Third-Order Differential Subordination for Analytic Functions Involving the Lommel Function of the First Kind

Abstract

This study investigates third-order differential subordination and its influence on classes of analytic functions associated with the Lommel function of the first kind. By employing a newly defined operator ${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)$, we identify and characterize the admissible function classes that satisfy the corresponding third-order differential subordinations. These admissibility conditions enable the derivation of several key results, including a sandwich-type theorem obtained as a direct consequence of the established framework. The findings contribute to a broader understanding of analytic functions governed by higher-order differential constraints and highlight the significant role played by the Lommel function in shaping these geometric properties.

1. Introduction

The development of complex analysis traces back to the attempts to solve cubic equations, eventually giving rise to a comprehensive theory shaped by Euler, Gauss, Cauchy, and Riemann. A decisive milestone appeared in 1851 when Riemann formulated his mapping theorem, laying the groundwork for what became Geometric Function Theory (GFT). Since then, GFT has evolved into a central area of modern mathematics with influential links to operator theory and differential inequalities.

Koebe’s refinement of the Riemann mapping theorem in 1907 [1] and Lindelöf’s introduction of subordination in 1909 [1] marked the early formation of the theory. A systematic framework for differential subordination was later established by Miller and Mocanu in their seminal works [2,3,4], culminating with the introduction of differential superordination in 2003 [5]. These dual concepts have become fundamental tools for deriving sandwich-type theorems in the study of analytic and meromorphic functions. For further related developments involving third-order differential subordination and superordination, fractional operators, and applications to integral transforms, the reader may consult [6,7,8,9,10].

However, a key distinction exists between our study and the aforementioned works: we focus on analytic functions involving the Lommel function, a class for which third-order differential subordination has not yet been established. Our research addresses this gap by deriving the third-order differential subordination results, providing special cases that reduce to the original function and its derivative, and presenting novel applications and illustrative examples that go beyond the conventional framework.

The theory continued to expand in the twenty-first century. Antonino and Miller [11] extended differential subordination to the third order, while Tang et al. [12] developed corresponding third-order superordination results. Further noteworthy contributions include those of Ibrahim et al. [13], Morais and Zayed [14], and Lupas and Oros [15], whose works integrate fractional calculus with subordination and superordination techniques. Recent studies have continued to advance these themes [16,17,18,19,20].

Let $\nabla =\{z\in \mathrm{C}:|z|<1\}$ be a symmetric domain with respect to both the x-axis and the y-axis. We denote by $\mathcal{K}(\nabla )$ the class of analytic functions defined in ∇. For any $a\in \mathrm{C}$ and $\mathrm{b}\in \mathbb{N}$, define

$$\mathcal{K}[a,\mathrm{b}]=\left\{f\in \mathcal{K}(\nabla ):f\left(z\right)=a+a_{\mathrm{b}}{z}^{\mathrm{b}}+a_{\mathrm{b}+1}{z}^{\mathrm{b}+1}+\cdots \right\}.$$

In particular, ${\mathcal{K}}_0=\mathcal{K}[0,1]$ and ${\mathcal{K}}_1=\mathcal{K}[1,1]$.

The class $\mathcal{A}$ consists of normalized analytic functions in ∇ of the form

$$f\left(z\right)=z+\sum _{\mathrm{b}=2}^{\infty }{a}_{\mathrm{b}}{z}^{\mathrm{b}},z\in \nabla .$$

(1)

For two analytic functions $f_1$ and $f_2$ in ∇, we write $f_1\prec f_2$ whenever there exists an analytic function v in ∇ satisfying $v\left(0\right)=0$ and $\left|v\right(z\left)\right|<1$ such that

$$f_1\left(z\right)=f_2\left(v\left(z\right)\right),z\in \nabla .$$

Moreover, if $f_2$ is univalent in ∇, then the subordination relation $f_1\prec f_2$ is equivalent to the conditions $f_1\left(0\right)=f_2\left(0\right)$ and $f_1(\nabla )\subset f_2(\nabla )$.

Definition 1

([11]). Let $\mathcal{L}:{\mathrm{C}}^4\times \nabla \to \mathrm{C}$ be an operator, and let $h\left(z\right)$ be univalent in ∇. A function $\mathrm{s}\left(z\right)$ analytic in ∇ is said to satisfy a third-order differential subordination whenever

$$\mathcal{L}\left(\mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z\right)\prec h\left(z\right).$$

In this case, the function $\mathrm{s}\left(z\right)$ is regarded as a solution of the differential subordination.

A univalent function $\mathrm{t}\left(z\right)$ is called a dominant if every function $\mathrm{s}\left(z\right)$ satisfying the above condition also satisfies $\mathrm{s}\left(z\right)\prec \mathrm{t}\left(z\right)$. If there exists a dominant $\tilde{\mathrm{t}}\left(z\right)$ such that $\tilde{\mathrm{t}}\left(z\right)\prec \mathrm{t}\left(z\right)$ for all other dominants $\mathrm{t}\left(z\right)$, then $\tilde{\mathrm{t}}\left(z\right)$ is referred to as the best dominant.

Definition 2

([12]). Consider the operator $\mathcal{L}:{\mathrm{C}}^4\times \nabla \to \mathrm{C}$, and assume that $h\left(z\right)$ is univalent in ∇. Suppose that the function $\mathrm{s}\left(z\right)$ and the quantity $\mathcal{L}\left(\mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z\right)$ are both univalent in ∇ and satisfy the third-order differential superordination

$$\mathrm{t}\left(z\right)\prec \mathcal{L}\left(\mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z\right).$$

(2)

In this case, the function $\mathrm{s}\left(z\right)$ is viewed as a solution of the differential superordination associated with (2).

Furthermore, an analytic function $\mathrm{t}\left(z\right)$ is called a subordinant of the solutions of (2) if $\mathrm{t}\left(z\right)\prec \mathrm{s}\left(z\right)$ for every function $\mathrm{s}\left(z\right)$ fulfilling (2).

The best subordinant associated with the differential superordination in (2) is the univalent function $\mathrm{t}\left(z\right)$ that lies below every other subordinant of (2), that is, $\mathrm{t}\left(z\right)\prec \tilde{\mathrm{t}}\left(z\right)$ for all subordinants $\mathrm{t}\left(z\right)$ corresponding to (2).

Motivated by recent developments involving special functions, we investigate certain integral operators constructed from the Lommel function of the first kind. This function possesses a convenient hypergeometric-type expansion, making it suitable for analyzing mapping behavior and geometric aspects of related operators [21].

$$l_{\vartheta ,\gamma }\left(z\right)={\displaystyle \frac{z^{\vartheta +1}}{(\vartheta -\gamma +1)(\vartheta +\gamma +1)}}{}_{1}F_2\left(1;{\displaystyle \frac{\vartheta -\gamma +3}{2}},{\displaystyle \frac{\vartheta +\gamma +3}{2}};-{\displaystyle \frac{z^2}{4}}\right),$$

where $\gamma$ is not a negative odd integer. The Lommel function $l_{\vartheta ,\gamma }$ appears as a particular solution of the nonhomogeneous Bessel equation (see [22]):

$$z^2{w}^{″}\left(z\right)+z{w}^{\prime }\left(z\right)+(z^2-{\gamma }^2)w\left(z\right)=z^{\vartheta +1},$$

(3)

It is evident that $l_{\vartheta ,\gamma }\left(z\right)$ does not lie in the normalized class $\mathcal{A}$. Recently, Yağmur [23] and Baricz et al. [24] introduced the function

$$h_{\vartheta ,\gamma }\left(z\right)=(\vartheta -\gamma +1)(\vartheta +\gamma +1)z^{(1-\vartheta )/2}{l}_{\vartheta ,\gamma }\left(\sqrt{z}\right),$$

and investigated several geometric features of $h_{\vartheta ,\gamma }$. For further details regarding properties of the Lommel function, see [25,26].

The function $h_{\vartheta ,\gamma }\left(z\right)$ belongs to the class $\mathcal{A}$ of normalized analytic functions and is given by (see [27])

$$h_{\vartheta ,\gamma }\left(z\right)=\sum _{\mathrm{b}=1}^{\infty }{\displaystyle \frac{{(-1/4)}^{\mathrm{b}-1}}{{\left(\frac{\vartheta -\gamma +3}{2}\right)}_{\mathrm{b}-1}{\left(\frac{\vartheta +\gamma +3}{2}\right)}_{\mathrm{b}-1}}}{z}^{\mathrm{b}}=z+\sum _{\mathrm{b}=2}^{\infty }{\displaystyle \frac{{(-1/4)}^{\mathrm{b}-1}}{{\left(\frac{\vartheta -\gamma +3}{2}\right)}_{\mathrm{b}-1}{\left(\frac{\vartheta +\gamma +3}{2}\right)}_{\mathrm{b}-1}}}{z}^{\mathrm{b}}$$

(4)

where ${(⋉)}_{\mathrm{b}}$ denotes the Pochhammer symbol, defined by

$${(⋉)}_0=1,{(⋉)}_{\mathrm{b}}={\displaystyle \frac{\Gamma (⋉+\mathrm{b})}{\Gamma (⋉)}}(\mathrm{b}\in \mathbb{N}).$$

By choosing special values of $(\vartheta ,\gamma )$, the function takes familiar closed forms:

$$h_{1/2,1/2}\left(z\right)=2\left(1-cos\left(\sqrt{z}\right)\right),h_{3/2,1/2}\left(z\right)=6\left(1-{\displaystyle \frac{sin\left(\sqrt{z}\right)}{\sqrt{z}}}\right),$$

and

$$h_{5/2,1/2}\left(z\right)=12\left({\displaystyle \frac{z+2cos\left(\sqrt{z}\right)-2}{z}}\right).$$

The following Mathematica plots depict the modulus and the imaginary part of $h_{\vartheta ,\gamma }\left(z\right)$ over the unit disk (symmetric domain).

The three-dimensional visualization provides deeper insight into the geometric behavior of the function by illustrating how its magnitude and imaginary part vary over the complex plane, revealing features such as growth, symmetry, and mapping properties that are not evident in two-dimensional plots.

Definition 3.

The mapping ${\mathbb{L}}_{\vartheta ,\gamma }$ is defined via the Hadamard product with the Lommel-type function $h_{\vartheta ,\gamma }\left(z\right)$ as

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)=f\left(z\right)\ast h_{\vartheta ,\gamma }\left(z\right),$$

where, for notational convenience, we set

$$\mathrm{j}={\displaystyle \frac{\vartheta -\gamma +3}{2}},\mathrm{w}={\displaystyle \frac{\vartheta +\gamma +3}{2}}.$$

(5)

The parameters ϑ and γ are assumed to satisfy the restriction

$${\displaystyle \frac{-\vartheta \pm \gamma -3}{2}}\notin \mathbb{N}:=\{1,2,\dots \},$$

which ensures that $\mathrm{j}$ and $\mathrm{w}$ do not take values that lead to singularities in the associated hypergeometric representation of $h_{\vartheta ,\gamma }\left(z\right)$.

Equivalently, for a function $f\in \mathcal{A}$ with Taylor expansion as in (1), the operator ${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}$ takes the form

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)=z+\sum _{\mathrm{b}=2}^{\infty }{\displaystyle \frac{{(-1/4)}^{\mathrm{b}-1}}{{\left(\mathrm{j}\right)}_{\mathrm{b}-1}{\left(\mathrm{w}\right)}_{\mathrm{b}-1}}}{a}_{\mathrm{b}}{z}^{\mathrm{b}}$$

(6)

which ensures that ${\mathbb{L}}_{\vartheta ,\gamma }$ preserves the class $\mathcal{A}$. Moreover, using the identity associated with the operator ${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}$, we obtain

$$z{\left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\right)}^{\prime }=(\mathrm{j}+\mathrm{w}){\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)-(\mathrm{j}+\mathrm{w}-1){\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)$$

(7)

Definition 4

([11]). Let $\mathbb{M}$ denote the class of all functions $\mathrm{t}$ that are analytic and one-to-one in $\nabla \cup \left(\partial \nabla \setminus E\left(\mathrm{t}\right)\right)$, where

$$E\left(\mathrm{t}\right)=\left\{\upsilon \in \partial \nabla :\underset{z\to \upsilon }{lim}\mathrm{t}\left(z\right)=\infty \right\}.$$

(8)

Moreover, it is assumed that

$$\underset{\upsilon \in \partial \nabla \setminus E\left(\mathrm{t}\right)}{min}\left|{\mathrm{t}}^{\prime }\left(\upsilon \right)\right|=\rho >0.$$

Let $\mathbb{M}\left(a\right)$ denote the subclass of $\mathbb{M}$ consisting of functions $\mathrm{t}$ satisfying $\mathrm{t}\left(0\right)=a$. In particular, we write $\mathbb{M}\left(1\right)\equiv {\mathbb{M}}_1$.

Definition 5

([11]). Let $\Theta \subset \mathrm{C}$, let $\mathrm{t}\in \mathbb{M}$, and let $\mathrm{b}\in \mathbb{N}\setminus \left\{1\right\}$. The class of admissible functions ${\mathcal{L}}_{\mathrm{b}}[\Theta ,\mathrm{t}]$ consists of all functions $\mathcal{L}:{\mathrm{C}}^4\times \nabla \to \mathrm{C}$ satisfying the admissibility condition

$$\mathcal{L}({\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4;z)\notin \Theta$$

whenever

$${\eta }_1=\mathrm{t}\left(\upsilon \right),{\eta }_2=\mathsf{\ell }\upsilon {\mathrm{t}}^{\prime }\left(\upsilon \right),$$

and

$$\Re \left({\displaystyle \frac{{\eta }_3}{{\eta }_2}}+1\right)\ge \mathsf{\ell }\Re \left(1+{\displaystyle \frac{\upsilon {\mathrm{t}}^{″}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right),\Re \left({\displaystyle \frac{{\eta }_4}{{\eta }_2}}\right)\ge {\mathsf{\ell }}^2\Re \left({\displaystyle \frac{{\upsilon }^2{\mathrm{t}}^{‴}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right),$$

for $z\in \nabla$, $\upsilon \in \partial \nabla \setminus E\left(\mathrm{t}\right)$, and $\mathsf{\ell }\ge \mathrm{b}$.

The following lemmas serve as the foundation of third-order differential subordination and superordination theory.

Lemma 1

([11]). Let $\mathrm{s}\in \mathcal{K}[a,\mathrm{b}]$ with $\mathrm{b}\ge 2$, and let $\mathrm{t}\in \mathbb{M}\left(a\right)$ satisfy

$$\Re \left({\displaystyle \frac{\upsilon {\mathrm{t}}^{″}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right)\ge 0,\left|{\displaystyle \frac{z{\mathrm{s}}^{\prime }\left(z\right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right|\le \mathsf{\ell },$$

for $z\in \nabla$, $\upsilon \in \partial \nabla \setminus E\left(\mathrm{t}\right)$, and $\mathsf{\ell }\ge \mathrm{b}$. If $\Theta \subset \mathrm{C}$ and $\mathcal{L}\in {\mathcal{L}}_{\mathrm{b}}[\Theta ,\mathrm{t}]$ satisfy

$$\mathcal{L}\left(\mathrm{t}\left(z\right),z{\mathrm{t}}^{\prime }\left(z\right),z^2{\mathrm{t}}^{″}\left(z\right),z^3{\mathrm{t}}^{‴}\left(z\right);z\right)\in \Theta ,$$

then

$$\mathrm{s}\left(z\right)\prec \mathrm{t}\left(z\right)(z\in \nabla ).$$

Definition 6

([12]). Let Θ be a subset of $\mathrm{C}$, and let $\mathrm{t}\in \mathcal{K}[a,\mathrm{b}]$ with ${\mathrm{t}}^{\prime }\left(z\right)\ne 0$ and $\mathrm{b}\in \mathbb{N}\setminus \left\{1\right\}$. Define $\overline{\nabla }=\{z\in \mathrm{C}:|z|\le 1\}$. The class ${\mathcal{L}}_{\mathrm{b}}[\Theta ,\mathrm{t}]$ is the set of all functions

$$\mathcal{L}:{\mathrm{C}}^4\times \overline{\nabla }\to \mathrm{C}$$

satisfying

$$\mathcal{L}({\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4;\upsilon )\in \Theta$$

whenever

$${\eta }_1=\mathrm{t}\left(z\right),{\eta }_2={\displaystyle \frac{z{\mathrm{t}}^{\prime }\left(z\right)}{\mathsf{\ell }}},$$

and

$$\Re \left({\displaystyle \frac{{\eta }_3}{{\eta }_2}}+1\right)\le {\displaystyle \frac{1}{\mathsf{\ell }}}\Re \left(1+{\displaystyle \frac{z{\mathrm{t}}^{″}\left(z\right)}{{\mathrm{t}}^{\prime }\left(z\right)}}\right),$$

$$\Re \left({\displaystyle \frac{{\eta }_4}{{\eta }_2}}\right)\le {\displaystyle \frac{1}{{\mathsf{\ell }}^2}}\Re \left({\displaystyle \frac{z^2{\mathrm{t}}^{‴}\left(z\right)}{{\mathrm{t}}^{\prime }\left(z\right)}}\right),$$

for $z\in \nabla$, $\upsilon \in \partial \nabla$, and $\mathsf{\ell }\ge \mathrm{b}$.

Lemma 2

([12]). Let $\mathcal{L}\in {\mathcal{L}}_{\mathrm{b}}^{\prime }[\Theta ,\mathrm{t}]$ and assume that

$$\mathcal{L}\left(\mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z\right)$$

is univalent in ∇, where $\mathrm{s}\in \mathbb{M}\left(a\right)\cap \mathcal{K}[a,\mathrm{b}]$ satisfies

$$\Re \left\{{\displaystyle \frac{\upsilon {\mathrm{t}}^{″}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right\}\ge 0,\left|{\displaystyle \frac{z{\mathrm{s}}^{\prime }\left(z\right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right|\le \mathsf{\ell },$$

for $z\in \nabla$, $\upsilon \in \partial \nabla$, and $\mathsf{\ell }\ge \mathrm{b}\ge 2$. If

$$\Theta \subset \left\{\mathcal{L}\left(\mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z\right):z\in \nabla \right\},$$

then

$$\mathrm{t}\left(z\right)\prec \mathrm{s}\left(z\right)(z\in \nabla ).$$

In what follows, we take the Lommel function of the first kind as the main tool and introduce an operator acting on normalized analytic functions that incorporates this function. Using differential subordination ideas, we set criteria that guarantee a sandwich-type inclusion of the form

$${\mathrm{t}}_1\left(z\right)\prec {\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec {\mathrm{t}}_2\left(z\right),z\in \nabla ,$$

where ${\mathrm{t}}_1$ and ${\mathrm{t}}_2$ are univalent in ∇, and ${\mathbb{L}}_{\vartheta ,\gamma }$ is the operator determined by the Lommel function of the first kind. For broader context and recent contributions in this direction, see [28,29,30,31].

2. Comprehensive Results on Third-Order Differential Subordination

In this section, we start with a chosen set $\Theta$ and a function $\mathrm{t}$, and our goal is to determine a group of admissible operators $\mathcal{L}$ for which the requirement in (1) is satisfied. To build the basic third-order differential subordination results linked to the operator ${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)$, given in (5), we introduce a new class of admissible functions that will serve as a key ingredient in what follows.

Definition 7.

Let $\Theta \subset \mathrm{C}$ and let $\mathrm{t}\in \mathcal{K}[a,\mathrm{b}]$ with ${\mathrm{t}}^{\prime }\left(z\right)\ne 0$, where $\mathrm{b}\in {\mathbb{M}}_0\cap {\mathcal{K}}_0$. The class of admissible functions ${\beth }_j[\Theta ,\mathrm{t}]$ is defined as the set of all mappings $\psi :{\mathrm{C}}^4\times \nabla \to \mathrm{C}$ that satisfy the corresponding admissibility condition.

$$\psi ({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4;z)\notin \Theta ,$$

whenever

$${\lambda }_1=\mathrm{t}\left(\upsilon \right),{\lambda }_2={\displaystyle \frac{(\mathrm{j}+\mathrm{b})\mathrm{t}\left(\upsilon \right)-\mathsf{\ell }\upsilon {\mathrm{t}}^{\prime }\left(\upsilon \right)}{\mathrm{j}+\mathrm{b}-1}},$$

$$\Re \left({\displaystyle \frac{{(\mathrm{j}+\mathrm{b})}^2{\lambda }_1+(\mathrm{j}+\mathrm{b}-1){\lambda }_2-(\mathrm{j}+\mathrm{b})(\mathrm{j}+\mathrm{b}-1){\lambda }_3}{(\mathrm{j}+\mathrm{b}){\lambda }_1-(\mathrm{j}+\mathrm{b}-1){\lambda }_2}}\right)\ge \mathsf{\ell }\Re \left(1+{\displaystyle \frac{\upsilon {\mathrm{t}}^{″}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right),$$

and

$$\Re \left\{{\displaystyle \frac{\begin{array}{cc}& (\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b})(\mathrm{j}+\mathrm{b}-1){\lambda }_4-(\mathrm{j}+\mathrm{b})(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b}-2){\lambda }_3\hfill \\ & -(\mathrm{j}+\mathrm{b}-1)({\mathrm{j}}^2+{\mathrm{b}}^2+2\mathrm{b}\mathrm{j}+\mathrm{j}+\mathrm{b}+6){\lambda }_2+(\mathrm{j}+\mathrm{b})[(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b}-2)\hfill \\ & +({\mathrm{j}}^2+{\mathrm{b}}^2+2\mathrm{b}\mathrm{j}+3\mathrm{b}+3\mathrm{j}+2)-(\mathrm{j}+\mathrm{b}+2)(\mathrm{j}+\mathrm{b}+1)]{\lambda }_1\hfill \end{array}}{(\mathrm{j}+\mathrm{b}){\lambda }_1-(\mathrm{j}+\mathrm{b}-1){\lambda }_2}}\right\}\le {\mathsf{\ell }}^2\Re \left({\displaystyle \frac{z^2{\mathrm{t}}^{‴}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right),$$

for all $z\in \nabla$, $\upsilon \in \partial \nabla -E\left(\mathrm{t}\right)$, and $\mathsf{\ell }\ge 2$.

Theorem 1.

Let $\psi \in {\beth }_1(\Theta ,\mathrm{t})$. If $f\in \mathcal{A}$ and $\mathrm{t}\in {\mathbb{M}}_0$ attain the following conditions:

$$\Re \left\{{\displaystyle \frac{\upsilon {\mathrm{t}}^{″}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right\}\ge 0,\left|{\displaystyle \frac{{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right|\le \mathsf{\ell },$$

(9)

and

$$\left\{\psi \left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\in \nabla \right)\right\}\subset \Theta ,$$

(10)

then

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec \mathrm{t}\left(z\right).$$

Proof. 

Let $\mathrm{s}\left(z\right)$ be an analytic function defined on the symmetric domain ∇ by

$$\mathrm{s}\left(z\right)={\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right).$$

(11)

It follows from (7) and (11) that, by multiplying both sides of (11) by z, differentiating, and then using the relation in (7), we obtain

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)={\displaystyle \frac{(\mathrm{j}+\mathrm{b})\mathrm{s}\left(z\right)-z{\mathrm{s}}^{\prime }\left(z\right)}{\mathrm{j}+\mathrm{b}-1}}.$$

(12)

Similarly, by multiplying both sides of (12) by z, differentiating, and applying (7), we obtain

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right)={\displaystyle \frac{(\mathrm{j}+\mathrm{b}+2)(\mathrm{j}+\mathrm{b})\mathrm{s}\left(z\right)-2z{\mathrm{s}}^{\prime }\left(z\right)-z^2{\mathrm{s}}^{″}\left(z\right)}{(\mathrm{j}+\mathrm{b})(\mathrm{j}+\mathrm{b}-2)}},$$

(13)

and

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right)={\displaystyle \frac{\begin{array}{cc}\hfill & \left((\mathrm{j}+\mathrm{b}+2)(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b})\mathrm{s}\left(z\right)\right)-({\mathrm{j}}^2+{\mathrm{b}}^2+2\mathrm{b}\mathrm{j}+3\mathrm{b}+3\mathrm{j}+2)z{\mathrm{s}}^{\prime }\left(z\right)\hfill \\ \hfill & -(\mathrm{j}+\mathrm{b}-2)z^2{\mathrm{s}}^{″}\left(z\right)+z^3{\mathrm{s}}^{‴}\left(z\right)\hfill \end{array}}{ (\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}}.$$

(14)

Now, we define a transformation (see Definition 6) from ${\mathrm{C}}^4$ to $\mathrm{C}$ as follows:

$${\lambda }_1={\eta }_1,{\lambda }_2={\displaystyle \frac{(\mathrm{j}+\mathrm{b}){\eta }_1-{\eta }_2}{\mathrm{j}+\mathrm{b}-1}},{\lambda }_3={\displaystyle \frac{(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}){\eta }_1-2{\eta }_2-{\eta }_3}{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}}.$$

(15)

and

$${\lambda }_4={\displaystyle \frac{(\mathrm{j}+\mathrm{b}+2)(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}){\eta }_1-({\mathrm{j}}^2+{\mathrm{b}}^2+2\mathrm{b}\mathrm{j}+3\mathrm{b}+3\mathrm{j}+2){\eta }_2-(\mathrm{j}+\mathrm{b}-2){\eta }_3+{\eta }_4}{(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}}$$

(16)

Let $\mathcal{L}({\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4;z)=\psi ({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4;z)$.

$$=\psi \left(\begin{array}{c}{\eta }_1,{\displaystyle \frac{(\mathrm{j}+\mathrm{b}){\eta }_1-{\eta }_2}{\mathrm{j}+\mathrm{b}-1}},{\displaystyle \frac{(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}){\eta }_1-2{\eta }_2-{\eta }_3}{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}},\hfill \\ {\displaystyle \frac{(\mathrm{j}+\mathrm{b}+2)(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}){\eta }_1-({\mathrm{j}}^2+{\mathrm{b}}^2+2\mathrm{b}\mathrm{j}+3\mathrm{b}+3\mathrm{j}+2){\eta }_2-(\mathrm{j}+\mathrm{b}-2){\eta }_3+{\eta }_4}{(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}}\hfill \end{array};z\right)$$

(17)

Employing (11) to (14), we yield

$$\mathcal{L}\left(\mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z\right)=\psi \left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\right).$$

(18)

$$\mathcal{L}\left(\mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z\right)\in \nabla$$

Therefore, from the previous results, Equation (10) can be expressed as

$${\displaystyle \frac{{\eta }_3}{{\eta }_2}}+1={\displaystyle \frac{(\mathrm{j}+\mathrm{b}){\lambda }_1+(\mathrm{j}+\mathrm{b}-1){\lambda }_2-(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b}){\lambda }_3}{(\mathrm{j}+\mathrm{b}){\lambda }_1-(\mathrm{j}+\mathrm{b}-1){\lambda }_2}},$$

(19)

where the parameters are given by

$$\begin{array}{cc}\hfill {\eta }_1& \to {\lambda }_1,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\eta }_2& \to (\mathrm{j}+\mathrm{b})({\lambda }_1-{\lambda }_2)+{\lambda }_2,\hfill \end{array}$$

(21)

$${\eta }_3\to (\mathrm{j}+\mathrm{b}-1)\left(2{\lambda }_2+(\mathrm{j}+\mathrm{b})({\lambda }_1-{\lambda }_3)\right),$$

(22)

$$\begin{array}{cc}\hfill {\eta }_4& \to (\mathrm{j}+\mathrm{b}-1)(-6{\lambda }_2+(\mathrm{j}+\mathrm{b})((\mathrm{j}+\mathrm{b}-2){\lambda }_1-{\lambda }_2+2{\lambda }_3\hfill \\ \hfill & -(\mathrm{j}+\mathrm{b})({\lambda }_2+{\lambda }_3-{\lambda }_4)+{\lambda }_4\left)\right).\hfill \end{array}$$

(23)

Consequently, we have

$${\displaystyle \frac{{\eta }_4}{{\eta }_2}}={\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)\left(-6{\lambda }_2+(\mathrm{j}+\mathrm{b})\left((\mathrm{j}+\mathrm{b}-2){\lambda }_1-{\lambda }_2+2{\lambda }_3-(\mathrm{j}+\mathrm{b})({\lambda }_2+{\lambda }_3-{\lambda }_4)+{\lambda }_4\right)\right)}{(\mathrm{j}+\mathrm{b}){\lambda }_1-(\mathrm{j}+\mathrm{b}-1){\lambda }_2}}.$$

(24)

Hence, the admissibility condition for $\psi \in {\beth }_1[\Theta ,\mathrm{t}]$ in Definition 6 is equivalent to the admissibility condition for $\mathcal{L}\in {\beth }_{\mathrm{b}}[\Theta ,\mathrm{t}]$. Thus, by applying Definition 4 together with Lemma 1 for $\mathrm{b}=2$, and utilizing Equation (9), we obtain

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec \mathrm{t}\left(z\right).$$

(25)

This completes the theorem. □

The next outcome is an extension of Theorem 1 to the case when the behavior of $\mathrm{t}\left(z\right)$ is on $\partial \nabla$.

Corollary 1.

Let $\Theta \subset \mathrm{C}$ and let $\mathrm{t}\left(z\right)$ be univalent in ∇ with $\mathrm{t}\left(0\right)=0$. Let $\psi \in {\beth }_1[\Theta ,{\mathrm{t}}_{\epsilon }]$ for some $\epsilon \in (0,1)$, where ${\mathrm{t}}_{\epsilon }\left(z\right)=\mathrm{t}\left(\epsilon z\right)$. If $\psi \in \mathcal{A}$ attains

$$\Re \left(\upsilon {\displaystyle \frac{{\mathrm{t}}_{\epsilon }^{″}\left(\upsilon \right)}{{\mathrm{t}}_{\epsilon }^{\prime }\left(\upsilon \right)}}\right)\ge 0,\left|{\displaystyle \frac{z{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)}{{\mathrm{t}}_{\epsilon }^{\prime }\left(\upsilon \right)}}\right|\le \mathsf{\ell },$$

and

$$\psi \left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\right)\subset \Theta ,$$

then

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec \mathrm{t}\left(z\right),z\in \nabla .$$

Proof. 

Theorem 1 yields

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec {\mathrm{t}}_{\epsilon }\left(z\right).$$

Therefore,

$${\mathrm{t}}_{\epsilon }\left(z\right)\prec \mathrm{t}\left(z\right),z\in \nabla .$$

This completes the proof. □

If $\Theta \ne \mathrm{C}$ is a simply connected domain, then $\Theta =h(\nabla )$ for some conformal mapping $h\left(z\right)$ of ∇ onto $\Theta$. In this setting, the class ${\beth }_1[h(\nabla ),\mathrm{t}]$ is denoted by ${\beth }_1[h,\mathrm{t}]$. The next result is an immediate implication of Theorem 1.

Theorem 2.

Let $\psi \in {\beth }_j[h,\mathrm{t}]$. If $f\in \mathcal{A}$ and $\mathrm{t}\in {\mathbb{M}}_1$ satisfy

$$\Re \left(\upsilon {\displaystyle \frac{{\mathrm{t}}_{\epsilon }^{″}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right)\ge 0,\left|{\displaystyle \frac{{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right|\le \mathsf{\ell }$$

(26)

and if

$$\psi \left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\right)\prec h\left(z\right)$$

(27)

then

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec \mathrm{t}\left(z\right),z\in \nabla .$$

Corollary 2.

Assume that $\Theta \subset \mathrm{C}$ and that $\mathrm{t}\left(z\right)$ is univalent in ∇ with $\mathrm{t}\left(0\right)=0$. Let $\psi \in {\beth }_1[h,{\mathrm{t}}_{\epsilon }]$ for some $\epsilon \in (0,1)$, where ${\mathrm{t}}_{\epsilon }\left(z\right)=\mathrm{t}\left(\epsilon z\right)$. If $f\in \mathcal{A}$ satisfies, for every $z\in \nabla$ and every $\upsilon \in \partial U\setminus E\left({\mathrm{t}}_{\epsilon }\right)$,

$$\Re \left(\upsilon {\displaystyle \frac{{\mathrm{t}}_{\epsilon }^{″}\left(\upsilon \right)}{{\mathrm{t}}_{\epsilon }^{\prime }\left(\upsilon \right)}}\right)\ge 0,\left|{\displaystyle \frac{{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)}{{\mathrm{t}}_{\epsilon }^{\prime }\left(\upsilon \right)}}\right|\le \mathsf{\ell },$$

and if

$$\psi \left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\right)\prec h\left(z\right),$$

then the subordination

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec \mathrm{t}\left(z\right)(z\in \nabla )$$

holds.

Theorem 3.

The following result establishes the best dominant associated with the differential subordination given in Equation (27).

Let h be a univalent function in ∇. Moreover, let $\psi :{\mathrm{C}}^4\times \nabla \to \mathrm{C}$ and the operator $\mathcal{L}$ be defined as in (17). Consider the differential equation

$$\mathcal{L}\left(\mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z\right)=h\left(z\right),$$

(28)

which admits a solution $\mathrm{t}\left(z\right)$ satisfying $\mathrm{t}\left(0\right)=0$ and fulfilling condition (9). If $f\in \mathcal{A}$ satisfies the differential subordination (27) and if

$$\psi \left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\right)$$

is analytic in ∇, then the subordination

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec \mathrm{t}\left(z\right)(z\in \nabla )$$

holds true, and $\mathrm{t}\left(z\right)$ serves as the best dominant.

Proof. 

By virtue of Theorem 1, it follows that $\mathrm{t}$ is a dominant for the differential subordination given in (27). Furthermore, since $\mathrm{t}$ satisfies Equation (28), it also constitutes a solution of (27). Consequently, $\mathrm{t}$ is subordinate to every other dominant of (27). Hence, $\mathrm{t}$ is identified as the best dominant. □

In view of Definition 6 and by considering the particular choice $\mathrm{t}\left(z\right)=1+\mathcal{J}z$ with $\mathcal{J}>0$, the admissible class of functions ${\beth }_1[\Theta ,\mathrm{t}]$, which is denoted by ${\beth }_1[\Theta ,\mathcal{J}]$, can be formulated as follows.

Definition 8.

Let Θ be a set in $\mathrm{j}$ and $\mathcal{J}>0$. The admissible class of functions ${\beth }_j[\Theta ,\mathcal{J}]$ includes all $\psi :{\mathrm{j}}^4\times \nabla \to \mathrm{j}$ which attain

$$\psi \left(\begin{array}{c}1+\mathcal{J}{e}^{i\theta },\hfill \\ {\displaystyle \frac{(\mathrm{j}+\mathrm{b}-\mathsf{\ell })\mathcal{J}{e}^{i\theta }-(\mathrm{j}+\mathrm{b})}{\mathrm{j}+\mathrm{b}}},\hfill \\ {\displaystyle \frac{(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b})(1+\mathcal{J}{e}^{i\theta })-2\mathsf{\ell }\mathcal{J}{e}^{i\theta }-L}{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}},\hfill \\ {\displaystyle \frac{\begin{array}{c}\mathrm{b}+(\mathrm{j}+\mathrm{b}-2)L\hfill \\ -({\mathrm{j}}^2+{\mathrm{b}}^2+2\mathrm{b}\mathrm{j}+3\mathrm{b}+3\mathrm{j}+2)\mathsf{\ell }\mathcal{J}{e}^{i\theta }\hfill \\ +(\mathrm{j}+\mathrm{b}+2)(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b})(1+\mathcal{J}{e}^{i\theta })\hfill \end{array}}{(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}}\hfill \end{array};z\right)$$

(29)

where $z\in \nabla$,

$$\Re \left(L{e}^{-i\theta }\right)\ge \mathsf{\ell }(\mathsf{\ell }-1)\mathcal{J},\Re \left(N{e}^{-i\theta }\right)\ge 0,$$

for all real $\theta \in \mathbb{R}$ and $\mathsf{\ell }\ge 2$.

Corollary 3.

Let $\psi \in {\beth }_j[\Theta ,\mathcal{J}]$. If $f\in \mathcal{A}$ attains

$$|{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)|\le \mathsf{\ell }\mathcal{J},z\in \nabla ;\mathsf{\ell }\ge 2,\mathcal{J}>0,$$

and

$$\psi \left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\right)\in \Theta ,$$

then

$$|{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)|<\mathcal{J}.$$

In the specific case $\Theta =\mathrm{t}(\nabla )=\{\tau :|\tau |<\mathcal{J}\}$$(\mathcal{J}>0)$, the class ${\beth }_j[\Theta ,\mathcal{J}]$ is simply denoted by ${\beth }_j\left[\mathcal{J}\right]$.

Corollary 4.

Let $\psi \in {\beth }_j\left[\mathcal{J}\right]$. If $f\in \mathcal{A}$ attains

$$|{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)|\le \mathsf{\ell }\mathcal{J},$$

and

$$|\psi \left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\right);z|<\mathcal{J},$$

we obtain

$$|{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)|<\mathcal{J}.$$

Corollary 5.

Let $\mathsf{\ell }\ge 2$ and $\mathcal{J}>0$. If $f\in \mathcal{A}$ satisfies

$$\left|{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)\right|\le \mathsf{\ell }\mathcal{J},$$

and also

$$\left|{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)-{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\right|\le {\displaystyle \frac{\mathcal{J}}{\mathrm{j}+\mathrm{b}}},$$

then

$$\left|{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\right|<\mathcal{J}.$$

Proof. 

Since ${\lambda }_1=1+\mathcal{J}{e}^{i\theta }$ and ${\lambda }_2={\lambda }_1+{\displaystyle \frac{\mathsf{\ell }\mathcal{J}{e}^{i\theta }}{\mathrm{j}+\mathrm{b}}},$ let

$$\psi ({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4;z)={\lambda }_2-{\lambda }_1,$$

and take $\Theta \in h(\nabla )$ defined by

$$h\left(z\right)={\displaystyle \frac{\mathcal{J}z}{\mathrm{j}+\mathrm{b}}},\mathcal{J}>0.$$

Applying Corollary (3) shows that $\psi \in {\beth }_{\iota }[\Theta ,\mathcal{J}]$. The admissibility condition yields

$$\psi ({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4;z)=\left|\mathsf{\ell }{\displaystyle \frac{\mathcal{J}{e}^{i\theta }}{\mathrm{j}+\mathrm{b}}}\right|\ge {\displaystyle \frac{\mathcal{J}}{|\mathrm{j}+\mathrm{b}|}}.$$

For $z\in \nabla$, $\mathsf{\ell }\ge 2$, and $\theta \in \mathbb{R}$, the desired conclusion follows from Corollary (3). □

Definition 9.

Let $\Theta \subset \mathrm{C}$ be a set and let $\mathrm{t}\in {\mathbb{M}}_1\cap {\mathcal{K}}_1$. The class ${\beth }_{j,1}[\Theta ,\mathrm{t}]$ consists of all functions $\psi :{\mathrm{C}}^4\times \nabla \to \mathrm{C}$ satisfying the admissibility condition

$$\psi ({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4;z)\notin \Theta ,$$

where

$${\lambda }_1=\mathrm{t}\left(\upsilon \right),{\lambda }_2={\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)\mathrm{t}\left(\upsilon \right)-B\upsilon {\mathrm{t}}^{\prime }\left(\upsilon \right)}{\mathrm{j}+\mathrm{b}-1}},$$

Moreover, the condition

$$\Re \left({\displaystyle \frac{(\mathrm{j}+\mathrm{b}){\lambda }_1-(2\mathrm{b}+2\mathrm{j}-1){\lambda }_2+(\mathrm{j}+\mathrm{b}-1){\lambda }_3}{{\lambda }_1-{\lambda }_2}}\right)\ge \mathsf{\ell }\Re \left(1+{\displaystyle \frac{\upsilon {\mathrm{t}}^{″}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right),$$

and also

$$\Re \left\{{\displaystyle \frac{\begin{array}{cc}& (\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}){\lambda }_4-[3(\mathrm{j}+\mathrm{b}-1)-(\mathrm{j}+\mathrm{b}+1)](\mathrm{j}+\mathrm{b}){\lambda }_3\hfill \\ & -[(2-\mathrm{j}-\mathrm{b})+{(\mathrm{j}+\mathrm{b}-1)}^2]{\lambda }_2-2\left[(\mathrm{j}+\mathrm{b}-1)(2-\mathrm{j}-\mathrm{b}))\right]{\lambda }_1\hfill \end{array}}{{\lambda }_1-{\lambda }_2}}\right\}\le {\mathsf{\ell }}^2\Re \left({\displaystyle \frac{z^2{\mathrm{t}}^{‴}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right),$$

hold whenever $z\in \nabla$, $\upsilon \in \partial \nabla \setminus E\left(\mathrm{t}\right)$, and $\mathsf{\ell }\ge 2$.

Theorem 4.

Let $\psi \in {\beth }_2[\Theta ,\mathrm{t}]$. If $f\in \mathcal{A}$ and $\mathrm{t}\in {\mathbb{M}}_1$ satisfy

$$\Re \left(\upsilon {\displaystyle \frac{{\mathrm{t}}^{″}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right)\ge 0,\left|{\displaystyle \frac{{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right|\le \mathsf{\ell }$$

(30)

and

$$\psi \left({\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\in \nabla \right)\subset \Theta$$

(31)

for all $z\in \nabla$, then

$${\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec \mathrm{t}\left(z\right).$$

Proof. 

Define

$$\mathrm{s}\left(z\right)=z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right).$$

(32)

From (7) and the above definition, we obtain

$$z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)={\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)\mathrm{s}\left(z\right)-z{\mathrm{s}}^{\prime }\left(z\right)}{\mathrm{j}+\mathrm{b}-1}}.$$

(33)

Similarly,

$$z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right)={\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})\mathrm{s}\left(z\right)-2(\mathrm{j}+\mathrm{b}-1)z{\mathrm{s}}^{\prime }\left(z\right)-z^2{\mathrm{s}}^{″}\left(z\right)}{(\mathrm{j}+\mathrm{b})(\mathrm{j}+\mathrm{b}-1)}}$$

(34)

and

$$z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right)={\displaystyle \frac{\begin{array}{c}\left((\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b})\mathrm{s}\left(z\right)\right)+(2-k-\mathrm{b})(\mathrm{j}+\mathrm{b}-1)z{\mathrm{s}}^{\prime }\left(z\right)\hfill \\ +3(\mathrm{j}+\mathrm{b}-1)z^2{\mathrm{s}}^{″}\left(z\right)-z^3{\mathrm{s}}^{‴}\left(z\right)\hfill \end{array}}{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b})}}.$$

(35)

Consider the transformation from ${\mathrm{j}}^4$ to $\mathrm{j}$ given by

$${\lambda }_1={\eta }_1$$

(36)

$${\lambda }_2={\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1){\eta }_1-{\eta }_2}{\mathrm{j}+\mathrm{b}-1}}$$

(37)

$${\lambda }_3={\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b}){\eta }_1-2(\mathrm{j}+\mathrm{b}-1){\eta }_2+{\eta }_3}{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}}$$

(38)

and

$${\lambda }_4={\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}){\eta }_1+(2-k-\mathrm{b})(\mathrm{j}+\mathrm{b}-1){\eta }_2+3(\mathrm{j}+\mathrm{b}-1){\eta }_3-{\eta }_4}{(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}}$$

(39)

Define

$$\mathcal{L}({\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4;z)=\psi ({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4;z).$$

$$=\psi \left(\begin{array}{c}{\eta }_1,{\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1){\eta }_1-{\eta }_2}{\mathrm{j}+\mathrm{b}-1}},{\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b}){\eta }_1-2(\mathrm{j}+\mathrm{b}-1){\eta }_2+{\eta }_3}{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}},\hfill \\ {\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}){\eta }_1+(2-\mathrm{j}-\mathrm{b})(\mathrm{j}+\mathrm{b}-1){\eta }_2+3(\mathrm{j}+\mathrm{b}-1){\eta }_3-{\eta }_4}{(\mathrm{j}+\mathrm{b}+1)(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}}\hfill \end{array};z\right)$$

(40)

Using (32)–(35) together with (40), we have

$$\begin{array}{cc}\hfill \mathcal{L}(& \mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z)\hfill \\ \hfill & =\psi \left(z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\in \nabla \right)\hfill \end{array}$$

(41)

Thus, (31) becomes

$$\mathcal{L}\left(\mathrm{s}\left(z\right),z{\mathrm{s}}^{\prime }\left(z\right),z^2{\mathrm{s}}^{″}\left(z\right),z^3{\mathrm{s}}^{‴}\left(z\right);z\right)\in \Theta .$$

Moreover,

$${\displaystyle \frac{{\eta }_3}{{\eta }_2}}+1={\displaystyle \frac{(\mathrm{j}+\mathrm{b}){\lambda }_3-(2\mathrm{j}+2\mathrm{b}-1){\lambda }_2+(\mathrm{j}+\mathrm{b}-1){\lambda }_1}{{\lambda }_2-{\lambda }_1}},$$

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\eta }_4}{{\eta }_2}}& ={\displaystyle \frac{(\mathrm{j}+\mathrm{b})\left(3(-1+\mathrm{j}+\mathrm{b}){\lambda }_3-(1+\mathrm{j}+\mathrm{b}){\lambda }_4-5{\lambda }_1+3(\mathrm{j}+\mathrm{b}){\lambda }_1\right)}{{\lambda }_1-{\lambda }_2}}\hfill \\ \hfill & +{\displaystyle \frac{(-1+\mathrm{j}+\mathrm{b})(-4+5\mathrm{j}+5\mathrm{b}){\lambda }_2-4{\lambda }_1}{{\lambda }_2-{\lambda }_1}}.\hfill \end{array}$$

Hence, the admissibility condition for $\psi \in {\beth }_{j,1}[\Theta ,\mathrm{t}]$ is equivalent to the admissibility condition for $\mathcal{L}\in {\mathcal{L}}_2[\Theta ,\mathrm{t}]$. By Definition 4 and Lemma 1, we obtain

$$z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec \mathrm{t}\left(z\right).$$

The next result generalizes Theorem 4. □

Theorem 5.

Let $\psi \in {\beth }_2[\lambda ,\mathrm{t}]$. Suppose that $\psi \in \mathcal{A}$ and $\mathrm{t}\in {\mathrm{t}}_1$ satisfy

$$\Re \left(\upsilon {\displaystyle \frac{{\mathrm{t}}^{″}\left(\upsilon \right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right)\ge 0,\left|{\displaystyle \frac{{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)}{\upsilon {\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right|\le \mathsf{\ell },$$

(42)

and

$$\psi \left(z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\in \nabla \right)\prec h\left(z\right).$$

(43)

Then,

$$z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\prec \mathrm{t}\left(z\right),(z\in \nabla ).$$

Using Definition 9 and $\mathrm{t}\left(z\right)=1+\mathcal{J}z$$(\mathcal{J}>0)$, the admissible class ${\beth }_{j,\iota }[\Theta ,\mathcal{J}]$ is given as follows.

Definition 10

([12]). Let $\Theta \subset \mathrm{C}$ and $\mathcal{J}>0$. A function $\psi :{\mathrm{C}}^4\times \nabla \to \mathrm{C}$ belongs to ${\beth }_{j,\iota }[\Theta ,\mathcal{J}]$ if

$$\psi \left(\begin{array}{c}1+\mathcal{J}{e}^{i\theta },\hfill \\ {\displaystyle \frac{(\mathrm{j}+\mathrm{b}-\mathsf{\ell }-1)\mathcal{J}{e}^{i\theta }-(\mathrm{j}+\mathrm{b}-1)}{\mathrm{j}+\mathrm{b}-1}},\hfill \\ {\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)\left[(\mathrm{j}+\mathrm{b}-2)\mathcal{J}{e}^{i\theta }+(\mathrm{j}+\mathrm{b})\right]+L}{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})}},\hfill \\ {\displaystyle \frac{(\mathrm{j}+\mathrm{b}-1)\left[({\mathrm{j}}^2+{\mathrm{b}}^2+2\mathrm{b}\mathrm{j}+2)\mathcal{J}{e}^{i\theta }+(\mathrm{j}+\mathrm{b})(\mathrm{j}+\mathrm{b}+1)+3L\right]-N}{(\mathrm{j}+\mathrm{b}-1)(\mathrm{j}+\mathrm{b})(\mathrm{j}+\mathrm{b}+1)}}\hfill \end{array};z\right)$$

(44)

for all real θ and all $\mathsf{\ell }\ge 2$, with

$$\Re \left(N{e}^{-i\theta }\right)\ge 0,\Re \left(L{e}^{-i\theta }\right)\ge \mathsf{\ell }.$$

Corollary 6.

Let $\psi \in {\beth }_2[\Theta ,\mathrm{t}]$. If $f\in \mathcal{A}$ and

$$\left|z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)\right|\le \mathsf{\ell }\mathcal{J},(z\in \nabla ,\mathsf{\ell }\ge 2,\mathcal{J}>0),$$

and

$$\psi \left(z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\in \nabla \right)\in \Theta ,$$

then

$$\left|z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\right|<\mathcal{J},(z\in \nabla ).$$

Corollary 7.

Let $\psi \in {\beth }_2[\Theta ,\mathrm{t}]$. If $f\in \mathcal{A}$ and $\mathrm{t}\in {\mathrm{t}}_1$ satisfy

$$\left|{\displaystyle \frac{{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right)}{{\mathrm{t}}^{\prime }\left(\upsilon \right)}}\right|\le \mathsf{\ell }\mathcal{J},(z\in \nabla ,\mathsf{\ell }\ge 2,\mathcal{J}>0),$$

and

$$\left|\psi \left(z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+1}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+2}f\left(z\right),z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}+3}f\left(z\right);z\in \nabla \right)\right|<\mathcal{J},(z\in \nabla ),$$

then

$$\left|z^{-1}{\mathbb{L}}_{\mathrm{w}}^{\mathrm{j}}f\left(z\right)\right|<\mathcal{J}(z\in \nabla ).$$

3. Conclusions

The results obtained in this work significantly advance the study of third-order differential subordination involving an operator associated with the Lommel function. The framework developed here extends several classical concepts in geometric function theory by establishing precise admissibility conditions that govern analytic functions within the unit disk, providing refined tools for deriving sandwich-type results and analyzing geometric properties of higher-order differential operators. Furthermore, unlike the studies in [8,9,10,15,29], which focus on different classes of functions such as generalized Bessel, Mittag–Leffler, and fractional derivative operators, our work addresses a gap by considering analytic functions involving the Lommel function, for which third-order differential subordinations had not been previously established. Special cases reducing to the original function and its derivative, along with the novel applications and illustrative examples presented here, demonstrate the versatility of the approach and open new directions for both theoretical and applied investigations in mathematics, physics, and engineering.