \right)=\mathrm{{\rm Y}}\left(\epsilon \varsigma \right)$. If $f\in \mathsf{\Lambda }\left(\mathsf{p}\right)$ and ${\mathrm{{\rm Y}}}_{\epsilon }$ satisfy

$$\Re \left\{{\displaystyle \frac{\varrho {\mathrm{{\rm Y}}}^{″}\left(\varrho \right)}{{\mathrm{{\rm Y}}}^{\prime }\left(\varrho \right)}}\right\}\ge 0and\left|z{\left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }\right|\le m\left|{\mathrm{{\rm Y}}}_{\epsilon }^{\prime }\left(\varrho \right)\right|,$$

(23)

then

$$\left\{\varphi \left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +2}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +1}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right);\varsigma \right):\varsigma \in \mathsf{\Theta }\right\}\subset \aleph ,$$

and consequently,

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right).$$

If $\aleph \ne \mathbb{C}$ is a simply connected domain, then there exists a conformal mapping ℓ from $\mathsf{\Theta }$ onto ℵ, that is, $\ell \left(\mathsf{\Theta }\right)=\aleph$. In this case, the class ${\psi }_2[\ell \left(\mathsf{\Theta }\right),\mathrm{{\rm Y}}]$ will be denoted by ${\psi }_2[\ell ,\mathrm{{\rm Y}}]$. The following two corollaries follow directly from Theorem (2) and Corollary (6).

Corollary 7. 

Let ℓ be univalent in Θ and let $\varphi \in {\psi }_2[\ell ,\mathrm{{\rm Y}}]$. If $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1$ satisfies condition (8), then

$$\varphi \left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +2}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +1}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right);\varsigma \right)\prec \ell \left(\varsigma \right),$$

(24)

which yields

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right).$$

Corollary 8. 

Let Υ be univalent in Θ with $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1$, and let $\varphi \in {\psi }_2[h,{\mathrm{{\rm Y}}}_{\epsilon }]$ for some $\epsilon \in (0,1)$, where ${\mathrm{{\rm Y}}}_{\epsilon }\left(\varsigma \right)=\mathrm{{\rm Y}}\left(\epsilon \varsigma \right)$. If ${\mathrm{{\rm Y}}}_{\epsilon }$ satisfies condition (23), then the subordination (24) implies

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\epsilon \varsigma \right).$$

The next Corollary establishes the relationship between the best dominant of a third-order differential subordination and the solution of the corresponding third-order differential equation.

Corollary 9. 

Let ℓ be univalent in Θ and let ℶ be defined by (16) with $\varphi \in {\psi }_2[\ell ,\mathrm{{\rm Y}}]$. If the differential equation

$$\beth \left(\mathrm{{\rm Y}}\left(\varsigma \right),z{\mathrm{{\rm Y}}}^{\prime }\left(\varsigma \right),{\varsigma }^2{\mathrm{{\rm Y}}}^{″}\left(\varsigma \right),{\varsigma }^3{\mathrm{{\rm Y}}}^{‴}\left(\varsigma \right);\varsigma \right)=\ell \left(\varsigma \right)$$

admits a solution $\mathrm{{\rm Y}}\in {\mathbb{Q}}_1$ satisfying condition (21), then the subordination (24) implies

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\prec \mathrm{{\rm Y}}\left(\varsigma \right),$$

and Υ is the best dominant of (24).

Theorem (2) is applicable for the choice $\mathrm{{\rm Y}}\left(\varsigma \right)=1+\mu \varsigma$, where $\mu >0$. According to Definition (9), the corresponding class of admissible functions ${\psi }_2[\aleph ,\mathrm{{\rm Y}}]$, denoted here by ${\psi }_2[\aleph ,\mu ]$, is defined as follows.

Definition 10. 

Let $\aleph \subset \mathbb{C}$ and $\mu >0$. The class ${\psi }_2[\aleph ,\mu ]$ consists of all functions $\varphi :{\mathbb{C}}^4\times \mathsf{\Theta }\to \mathbb{C}$ such that

$$\begin{array}{cc}\hfill & \varphi (1+\mu e^{i\vartheta },1+e^{i\vartheta }\left(\mu +{\displaystyle \frac{m\mu }{2+\mathsf{p}+\lambda }}\right),1+{\displaystyle \frac{L}{(1+\mathsf{p}+\lambda )(2+\mathsf{p}+\lambda )}}+e^{i\vartheta }\left(\mu +{\displaystyle \frac{2m\mu }{1+\mathsf{p}+\lambda }}\right),\hfill \\ \hfill & 1+e^{i\vartheta }\left(\mu +{\displaystyle \frac{3m\mu }{\mathsf{p}+\lambda }}\right)+{\displaystyle \frac{N+3L(2+\mathsf{p}+\lambda )}{(\mathsf{p}+\lambda )(1+\mathsf{p}+\lambda )(2+\mathsf{p}+\lambda )}};z)\notin \aleph ,\hfill \end{array}$$

whenever $\varsigma \in \mathsf{\Theta }$, $\Re \left\{L{e}^{-i\vartheta }\right\}\ge m(m-1)\mu$, $\Re \left\{N{e}^{-i\vartheta }\right\}\ge 0$, for every $\theta \in [0,2\pi ]$ and $m\ge 2$.

Using this Definition together with Theorem (2), we obtain the following result.

Corollary 10. 

Let $\aleph \subset \mathbb{C}$ and $\varphi \in {\psi }_2[\aleph ,\mu ]$. If

$$\left|z{\left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }\right|\le m\mu ,$$

(25)

then

$$\varphi \left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +2}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +1}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right);\varsigma \right)\in \aleph (\varsigma \in \mathsf{\Theta }),$$

and consequently,

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\prec 1+\mu \varsigma .$$

4. Applications and Examples

If we take $\aleph =\{\nu \in \mathbb{C}:|\nu -1|<\mu \}$, and denote ${\psi }_1[\aleph ,\mu ]$ simply by ${\psi }_1\left[\mu \right]$, then Corollary (5) reduces to the following result:

Corollary 11. 

Let $\varphi \in {\psi }_1\left[\mu \right]$ and suppose that

$$|\varsigma {\left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }|\le m\mu ,$$

(26)

then

$$\left|\varphi \right({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +1}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +2}f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta +3}f\left(\varsigma \right):\varsigma )-1|<\mu (\varsigma \in \mathsf{\Theta }),$$

(27)

which implies

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec 1+\mu \varsigma .$$

(28)

Corollary 12. 

Suppose that

$$|\varsigma {\left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }|\le m\mu ,$$

(29)

then

$$|{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)-1|<\mu (\varsigma \in \mathsf{\Theta }),$$

(30)

which implies

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)\prec 1+\mu \varsigma .$$

(31)

Proof. 

The result follows from Corollary (11) by choosing $\varphi ({\mathsf{c}}_1,{\mathsf{c}}_2,{\mathsf{c}}_3,{\mathsf{c}}_4:\varsigma )={\mathsf{c}}_2$. □

Putting $\lambda =\eta =1,$ in Corollary (12), and noting that

$${\mathbb{T}}_1^{\mathsf{p},1}f\left(\varsigma \right)=f\left(\varsigma \right),$$

(32)

we obtain the following result:

Example 3. 

If $f\in \mathsf{\Lambda }\left(\mathsf{p}\right)$ satisfies the following conditions

$$|\varsigma {\left({\varsigma }^{-\mathsf{p}}f\left(\varsigma \right)\right)}^{\prime }|\le m\mu ,$$

(33)

and

$$|{\varsigma }^{-\mathsf{p}+1}{f}^{\prime }\left(\varsigma \right)-\mathsf{p}{\varsigma }^{-\mathsf{p}-1}f\left(\varsigma \right)-1|<\mu ,$$

(34)

then

$$|{\varsigma }^{-\mathsf{p}}f\left(\varsigma \right)-1|<\mu .$$

(35)

Example 4. 

Application of Example 3. Consider the following function. We first define the function and compute its derivative:

$$f\left(\varsigma \right)={\varsigma }^{\mathsf{p}}+2{\varsigma }^{\mathsf{p}+1},f^{\prime }\left(\varsigma \right)=\mathsf{p}{\varsigma }^{\mathsf{p}-1}+2(\mathsf{p}+1){\varsigma }^{\mathsf{p}}.$$

Then

$$|\varsigma {\left({\varsigma }^{-\mathsf{p}}f\left(\varsigma \right)\right)}^{\prime }|=2|\varsigma |,|{\varsigma }^{-\mathsf{p}+1}{f}^{\prime }\left(\varsigma \right)-\mathsf{p}{\varsigma }^{-\mathsf{p}-1}f\left(\varsigma \right)-1|=2|\varsigma |,$$

so that both conditions (33) and (34) hold for $\mu >2\left|\varsigma \right|$. Hence, from (35) we obtain

$$|{\varsigma }^{-\mathsf{p}}f\left(\varsigma \right)-1|=2|\varsigma |<\mu ,$$

confirming that f satisfies the requirements of Example 3.

Furthermore, by putting $\lambda =\eta =1$ and $\mathsf{p}=1$ in Corollary (12), we obtain the following result:

Example 5. 

If $f\in \mathsf{\Lambda }\left(\mathsf{1}\right)$ satisfies the inequalities

$$|\varsigma {\left({\varsigma }^{-1}f\left(\varsigma \right)\right)}^{\prime }|\le m\mu ,$$

(36)

and

$$|f^{\prime }\left(\varsigma \right)+{\displaystyle \frac{1}{\varsigma }}f\left(\varsigma \right)-1|<\mu ,$$

(37)

then

$$|{\varsigma }^{-1}f\left(\varsigma \right)-1|<\mu .$$

(38)

Example 6. 

Application of Example 5. Let us apply the famous Koebe function. Consider

$$f\left(\varsigma \right)={\displaystyle \frac{\varsigma }{{(1-\varsigma )}^2}},f^{\prime }\left(\varsigma \right)={\displaystyle \frac{1+\varsigma }{{(1-\varsigma )}^3}}.$$

First, we verify (36):

$$|\varsigma {\left({\varsigma }^{-1}f\left(\varsigma \right)\right)}^{\prime }|=|{\left({\varsigma }^{-1}f\left(\varsigma \right)\right)}^{\prime }·\varsigma |=\left|{\displaystyle \frac{d}{d\varsigma }}{\displaystyle \frac{1}{{(1-\varsigma )}^2}}·\varsigma \right|=\left|{\displaystyle \frac{2\varsigma }{{(1-\varsigma )}^3}}\right|.$$

Next, we check (37):

$$\left|f^{\prime }\left(\varsigma \right)+{\displaystyle \frac{1}{\varsigma }}f\left(\varsigma \right)-1\right|=\left|{\displaystyle \frac{1+\varsigma }{{(1-\varsigma )}^3}}+{\displaystyle \frac{1}{{(1-\varsigma )}^2}}-1\right|=\left|{\displaystyle \frac{2\varsigma }{{(1-\varsigma )}^3}}\right|.$$

Hence, both inequalities hold for $\mu >{\displaystyle \frac{2\left|\varsigma \right|}{{|1-\varsigma |}^3}}$, and from (38) we obtain

$$\left|{\varsigma }^{-1}f\left(\varsigma \right)-1\right|=\left|{\displaystyle \frac{1}{{(1-\varsigma )}^2}}-1\right|<\mu .$$

This confirms that the Koebe function $f\left(\varsigma \right)={\displaystyle \frac{\varsigma }{{(1-\varsigma )}^2}}$ satisfies the conditions of Example 5.

Putting $\aleph =\{\nu \in \mathbb{C}:|\nu -1|<\mu \}$ as a special case, and denoting ${\psi }_2[\aleph ,\mu ]$ simply by ${\psi }_2\left[\mu \right]$, Corollary (10) reduces to the following result.

Corollary 13. 

Let $\varphi \in {\psi }_2\left[\mu \right]$ and suppose that

$$|\varsigma {\left({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\right)}^{\prime }|\le m\mu ,$$

(39)

then

$$\left|\varphi \right({\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +2}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +1}^{\mathsf{p},\eta }f\left(\varsigma \right),{\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right):\varsigma )-1|<\mu (\varsigma \in \mathsf{\Theta }),$$

(40)

which implies

$${\varsigma }^{-\mathsf{p}}{\mathbb{T}}_{\lambda +3}^{\mathsf{p},\eta }f\left(\varsigma \right)\prec 1+\mu \varsigma .$$

(41)

5. Conclusions

In this work, we have identified sufficient criteria for the classes ${\psi }_1[\aleph ,\mathrm{{\rm Y}}]$ and ${\psi }_2[\aleph ,\mathrm{{\rm Y}}]$ of admissible functions, enabling the derivation of notable results concerning third-order differential subordination for $\mathsf{p}$-valent analytic functions that incorporate the linear operator ${\mathbb{T}}_{\lambda }^{\mathsf{p},\eta }f\left(\varsigma \right)$ linked to the Tremblay Fractional Derivative Operator. Moreover, we have explored particular cases of these admissible function classes and established several important inequalities. The findings obtained here extend and complement prior studies in the field of differential subordination and superordination within Geometric Function Theory.