Stability and bifurcations in a delayed predator-prey system with prey-taxis and hunting cooperation functional response

Kimun Ryu; Wonlyul Ko; Kimun Ryu; Wonlyul Ko

doi:10.3934/math.2025576

AIMS Mathematics

2025, Volume 10, Issue 6: 12808-12840. doi: 10.3934/math.2025576

Previous Article Next Article

Research article

Stability and bifurcations in a delayed predator-prey system with prey-taxis and hunting cooperation functional response

Kimun Ryu ,
Wonlyul Ko ^,

Department of Mathematics Education, Cheongju University, Cheongju, Chungbuk 28503, Republic of Korea

Received: 24 February 2025 Revised: 16 April 2025 Accepted: 24 April 2025 Published: 03 June 2025
MSC : 35B32, 35K57, 92D25

In this paper, we studied a diffusive predator-prey system that incorporated three ecological features under homogeneous Neumann boundary conditions. These features included a cooperative hunting functional response, prey-taxis, and a time delay effect in the predator growth rate, all of which influenced predator-prey interactions. These three features, respectively, indicated that predators cooperated while hunting their prey, that predators tended to move in the direction of an increasing prey density gradient, and that a certain amount of time was required for the conversion of captured prey into predator growth. We first analyzed the occurrence of steady-state bifurcation by examining the role of the prey-taxis rate $\chi$ . Furthermore, we examined the impact of the time delay effect on the occurrence of Hopf bifurcation, which involved the emergence of spatially homogeneous or nonhomogeneous periodic solutions, as well as stability switches at a positive constant steady state.

Keywords:

Citation: Kimun Ryu, Wonlyul Ko. Stability and bifurcations in a delayed predator-prey system with prey-taxis and hunting cooperation functional response[J]. AIMS Mathematics, 2025, 10(6): 12808-12840. doi: 10.3934/math.2025576

Related Papers:

[1]	Shuhai Li, Lina Ma, Huo Tang . Meromorphic harmonic univalent functions related with generalized (p, q)-post quantum calculus operators. AIMS Mathematics, 2021, 6(1): 223-234. doi: 10.3934/math.2021015
[2]	Bakhtiar Ahmad, Muhammad Ghaffar Khan, Basem Aref Frasin, Mohamed Kamal Aouf, Thabet Abdeljawad, Wali Khan Mashwani, Muhammad Arif . On $q$ -analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain. AIMS Mathematics, 2021, 6(4): 3037-3052. doi: 10.3934/math.2021185
[3]	Ying Yang, Jin-Lin Liu . Some geometric properties of certain meromorphically multivalent functions associated with the first-order differential subordination. AIMS Mathematics, 2021, 6(4): 4197-4210. doi: 10.3934/math.2021248
[4]	Hari Mohan Srivastava, Muhammad Arif, Mohsan Raza . Convolution properties of meromorphically harmonic functions defined by a generalized convolution $q$ -derivative operator. AIMS Mathematics, 2021, 6(6): 5869-5885. doi: 10.3934/math.2021347
[5]	Tao He, Shu-Hai Li, Li-Na Ma, Huo Tang . Closure properties of generalized $\lambda$ -Hadamard product for a class of meromorphic Janowski functions. AIMS Mathematics, 2021, 6(2): 1715-1726. doi: 10.3934/math.2021102
[6]	Zhuo Wang, Weichuan Lin . The uniqueness of meromorphic function shared values with meromorphic solutions of a class of q-difference equations. AIMS Mathematics, 2024, 9(3): 5501-5522. doi: 10.3934/math.2024267
[7]	Pinhong Long, Xing Li, Gangadharan Murugusundaramoorthy, Wenshuai Wang . The Fekete-Szegö type inequalities for certain subclasses analytic functions associated with petal shaped region. AIMS Mathematics, 2021, 6(6): 6087-6106. doi: 10.3934/math.2021357
[8]	Ekram E. Ali, Nicoleta Breaz, Rabha M. El-Ashwah . Subordinations and superordinations studies using $q$ -difference operator. AIMS Mathematics, 2024, 9(7): 18143-18162. doi: 10.3934/math.2024886
[9]	Erhan Deniz, Hatice Tuǧba Yolcu . Faber polynomial coefficients for meromorphic bi-subordinate functions of complex order. AIMS Mathematics, 2020, 5(1): 640-649. doi: 10.3934/math.2020043
[10]	Huo Tang, Muhammad Arif, Khalil Ullah, Nazar Khan, Bilal Khan . Majorization results for non vanishing analytic functions in different domains. AIMS Mathematics, 2022, 7(11): 19727-19738. doi: 10.3934/math.20221081

Abstract

1. Introduction

Let $\sum\limits$ denote the class of meromorphic function of the form:

$\begin{equation} \lambda \left( \omega \right) = \frac{1}{\omega }+\underset{t = 0}{\overset{ \infty }{\sum\limits }}a_{t}\omega ^{t}, \end{equation}$

(1.1)

which are analytic in the punctured open unit disc $U^{\ast } = \{\omega :\omega \in \mathbb{C}$ and $0 < \left\vert \omega \right\vert < 1\} = U-\{0\},$ where $U = U^{\ast }\cup \{0\}.$ Let $\delta \left(\omega \right) \in \sum\limits$ , be given by

$\begin{equation} \delta \left( \omega \right) = \frac{1}{\omega }+\underset{t = 0}{\overset{ \infty }{\sum\limits }}b_{t}\omega ^{t}, \end{equation}$

(1.2)

then the Convolution (Hadamard product) of $\lambda \left(\omega \right)$ and $\delta \left(\omega \right)$ is denoted and defined as:

$\begin{equation*} \left( \lambda \ast \delta \right) \left( \omega \right) = \frac{1}{\omega }+ \underset{t = 0}{\overset{\infty }{\sum\limits }}a_{t}b_{t}\omega ^{t}. \end{equation*}$

In 1967, MacGregor ^[17] introduced the concept of majorization as follows.

Definition 1. Let $\lambda$ and $\delta$ be analytic in $U^{\ast }$ . We say that $\lambda$ is majorized by $\delta$ in $U^{\ast }$ and written as $\lambda \left(\omega \right) \ll \delta \left(\omega \right) \; \omega \in U^{\ast },$ if there exists a function $\varphi \left(\omega \right),$ analytic in $U^{\ast },$ satisfying

$\begin{equation} \left \vert \varphi \left( \omega \right) \right \vert \leq 1,{ \ \ and \ \ }\lambda \left( \omega \right) = \varphi \left( \omega \right) \delta \left( \omega \right) ,{\rm{ \ }}\omega \in U^{\ast }\text{.} \end{equation}$

(1.3)

In 1970, Robertson ^[19] gave the idea of quasi-subordination as:

Definition 2. A function $\lambda \left(\omega \right)$ is subordinate to $\delta \left(\omega \right)$ in $U$ and written as: $\lambda \left(\omega \right) \prec \delta \left(\omega \right),$ if there exists a Schwarz function $k(\omega)$ , which is holomorphic in $U^{\ast }$ with $\left\vert k(\omega)\right\vert < 1$ , such that $\lambda \left(\omega \right) = \delta (k\left(\omega \right)).\$ Furthermore, if the function $\delta \left(\omega \right)$ is univalent in $U^{\ast },$ then we have the following equivalence $($ see ^[16] $)$ :

$\begin{equation} \lambda \left( \omega \right) \prec \delta \left( \omega \right)\;\;\; { and }\;\;\; \lambda \left( U\right) \subset \delta \left( U\right) \text{.} \end{equation}$

(1.4)

Further, $\lambda \left(\omega \right)$ is quasi-subordinate to $\delta \left(\omega \right)$ in $U^{\ast }$ and written is

$\begin{equation*} \lambda \left( \omega \right) \prec _{q}\delta \left( \omega \right) {\rm{ \ \ }}\left( {{\rm{\ }}}\omega \in U^{\ast }\right) \text{,} \end{equation*}$

if there exist two analytic functions $\varphi \left(\omega \right)$ and $k\left(\omega \right)$ in $U^{\ast }$ such that $\frac{\lambda \left(\omega \right) }{\varphi \left(\omega \right) }$ is analytic in $U^{\ast }$ and

$\begin{equation*} \left\vert \varphi \left( \omega \right) \right\vert \leq 1\;\;{ \ and \ }\;\; k\left( \omega \right) \leq \left\vert \omega \right\vert \lt 1{\rm{ \ \ }} \omega \in U^{\ast }\text{,} \end{equation*}$

satisfying

$\begin{equation} {\rm{\ \ }}\lambda \left( \omega \right) = \varphi \left( \omega \right) \delta \left( k\left( \omega \right) \right) {\rm{ \ \ }}\omega \in U^{\ast } \text{.} \end{equation}$

(1.5)

(ⅰ) For $\varphi \left(\omega \right) = 1$ in (1.5), we have

$\begin{equation*} {\rm{\ \ }}\lambda \left( \omega \right) = \delta \left( k\left( \omega \right) \right) {\rm{ \ \ }}\omega \in U^{\ast }, \end{equation*}$

and we say that the $\lambda$ function is subordinate to $\delta$ in $U^{\ast },$ denoted by $($ see ^[20] $)$

$\begin{equation*} \lambda \left( \omega \right) \prec \delta \left( \omega \right) {\rm{ \ \ }} \left( {{\rm{\ }}}\omega \in U^{\ast }\right) \text{.} \end{equation*}$

(ⅱ) If $k\left(\omega \right) = \omega$ , the quasi-subordination (1.5) becomes the majorization given in (1.3). For related work on majorization see ^[1,4,9,21].

Let us consider the second order linear homogenous differential equation (see, Baricz ^[6]):

$\begin{equation} \omega ^{2}k^{^{^{\prime \prime }}}\left( \omega \right) +\alpha \omega k^{^{^{\prime }}}\left( \omega \right) +\left[ \beta \omega ^{2}-\nu ^{2}+(1-\alpha )\right] k\left( \omega \right) = 0. \end{equation}$

(1.6)

The function $k_{\nu, \alpha, \beta }(\omega)$ , is known as generalized Bessel's function of first kind and is the solution of differential equation given in (1.6)

$\begin{equation} k_{\nu ,\alpha ,\beta }(\omega ) = \sum\limits\limits_{t = 0}^{\infty }\frac{(-\beta )^{t}}{\Gamma (t+1)\Gamma (t+\nu +1+\frac{\alpha +1}{2})}\left( \frac{\omega }{2}\right) ^{^{2t+\nu }}. \end{equation}$

(1.7)

Let us denote

$\begin{eqnarray*} \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) & = &\frac{ 2^{\nu }\Gamma (\nu +\frac{\alpha +1}{2})}{\omega ^{\frac{\nu }{2}+1}}k_{\nu ,\alpha ,\beta }(\omega ^{\frac{1}{2}}),\ \ \\ & = &\frac{1}{\omega }+\sum\limits\limits_{t = 0}^{\infty }\frac{(-\beta )^{t+1}\Gamma (\nu +\frac{\alpha +1}{2})}{4^{t+1}\Gamma (t+2)\Gamma (t+\nu +1+\frac{\alpha +1}{2})}(\omega )^{t}, \end{eqnarray*}$

where $\nu, \; \alpha$ and $\beta$ are positive real numbers. The operator $\mathcal{L}_{\nu, \alpha, \beta }$ is a variation of the operator introduced by Deniz ^[7] (see also Baricz et al. ^[5]) for analytic functions. By using the convolution, we define the operator $\mathcal{L}_{\nu, \alpha, \beta }$ as follows:

$\begin{eqnarray} (\ \mathcal{L}_{\nu ,\alpha ,\beta }\lambda )(\omega ) & = &\mathcal{L}_{\nu ,\alpha ,\beta }\left( \omega \right) \ast \lambda (\omega ), \\ & = &\frac{1}{\omega }+\sum\limits\limits_{t = 0}^{\infty }\frac{(-\beta )^{t+1}\Gamma (\nu +\frac{\alpha +1}{2})}{4^{t+1}\Gamma (t+2)\Gamma (t+\nu +1+\frac{\alpha +1}{2})}a_{t}(\omega )^{t}. \end{eqnarray}$

(1.8)

The operator $\ \mathcal{L}_{\nu, \alpha, \beta }$ was introduced and studied by Mostafa et al. ^[15] (see also ^[2]). From (1.8), we have

$\begin{equation} \omega \left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1} = \left( \nu -1+\frac{\alpha +1}{2}\right) \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}-\left( \nu +\frac{\alpha +1}{2}\right) \left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}. \end{equation}$

(1.9)

By taking $\alpha = \beta = 1,$ the above operator reduces to $(\ \mathcal{L} _{\nu }\lambda)(\omega)$ studied by Aouf et al. ^[2].

Definition 3. Let $-1\leq B < A\leq 1, \; \eta \in \mathbb{C} -\{0\}, \; j\in W$ and $\nu, \; \alpha, \; \beta > 0$ . A function $\lambda \in \sum\limits$ is said to be in the class $M_{\alpha, \beta }^{\nu, j}(\eta, \varkappa; A, B)$ of meromorphic functions of complex order $\eta \neq 0$ in $U^{\ast }$ if and only if

$\begin{equation} 1-\frac{1}{\eta }\left( \frac{\omega \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}}{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}}+\nu +j\right) -\varkappa \left \vert -\frac{1}{\eta }\left( \frac{\omega \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}}{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}} +\nu +j\right) \right \vert \prec \frac{1+A\omega }{1+B\omega }. \end{equation}$

(1.10)

Remark 1.

$(i).$ For $A = 1, \; B = -1$ and $\varkappa = 0,$ we denote the class

$\begin{equation*} M_{\alpha ,\beta }^{\nu ,j}(\eta ,0;1,-1) = M_{\alpha ,\beta }^{\nu ,j}(\eta ). \end{equation*}$

So, $\lambda \in M_{\alpha, \beta }^{\nu, j}(\eta, \varkappa; A, B)$ if and only if

$\begin{equation*} \Re \left[ 1-\frac{1}{\eta }\left( \frac{\omega \left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}}{\left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}} +\nu +j\right) \right] \gt 0. \end{equation*}$

$(ii).$ For $\alpha = 1, \; \beta = 1$ , $M_{1, 1}^{\nu, j}(\eta, 0;1, -1)$ reduces to the class $M^{\nu, j}(\eta).$

$\begin{equation*} \Re \left[ 1-\frac{1}{\eta }\left( \frac{\omega \left( \mathcal{L} _{\nu }\lambda \left( \omega \right) \right) ^{j+1}}{\left( \mathcal{L} _{\nu }\lambda \left( \omega \right) \right) ^{j}}+\nu +j\right) \right] \gt 0. \end{equation*}$

Definition 4. A function $\lambda \in \sum\limits$ is said to be in the class $\ N_{\alpha, \beta }^{\nu, j}(\theta, b;A, B)$ of meromorphic spirllike functions of complex order $b\neq 0$ in $U^{\ast },$ if and only if

$\begin{equation} 1-\frac{e^{i\theta }}{b\cos \theta }\left( \frac{\omega \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}}{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}} +j+1\right) \prec \frac{1+A\omega }{1+B\omega }, \end{equation}$

(1.11)

where,

$\begin{equation*} \left( -\frac{\pi }{2} \lt \theta \lt \frac{\pi }{2},\text{ }-1\leq \beta \lt A\leq 1,\eta \in \mathbb{C} -\{0\},\text{ }j\in W,\text{ }\nu ,\alpha ,\beta \gt 0\;{ and }\;\omega \in U^{\ast }\text{ }\right) . \end{equation*}$

$(i).$ For $A = 1$ and $B = -1,$ we set

$\begin{equation*} N_{\alpha ,\beta }^{\nu ,j}(\theta ,b;1,-1) = N_{\alpha ,\beta }^{\nu ,j}(\theta ,b), \end{equation*}$

where $N_{\alpha, \beta }^{\nu, j}(\theta, b)$ denote the class of functions $\lambda \in \sum\limits$ satisfying the following inequality:

$\begin{equation*} \Re \left[ \frac{e^{i\theta }}{b\cos \theta }\left( \frac{\omega \left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1} }{\left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}}+j+1\right) \right] \lt 1. \end{equation*}$

$(ii).$ For $\theta = 0$ and $\alpha = \beta = 1$ we write

$\begin{equation*} N_{1,1}^{\nu ,j}(0,b;1,-1) = N^{\nu ,j}(b), \end{equation*}$

where $N^{\nu, j}(b)$ denote the class of functions $\lambda \in \sum\limits$ satisfying the following inequality:

$\begin{equation*} \Re \left[ \frac{1}{b}\left( \frac{\omega \left( \mathcal{L}_{\nu }\lambda \left( \omega \right) \right) ^{j+1}}{\left( \mathcal{L}_{\nu }\lambda \left( \omega \right) \right) ^{j}}+j+1\right) \right] \lt 1. \end{equation*}$

A majorization problem for the normalized class of starlike functions has been examined by MacGregor ^[17] and Altintas et al. ^[3,4]. Recently, Eljamal et al. ^[8], Goyal et al. ^[12,13], Goswami et al. ^[10,11], Li et al. ^[14], Tang et al. ^[21,22] and Prajapat and Aouf ^[18] generalized these results for different classes of analytic functions.

The objective of this paper is to examined the problems of majorization for the classes $M_{\alpha, \beta }^{\nu, j}(\eta, \varkappa; A, B)$ and $\ N_{\alpha, \beta }^{\nu, j}(\theta, b;A, B)$ .

2. Majorization problem for the class $M_{\alpha, \beta }^{\nu, j}(\eta, \varkappa; A, B)$

In Theorem 1, we prove majorization property for the class $M_{\alpha, \beta }^{\nu, j}(\eta, \varkappa; A, B).$

Theorem 1. Let the function $\lambda \in \sum\limits$ and suppose that $\delta \in M_{\alpha , \beta }^{\nu, j}(\eta, \varkappa; A, B).$ If $\ \left(\mathcal{L} _{\nu , \alpha, \beta }\lambda \left(\omega \right) \right) ^{j}$ is majorized by $\left(\mathcal{L} _{\nu, \alpha, \beta }\delta \left(\omega \right) \right) ^{j}$ in $U^{\ast }$ , then

$\begin{equation} \left \vert \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}\right \vert \leq \left \vert \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}\right \vert , \left( \left \vert \omega \right \vert \lt r_{0}\right) , \end{equation}$

(2.1)

where $r_{0} = r_{0}(\eta, \varkappa, \nu, \alpha, \beta, A, B)$ is the smallest positive roots of the equation

$\begin{eqnarray} &&-\rho \left( \nu -1+\frac{\alpha +1}{2}\right) \left[ \frac{\left( A-B\right) \left \vert \eta \right \vert }{1-\varkappa }-\left( \frac{\alpha +1}{2}\right) \left \vert B\right \vert \right] r^{3}-\left( \nu -1+\frac{ \alpha +1}{2}\right) \left[ \rho \left( \frac{\alpha +1}{2}\right) +\rho ^{2}\left \vert B\right \vert -\left \vert B\right \vert \right] r^{2} \\ &&-\left( \nu -1+\frac{\alpha +1}{2}\right) \left[ \frac{\left( A-B\right) \left \vert \eta \right \vert }{1-\varkappa }-\left( \frac{\alpha +1}{2} \right) \left \vert B\right \vert +\rho ^{2}\left \vert B\right \vert -1 \right] r+ \\ &&\rho \left( \nu -1+\frac{\alpha +1}{2}\right) \left( \frac{\alpha +1}{2} \right) \\ & = &0. \end{eqnarray}$

(2.2)

Proof. Since $\delta \in M_{\alpha, \beta }^{\nu, j}(\eta, \varkappa; A, B),$ we have

$\begin{equation} 1-\frac{1}{\eta }\left( \frac{\omega \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}}{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}}+\nu +j\right) -\varkappa \left \vert -\frac{1}{\eta }\left( \frac{\omega \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1} }{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}}+\nu +j\right) \right \vert = \frac{1+Ak\left( \omega \right) }{ 1+Bk\left( \omega \right) }, \end{equation}$

(2.3)

where $k\left(\omega \right) = c_{1}\omega +c_{2}\omega ^{2}+...,$ is analytic and bounded functions in $U^{\ast }$ with

$\begin{equation} {{\rm{\ }}}\left \vert k\left( \omega \right) \right \vert \leq \left \vert \omega \right \vert \ \ \left( \omega \in U^{\ast }\right) . \end{equation}$

(2.4)

Taking

$\begin{equation} \S \left( \omega \right) = 1-\frac{1}{\eta }\left( \frac{\omega \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1} }{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}}+\nu +j\right) , \end{equation}$

(2.5)

In (2.3), we have

$\begin{equation*} \S \left( \omega \right) -\varkappa \left \vert \S \left( \omega \right) -1\right \vert = \frac{1+Ak\left( \omega \right) }{1+Bk\left( \omega \right) } , \end{equation*}$

which implies

$\begin{equation} \S \left( \omega \right) = \frac{1+\left( \frac{A-B\varkappa e^{-i\theta }}{ 1-\varkappa e^{-i\theta }}\right) k\left( \omega \right) }{1+Bk\left( \omega \right) }. \end{equation}$

(2.6)

Using (2.6) in (2.5), we get

$\begin{equation} \frac{\omega \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}}{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}} = -\frac{\nu +j+\left[ \frac{\left( A-B\right) \eta }{1-\varkappa e^{-i\theta }}+\left( \nu +j\right) B\right] k\left( \omega \right) }{1+Bk\left( \omega \right) }. \end{equation}$

(2.7)

Application of Leibnitz's Theorem on (1.9) gives

$\begin{equation} \omega \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1} = \left( \nu -1+\frac{\alpha +1}{2}\right) \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}-\left( \nu +j+\frac{\alpha +1}{2}\right) \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}. \end{equation}$

(2.8)

By using (2.8) in (2.7) and making simple calculations, we have

$\begin{equation} \frac{\left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}}{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}} = \frac{\frac{\alpha +1}{2}-\left[ \frac{\left( A-B\right) \eta }{1-\varkappa e^{-i\theta }}-\left( \frac{\alpha +1}{2} \right) B\right] k\left( \omega \right) }{\left( 1+Bk\left( \omega \right) \right) \left( \nu -1+\frac{\alpha +1}{2}\right) }. \end{equation}$

(2.9)

Or, equivalently

$\begin{equation} \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j} = \frac{\left( 1+Bk\left( \omega \right) \right) \left( \nu -1+ \frac{\alpha +1}{2}\right) }{\frac{\alpha +1}{2}-\left[ \frac{\left( A-B\right) \eta }{1-\varkappa e^{-i\theta }}-\left( \frac{\alpha +1}{2} \right) B\right] k\left( \omega \right) }\left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}. \end{equation}$

(2.10)

Since $\left \vert k\left(\omega \right) \right \vert \leq \left \vert \omega \right \vert$ , (2.10) gives us

$\begin{eqnarray} \left \vert \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}\right \vert &\leq &\frac{\left[ 1+\left \vert B\right \vert \left \vert \omega \right \vert \right] \left( \nu -1+\frac{\alpha +1}{ 2}\right) }{\frac{\alpha +1}{2}-\left \vert \frac{\left( A-B\right) \eta }{ 1-\varkappa e^{-i\theta }}-\left( \frac{\alpha +1}{2}\right) B\right \vert \left \vert \omega \right \vert }\left \vert \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}\right \vert \\ &\leq &\frac{\left[ 1+\left \vert B\right \vert \left \vert \omega \right \vert \right] \left( \nu -1+\frac{\alpha +1}{2}\right) }{\frac{\alpha +1}{2}- \left[ \frac{\left( A-B\right) \left \vert \eta \right \vert }{1-\varkappa } -\left( \frac{\alpha +1}{2}\right) \left \vert B\right \vert \right] \left \vert \omega \right \vert }\left \vert \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}\right \vert \end{eqnarray}$

(2.11)

Since $\left(\mathcal{L} _{\nu, \alpha, \beta }\lambda \left(\omega \right) \right) ^{j}$ is majorized by $\left(\mathcal{L} _{\nu, \alpha, \beta }\delta \left(\omega \right) \right) ^{j}$ in $U^{\ast }.$ So from (1.3), we have

$\begin{equation} \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j} = \varphi \left( \omega \right) \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}. \end{equation}$

(2.12)

Differentiating (2.12) with respect to $\omega$ then multiplying with $\omega$ , we get

$\begin{equation} \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j} = \omega \varphi ^{^{\prime }}\left( \omega \right) \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}+\omega \varphi \left( \omega \right) \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}. \end{equation}$

(2.13)

By using (2.8), (2.12) and (2.13), we have

$\begin{equation} \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1} = \frac{1}{\left( \nu -1+\frac{\alpha +1}{2}\right) }\omega \varphi ^{^{\prime }}\left( \omega \right) \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}+\varphi \left( \omega \right) \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}. \end{equation}$

(2.14)

On the other hand, noticing that the Schwarz function $\varphi$ satisfies the inequality

$\begin{equation} \left \vert \varphi ^{^{\prime }}\left( \omega \right) \right \vert \leq \frac{1-\left \vert \varphi \left( \omega \right) \right \vert ^{2}}{1-\left \vert \omega \right \vert ^{2}}{\rm{ \ \ \ }}\left( \omega \in U^{\ast }\right) . \end{equation}$

(2.15)

Using (2.8) and (2.15) in (2.14), we get

$\begin{eqnarray*} \left \vert \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}\right \vert &\leq &\left[ \left \vert \varphi \left( \omega \right) \right \vert +\frac{\omega \left( 1-\left \vert \varphi \left( \omega \right) \right \vert ^{2}\right) \left[ 1+\left \vert B\right \vert \left \vert \omega \right \vert \right] \left( \nu -1+\frac{\alpha +1}{ 2}\right) }{\left( \nu -1+\frac{\alpha +1}{2}\right) \left( 1-\left \vert \omega \right \vert ^{2}\right) \left( \frac{\alpha +1}{2}-\left[ \frac{ \left( A-B\right) \left \vert \eta \right \vert }{1-\varkappa }-\left( \frac{ \alpha +1}{2}\right) B\right] \left \vert \omega \right \vert \right) } \right] \\ &&\times \left \vert \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}\right \vert , \end{eqnarray*}$

By taking

$\begin{equation*} \left \vert \omega \right \vert = r,{\rm{ \ \ }}\left \vert \varphi \left( \omega \right) \right \vert = \rho {\rm{ \ \ \ \ }}\left( 0\leq \rho \leq 1\right) , \end{equation*}$

reduces to the inequality

$\begin{equation*} \left \vert \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}\right \vert \leq \frac{\Phi _{1}(\rho )}{\left( \nu -1+ \frac{\alpha +1}{2}\right) \left( 1-r^{2}\right) \left( \frac{\alpha +1}{2}- \left[ \frac{\left( A-B\right) \left \vert \eta \right \vert }{1-\varkappa } -\left( \frac{\alpha +1}{2}\right) B\right] r\right) }\left \vert \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}\right \vert , \end{equation*}$

where

$\begin{eqnarray} \Phi _{1}(\rho ) & = &\left[ \begin{array}{c} \rho \left( \nu -1+\frac{\alpha +1}{2}\right) \left( 1-r^{2}\right) \left( \frac{\alpha +1}{2}-\left[ \frac{\left( A-B\right) \left \vert \eta \right \vert }{1-\varkappa }-\left( \frac{\alpha +1}{2}\right) B\right] r\right) \\ +r\left( 1-\rho ^{2}\right) \left[ 1+\left \vert B\right \vert r\right] \left( \nu -1+\frac{\alpha +1}{2}\right) \end{array} \right] \\ & = &-r\left[ 1+\left \vert B\right \vert r\right] \left( \nu -1+\frac{\alpha +1}{2}\right) \rho ^{2}+\rho \left( \nu -1+\frac{\alpha +1}{2}\right) \left( 1-r^{2}\right) \\ &&\left( \frac{\alpha +1}{2}-\left[ \frac{\left( A-B\right) \left \vert \eta \right \vert }{1-\varkappa }-\left( \frac{\alpha +1}{2}\right) B\right] r\right) +r\left[ 1+\left \vert B\right \vert r\right] \left( \nu -1+\frac{ \alpha +1}{2}\right) ,{\rm{ \ \ \ \ \ \ \ \ \ \ }} \end{eqnarray}$

(2.16)

takes in maximum value at $\rho = 1$ with $r_{0} = r_{0}(\eta, \varkappa, \nu, \alpha, \beta, A, B)$ where $r_{0}$ is the least positive root of the (2.2). Furthermore, if $0\leq \xi _{0}\leq r_{0}(\eta, \varkappa, \nu, \alpha, \beta, A, B),$ then the function $\psi _{1}(\rho)$ defined by

$\begin{eqnarray} \psi _{1}(\rho ) & = &-\xi _{0}\left[ 1+\left \vert B\right \vert \xi _{0} \right] \left( \nu -1+\frac{\alpha +1}{2}\right) \rho ^{2}+\rho \left( \nu -1+\frac{\alpha +1}{2}\right) \left( 1-\xi _{0}^{2}\right) \\ &&\left( \frac{\alpha +1}{2}-\left[ \frac{\left( A-B\right) \left \vert \eta \right \vert }{1-\varkappa }-\left( \frac{\alpha +1}{2}\right) B\right] \xi _{0}\right) +\xi _{0}\left[ 1+\left \vert B\right \vert \xi _{0}\right] \left( \nu -1+\frac{\alpha +1}{2}\right) ,{\rm{ \ \ \ \ \ \ \ \ \ }} \end{eqnarray}$

(2.17)

is an increasing function on the interval $\left(0\leq \rho \leq 1\right),$ so that

$\begin{eqnarray*} \psi _{1}(\rho ) &\leq &\psi _{1}(1) = \left( \nu -1+\frac{\alpha +1}{2} \right) \left( 1-\xi _{0}^{2}\right) \left[ \frac{\alpha +1}{2}-\left( \frac{ \left( A-B\right) \left \vert \eta \right \vert }{1-\varkappa }-\left( \frac{ \alpha +1}{2}\right) B\right) \xi _{0}\right] \\ &&\left( 0\leq \rho \leq 1,\text{ }0\leq \xi _{0}\leq r_{0}(\eta ,\varkappa ,A,B)\right) . \end{eqnarray*}$

Hence, upon setting $\rho = 1$ in (2.17), we achieve (2.1).

Special Cases: Let $A = 1$ and $B = -1$ in Theorem 1, we obtain the following corollary.

Corollary 1. Let the function $\lambda \in \sum\limits$ and suppose that $\delta \in M_{\alpha , \beta }^{\nu, j}(\eta, \varkappa; A, B).$ If $\ \left(\mathcal{L} _{\nu , \alpha, \beta }\lambda \left(\omega \right) \right) ^{j}$ is majorized by $\left(\mathcal{L} _{\nu, \alpha, \beta }\delta \left(\omega \right) \right) ^{j}$ in $U^{\ast }$ , then

$\begin{equation*} \left \vert \left( \mathcal{L} _{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}\right \vert \leq \left \vert \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}\right \vert , \left( \left \vert \omega \right \vert \lt r_{1}\right) , \end{equation*}$

where $r_{1} = r_{1}(\eta, \varkappa, \nu, \alpha, \beta)$ is the least positive roots of the equation

$\begin{eqnarray} &&\rho \left( \nu -1+\frac{\alpha +1}{2}\right) \left[ \frac{2\left \vert \eta \right \vert }{1-\varkappa }-\left( \frac{\alpha +1}{2}\right) \right] r^{3}-\left( \nu -1+\frac{\alpha +1}{2}\right) \left[ \rho \left( \frac{ \alpha +1}{2}\right) +\rho ^{2}-1\right] r^{2}- \\ &&\left( \nu -1+\frac{\alpha +1}{2}\right) \left[ \rho \left \{ \frac{2\left \vert \eta \right \vert }{1-\varkappa }-\left( \frac{\alpha +1}{2}\right) \right \} +\rho ^{2}-1\right] r+\rho \left( \nu -1+\frac{\alpha +1}{2} \right) \left( \frac{\alpha +1}{2}\right) \\ & = &0. \end{eqnarray}$

(2.18)

Here, $r = -1$ is one of the roots (2.18) and the other roots are given by

$\begin{equation*} r_{1} = \frac{k_{0}- \sqrt{k_{0}^{2}-4\rho ^{2}\left( \nu -1+\frac{\alpha +1 }{2}\right) \left[ \frac{2\left \vert \eta \right \vert }{1-\varkappa } -\left( \frac{\alpha +1}{2}\right) \right] \left( \nu -1+\frac{\alpha +1}{2} \right) \left( \frac{\alpha +1}{2}\right) }}{2\rho \left( \nu -1+\frac{ \alpha +1}{2}\right) \left[ \frac{2\left \vert \eta \right \vert }{ 1-\varkappa }-\left( \frac{\alpha +1}{2}\right) \right] }, \end{equation*}$

where

$\begin{equation*} k_{0} = \left( \nu -1+\frac{\alpha +1}{2}\right) \left[ \rho \left \{ \frac{2\left \vert \eta \right \vert }{1-\varkappa }-2\left( \frac{\alpha +1 }{2}\right) \right \} +\rho ^{2}-1\right] . \end{equation*}$

Taking $\varkappa = 0$ in corollary 1, we state the following:

Corollary 2. Let the function $\lambda \in \sum\limits$ and suppose that $\delta \in M_{\alpha , \beta }^{\nu, j}(\eta, \varkappa; A, B).$ If $\ \left(\mathcal{L} _{\nu , \alpha, \beta }\lambda \left(\omega \right) \right) ^{j}$ is majorized by $\left(\mathcal{L} _{\nu, \alpha, \beta }\delta \left(\omega \right) \right) ^{j}$ in $U^{\ast }$ , then

$\begin{equation*} \left| {{{\left( {{{\cal L}_{v,\alpha ,\beta }}\lambda (\omega )} \right)}^{j + 1}}} \right| \le \left| {{{\left( {{{\cal L}_{v,\alpha ,\beta }}\delta (\omega )} \right)}^{j + 1}}} \right| ,\;\;\;\; \left( \left \vert \omega \right \vert \lt r_{2}\right) , \end{equation*}$

where $r_{2} = r_{2}(\eta, \nu, \alpha, \beta)$ is the lowest positive roots of the equation

$\begin{eqnarray} &&\rho \left( \nu -1+\frac{\alpha +1}{2}\right) \left[ 2\left \vert \eta \right \vert -\left( \frac{\alpha +1}{2}\right) \right] r^{3}-\left( \nu -1+ \frac{\alpha +1}{2}\right) \left[ \rho \left( \frac{\alpha +1}{2}\right) +\rho ^{2}-1\right] r^{2}- \\ &&\left( \nu -1+\frac{\alpha +1}{2}\right) \left[ \rho \left \{ 2\left \vert \eta \right \vert -\left( \frac{\alpha +1}{2}\right) \right \} +\rho ^{2}-1 \right] r+\rho \left( \nu -1+\frac{\alpha +1}{2}\right) \left( \frac{\alpha +1}{2}\right) \\ & = &0, \end{eqnarray}$

(2.19)

given by

$\begin{equation*} r_{2} = \frac{k_{1}- \sqrt{k_{1}^{2}-4\rho ^{2}\left( \nu -1+\frac{\alpha +1 }{2}\right) \left[ 2\left \vert \eta \right \vert -\left( \frac{\alpha +1}{2} \right) \right] \left( \nu -1+\frac{\alpha +1}{2}\right) \left( \frac{\alpha +1}{2}\right) }}{2\rho \left( \nu -1+\frac{\alpha +1}{2}\right) \left[ 2\left \vert \eta \right \vert -\left( \frac{\alpha +1}{2}\right) \right] }, \end{equation*}$

where

$\begin{equation*} k_{1} = \left( \nu -1+\frac{\alpha +1}{2}\right) \left[ \rho \left \{ 2\left \vert \eta \right \vert -2\left( \frac{\alpha +1}{2}\right) \right \} +\rho ^{2}-1\right] . \end{equation*}$

Taking $\alpha = \beta = 1$ in corollary 2, we get the following:

Corollary 3. Let the function $\lambda \in \sum\limits$ and suppose that $\delta \in M_{\alpha , \beta }^{\nu, j}(\eta, \varkappa; A, B).$ If $\ \left(\mathcal{L} _{\nu , \alpha, \beta }\lambda \left(\omega \right) \right) ^{j}$ is majorized by $\left(\mathcal{L} _{\nu, \alpha, \beta }\delta \left(\omega \right) \right) ^{j}$ in $U^{\ast }$ , then

where $r_{3} = r_{3}(\eta, \nu)$ is the lowest positive roots of the equation

$\begin{eqnarray} &&\rho \nu \left[ 2\left \vert \eta \right \vert -1\right] r^{3}-\nu \left[ \rho +\rho ^{2}-1\right] r^{2}-\nu \left[ \rho \left( 2\left \vert \eta \right \vert -1\right) +\rho ^{2}-1\right] r+\rho \nu \\ & = &0, \end{eqnarray}$

(2.20)

given by

$\begin{equation*} r_{3} = \frac{k_{2}- \sqrt{k_{2}^{2}-4\rho ^{2}\nu \left[ 2\left \vert \eta \right \vert -1\right] \nu }}{2\rho \nu \left[ 2\left \vert \eta \right \vert -1\right] }, \end{equation*}$

where

$\begin{equation*} k_{2} = \nu \left[ \rho \left \{ 2\left \vert \eta \right \vert -2\right \} +\rho ^{2}-1\right] . \end{equation*}$

3. Majorization problem for the class $N_{\alpha, \beta }^{\nu, j}(\theta, b;A, B)$

Secondly, we exam majorization property for the class $N_{\alpha, \beta }^{\nu, j}(\theta, b;A, B).$

Theorem 2. Let the function $\lambda \in \sum\limits$ and suppose that $\delta \in N_{\alpha , \beta }^{\nu, j}(\theta, b;A, B).$ If

$\begin{equation*} \left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}\ll \left( \mathcal{L}_{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j},\left( j\in 0,1,2...\right) , \end{equation*}$

then

$\begin{equation} \left\vert \left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}\right\vert \leq \left\vert \left( \mathcal{L}_{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}\right\vert , \left( \left\vert \omega \right\vert \lt r_{4}\right) , \end{equation}$

(3.1)

where $r_{4} = r_{4}(\theta, b, \nu, \alpha, \beta, A, B)$ is the smallest positive roots of the equation

$\begin{eqnarray} &&-\rho \left[ \left\vert \left( B-A\right) b\cos \theta +\left( \nu +\frac{ \alpha +1}{2}-1\right) \left\vert B\right\vert \right\vert \right] r^{3}- \\ &&\left[ \rho \left\{ \nu +\frac{\alpha +1}{2}-1\right\} -\left\vert B\right\vert \left( 1-\rho ^{2}\right) \left( \nu -1+\frac{\alpha +1}{2} \right) \right] r^{2} \\ &&+\left[ \rho \left\{ \left\vert \left( B-A\right) b\cos \theta +\left( \nu +\frac{\alpha +1}{2}-1\right) \left\vert B\right\vert \right\vert \right\} +\left( 1-\rho ^{2}\right) \left( \nu -1+\frac{\alpha +1}{2}\right) \right] r \\ &&+\rho \left[ \nu +\frac{\alpha +1}{2}-1\right] = 0, \\ &&\mathit{\text{}}\left( -\frac{\pi }{2} \lt \theta \lt \frac{\pi }{2},\mathit{\text{}}-1\leq \beta \lt A\leq 1,\eta \in \mathbb{C} -\{0\},\mathit{\text{}}\nu ,\alpha ,\beta \gt 0,\mathit{\text{and}}\;\omega \in U^{\ast }\mathit{\text{}} \right) . \end{eqnarray}$

(3.2)

Proof. Since $\delta \in N_{\alpha, \beta }^{\nu, j}(\theta, b;A, B)$ , so

(3.3)

where, $k\left(\omega \right)$ is defined as (2.4).

From (3.3), we have

$\begin{equation} \frac{\omega \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}}{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}} = \frac{\left[ \left( B-A\right) b\cos \theta -\left( j+1\right) Be^{i\theta }\right] k\left( \omega \right) -\left( j+1\right) e^{i\theta }}{e^{i\theta }\left( 1+Bk\left( \omega \right) \right) }. \end{equation}$

(3.4)

Now, using (2.8) in (3.4) and making simple calculations, we obtain

$\begin{equation} \frac{\left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}}{\left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}} = \frac{ \begin{array}{c} \left[ \left( B-A\right) b\cos \theta +\left( \nu +\frac{\alpha +1}{2} -1 \right) Be^{i\theta }\right] k\left( \omega \right) \\ +\left[ \left( \nu +j+\frac{\alpha +1}{2}\right) -1 \right] e^{i\theta } \end{array} }{e^{i\theta }\left( 1+Bk\left( \omega \right) \right) \left( \nu -1+\frac{ \alpha +1}{2}\right) }, \end{equation}$

(3.5)

which, in view of $\ \left \vert k\left(\omega \right) \right \vert \leq \left \vert \omega \right \vert \; \ \left(\omega \in U^{\ast }\right),$ immediately yields the following inequality

$\begin{eqnarray} \left \vert \left( \mathcal{L} _{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}\right \vert &\leq &\frac{\left \vert e^{i\theta }\right \vert \left( 1+\left \vert B\right \vert \left \vert k\left( \omega \right) \right \vert \right) \left( \nu -1+\frac{\alpha +1}{2}\right) }{ \begin{array}{c} \left[ \left \vert \left( B-A\right) b\cos \theta +\left( \nu +\frac{ \alpha +1}{2}-1 \right) Be^{i\theta }\right \vert \right] \left \vert k\left( \omega \right) \right \vert \\ +\left[ \left( \nu +\frac{\alpha +1}{2}\right) -1 \right] \left \vert e^{i\theta }\right \vert \end{array} } \\ &&\times \left \vert \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}\right \vert . \end{eqnarray}$

(3.6)

Now, using (2.15) and (3.6) in (2.14) and working on the similar lines as in Theorem 1, we have

$\begin{eqnarray*} \left \vert \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}\right \vert &\leq &\left[ \left \vert \varphi \left( \omega \right) \right \vert +\frac{\left \vert \omega \right \vert \left( 1-\left \vert \varphi \left( \omega \right) \right \vert ^{2}\right) \left( 1+\left \vert B\right \vert \left \vert \omega \right \vert \right) \left( \nu -1+\frac{\alpha +1}{2}\right) }{\left( 1-\left \vert \omega \right \vert ^{2}\right) \left[ \begin{array}{c} \left \{ \left \vert \left( B-A\right) b\cos \theta +\left( \begin{array}{c} \nu +\frac{\alpha +1}{2} -1 \end{array} \right) B\right \vert \right \} \left \vert \omega \right \vert \\ +\left[ \left( \nu +\frac{\alpha +1}{2}\right) -1 \right] \end{array} \right] }\right] \\ &&\times \left \vert \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}\right \vert . \end{eqnarray*}$

By setting $\left \vert \omega \right \vert = r, \;\; \left \vert \varphi \left(\omega \right) \right \vert = \rho \;\; \left(0\leq \rho \leq 1\right),$ leads us to the inequality

$\begin{eqnarray} \left \vert \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j}\right \vert &\leq &\left[ \frac{\Phi _{2}(\rho )}{ \left( 1-r^{2}\right) \left[ \begin{array}{c} \left \{ \left \vert \left( B-A\right) b\cos \theta +\left( \nu +\frac{ \alpha +1}{2}-1 \right) B\right \vert \right \} r \\ +\left( \nu +\frac{\alpha +1}{2}\right) -1 \end{array} \right] }\right] \\ &&\times \left \vert \left( \mathcal{L} _{\nu -1,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j}\right \vert , \end{eqnarray}$

(3.7)

where the function $\Phi _{2}(\rho)$ is given by

$\begin{eqnarray*} \Phi _{2}(\rho ) & = &\rho \left( 1-r^{2}\right) \left[ \begin{array}{c} \left \{ \left \vert \left( B-A\right) b\cos \theta +\left( \nu +\frac{ \alpha +1}{2}-1 \right) B\right \vert \right \} r \\ +\left( \nu +\frac{\alpha +1}{2}\right) -1 \end{array} \right] \\ &&+r\left( 1-\rho ^{2}\right) \left( 1+Br\right) \left( \nu -1+\frac{\alpha +1}{2}\right) . \end{eqnarray*}$

$\Phi _{2}(\rho)$ its maximum value at $\rho = 1$ with $r_{4} = r_{4}(\theta, b, \nu, \alpha, \beta, A, B)$ given in (3.2). Moreover if $0\leq \xi _{1}\leq r_{4}(\theta, b, \nu, \alpha, \beta, A, B),$ then the function.

$\begin{eqnarray*} \psi _{2}(\rho ) & = &\rho \left( 1-\xi _{1}^{2}\right) \left[ \begin{array}{c} \left \{ \left \vert \left( B-A\right) b\cos \theta +\left( \nu +\frac{ \alpha +1}{2}-1 \right) B\right \vert \right \} \xi _{1} \\ +\left( \nu +\frac{\alpha +1}{2}\right) -1 \end{array} \right] \\ &&+\xi _{1}\left( 1-\rho ^{2}\right) \left( 1+B\xi _{1}\right) \left( \nu -1+ \frac{\alpha +1}{2}\right) , \end{eqnarray*}$

increasing on the interval $0\leq \rho \leq 1,$ so that $\psi _{2}(\rho)$ does not exceed

$\begin{equation*} \psi _{2}(1) = \left( 1-\xi _{1}^{2}\right) \left[ \begin{array}{c} \left \{ \left \vert \left( B-A\right) b\cos \theta +\left( \nu +\frac{ \alpha +1}{2}-1 \right) B\right \vert \right \} \xi _{1} \\ +\left( \nu +\frac{\alpha +1}{2}\right) -1 \end{array} \right] . \end{equation*}$

Therefore, from this fact (3.7) gives the inequality (3.1). We complete the proof.

Special Cases: Let $A = 1$ and $B = -1$ in Theorem 2, we obtain the following corollary.

Corollary 4. Let the function $\lambda \in \sum\limits$ and suppose that $\delta \in N_{\alpha , \beta }^{\nu, j}(\theta, b;A, B).$ If

then

$\begin{equation*} \left\vert \left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}\right\vert \leq \left\vert \left( \mathcal{L}_{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}\right\vert , \left( \left\vert \omega \right\vert \lt r_{5}\right) , \end{equation*}$

where $r_{5} = r_{5}(\theta, b, \nu, \alpha, \beta)$ is the lowest positive roots of the equation

$\begin{eqnarray} &&-\rho \left[ \left\vert -2b\cos \theta +\left( \nu +\frac{\alpha +1}{2} -1\right) \right\vert \right] r^{3}- \\ &&\left[ \rho \left\{ \nu +\frac{\alpha +1}{2}-1\right\} -\left( 1-\rho ^{2}\right) \left( \nu -1+\frac{\alpha +1}{2}\right) \right] r^{2}+ \\ &&\left[ \rho \left\{ \left\vert -2b\cos \theta +\left( \nu +\frac{\alpha +1 }{2}-1\right) \right\vert \right\} +\left( 1-\rho ^{2}\right) \left( \nu -1+ \frac{\alpha +1}{2}\right) \right] r+ \\ &&\rho \left[ \nu +\frac{\alpha +1}{2}-1\right] = 0. \end{eqnarray}$

(3.8)

Where $r = -1$ is first roots and the other two roots are given by

$\begin{equation*} \mathit{\text{}}r_{5} = \frac{\kappa _{0}-\sqrt{\kappa _{0}^{2}+4\rho ^{2}\left[ \left\vert -2b\cos \theta +\left( \nu +\frac{\alpha +1}{2}-1\right) \right\vert \right] \left[ \nu +\frac{\alpha +1}{2}-1\right] }}{-2\rho \left[ \left\vert -2b\cos \theta +\left( \nu +\frac{\alpha +1}{2}-1\right) \right\vert \right] }\mathit{\text{,}} \end{equation*}$

and

$\begin{equation*} \kappa _{0} = \left[ \left( 1-\rho ^{2}\right) \left( \nu -1+\frac{\alpha +1}{2 }\right) -\rho \left\{ \left\vert -2b\cos \theta +2\left( \nu +\frac{\alpha +1}{2}-1\right) \right\vert \right\} \right] .\mathit{\text{}} \end{equation*}$

Which reduces to Corollary 4 for $\theta = 0$ .

Corollary 5. Let the function $\lambda \in \sum\limits$ and suppose that $\delta \in N_{\alpha , \beta }^{\nu, j}(\theta, b;A, B).$ If

then

$\begin{equation*} \left\vert \left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}\right\vert \leq \left\vert \left( \mathcal{L}_{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}\right\vert , \left( \left\vert \omega \right\vert \lt r_{6}\right) , \end{equation*}$

where $r_{6} = r_{6}(b, \nu, \alpha, \beta)$ is the least positive roots of the equation

$\begin{eqnarray} &&-\rho \left[ \left\vert -2b+\left( \nu +\frac{\alpha +1}{2}-1\right) \right\vert \right] r^{3}- \\ &&\left[ \rho \left\{ \nu +\frac{\alpha +1}{2}-1\right\} -\left( 1-\rho ^{2}\right) \left( \nu -1+\frac{\alpha +1}{2}\right) \right] r^{2}+ \\ &&\left[ \rho \left\{ \left\vert -2b+\left( \nu +\frac{\alpha +1}{2} -1\right) \right\vert \right\} +\left( 1-\rho ^{2}\right) \left( \nu -1+ \frac{\alpha +1}{2}\right) \right] r+ \\ &&\rho \left[ \nu +\frac{\alpha +1}{2}-1\right] = 0, \end{eqnarray}$

(3.9)

given by

$\begin{equation*} r_{6} = \frac{\kappa _{1}-\sqrt{\kappa _{1}^{2}+4\rho ^{2}\left[ \left\vert -2b+\left( \nu +\frac{\alpha +1}{2}-1\right) \right\vert \right] \left[ \nu + \frac{\alpha +1}{2}-1\right] }}{-2\rho \left[ \left\vert -2b+\left( \nu + \frac{\alpha +1}{2}-1\right) \right\vert \right] },\mathit{\text{}} \end{equation*}$

and

$\begin{equation*} \kappa _{1} = \left[ \left( 1-\rho ^{2}\right) \left( \nu -1+\frac{ \alpha +1}{2}\right) -\rho \left\{ \left\vert -2b+2\left( \nu +\frac{\alpha +1}{2}-1\right) \right\vert \right\} \right] .\mathit{\text{}} \end{equation*}$

Taking $\alpha = \beta = 1$ in corollary 5, we get.

Corollary 6. Let the function $\lambda \in \sum\limits$ and suppose that $\delta \in N_{\alpha , \beta }^{\nu, j}(\theta, b;A, B).$ If

then

$\begin{equation*} \left\vert \left( \mathcal{L}_{\nu ,\alpha ,\beta }\lambda \left( \omega \right) \right) ^{j+1}\right\vert \leq \left\vert \left( \mathcal{L}_{\nu ,\alpha ,\beta }\delta \left( \omega \right) \right) ^{j+1}\right\vert , \left( \left\vert \omega \right\vert \lt r_{7}\right) , \end{equation*}$

where $r_{7} = r_{7}(b, \nu)$ is the lowest positive roots of the equation

$\begin{eqnarray} &&-\rho \left\vert -2b+\nu \right\vert r^{3}-\left[ \rho \nu -\left( 1-\rho ^{2}\right) \nu \right] r^{2}+ \\ &&\left[ \rho \left\vert -2b+\nu \right\vert +\left( 1-\rho ^{2}\right) \nu \right] r+\rho \left[ \nu \right] = 0, \end{eqnarray}$

(3.10)

given by

$\begin{equation*} r_{7} = \frac{\kappa _{2}-\sqrt{\kappa _{2}^{2}+4\rho ^{2}\left[ \left\vert -2b+\nu \right\vert \right] \left[ \nu \right] }}{-2\rho \left[ \left\vert -2b+\nu \right\vert \right] },\mathit{\text{}} \end{equation*}$

and

$\begin{equation*} \kappa _{2} = \left[ \left( 1-\rho ^{2}\right) \nu -\rho \left\{ \left\vert -2b+2\nu \right\vert \right\} \right] . \end{equation*}$

4. Conclusion

In this paper, we explore the problems of majorization for the classes $M_{\alpha, \beta }^{\nu, j}(\eta, \varkappa; A, B)$ and $N_{\alpha, \beta }^{\nu, j}(\theta, b;A, B)$ by using a convolution operator $\mathcal{L} _{\nu, \alpha, \beta }$ . These results generalizes and unify the theory of majorization which is an active part of current ongoing research in Geometric Function Theory. By specializing different parameters like $\nu, \; \eta, \; \varkappa, \; \theta$ and $b$ , we obtain a number of important corollaries in Geometric Function Theory.

Acknowledgments

The work here is supported by GUP-2019-032.

Conflict of interest

The authors agree with the contents of the manuscript, and there is no conflict of interest among the authors.

References

[1]	M. T. Alves, F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13–22. https://doi.org/10.1016/j.jtbi.2017.02.002 doi: 10.1016/j.jtbi.2017.02.002
[2]	R. Arditi, L. R. Ginzburg, Couplng in predator-prey dynamics: ratio-dependence, J. Theor. Biol., 139 (1989), 311–326. https://doi.org/10.1016/S0022-5193(89)80211-5 doi: 10.1016/S0022-5193(89)80211-5
[3]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331–340. https://doi.org/10.2307/3866 doi: 10.2307/3866
[4]	E. Beretta, Y. Kuang, Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal.-Theor., 32 (1998), 381–408. https://doi.org/10.1016/S0362-546X(97)00491-4 doi: 10.1016/S0362-546X(97)00491-4
[5]	L. Berec, Impacts of foraging facilitation among predators on predator-prey dynamics, Bull. Math. Biol., 72 (2010), 94–121. https://doi.org/10.1007/s11538-009-9439-1 doi: 10.1007/s11538-009-9439-1
[6]	R. S. Cantrell, C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206–222. https://doi.org/10.1006/jmaa.2000.7343 doi: 10.1006/jmaa.2000.7343
[7]	J. Caperon, Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), 188–192. https://doi.org/10.2307/1934845 doi: 10.2307/1934845
[8]	C. Cosner, D. L. DeAngelis, J. S. Ault, D. B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65–75. https://doi.org/10.1006/tpbi.1999.1414 doi: 10.1006/tpbi.1999.1414
[9]	M. G. Crandall, P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321–340. https://doi.org/10.1016/0022-1236(71)90015-2 doi: 10.1016/0022-1236(71)90015-2
[10]	Y. Dai, Y. Zhao, B. Sang, Four limit cycles in a predator–prey system of Leslie type with generalized Holling type Ⅲ functional response, Nonlinear Anal.-Real, 50 (2019), 218–239. https://doi.org/10.1016/j.nonrwa.2019.04.003 doi: 10.1016/j.nonrwa.2019.04.003
[11]	Y. Dai, Y. Zhao, Hopf cyclicity and global dynamics for a predator–prey system of Leslie type with simplified Holling type Ⅳ functional response, Int. J. Bifurcat. Chaos, 28 (2018), 1850166. https://doi.org/10.1142/S0218127418501663 doi: 10.1142/S0218127418501663
[12]	D. L. DeAngelis, R. A. Goldstein, R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881–892. https://doi.org/10.2307/1936298 doi: 10.2307/1936298
[13]	Y. Enatsu, J. Roy, M. Banerjee, Hunting cooperation in a prey-predator model with maturation delay, J. Biol. Dyn., 18 (2024), 2332279. https://doi.org/10.1080/17513758.2024.2332279 doi: 10.1080/17513758.2024.2332279
[14]	S. Fu, H. Zhang, Effect of hunting cooperation on the dynamic behavior for a diffusive Holling tyep Ⅱ predator-prey model, Commun. Nonlinear Sci. Numer. Simulat., 99 (2021), 105807. https://doi.org/10.1016/j.cnsns.2021.105807 doi: 10.1016/j.cnsns.2021.105807
[15]	C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, The Memoirs of the Entomological Society of Canada, 97 (1995), 5–60. https://doi.org/10.4039/entm9745fv doi: 10.4039/entm9745fv
[16]	P. Kareiva, G. Odell, Swarms of predators exhibit "prey-taxis" if individual predators use are-restricted search, Amer. Nat., 130 (1987), 233–270. https://doi.org/10.1086/284707 doi: 10.1086/284707
[17]	W. Ko, K. Ryu, A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: Ⅰ global existence and stability, J. Math. Anal. Appl., 543 (2025), 129005. https://doi.org/10.1016/j.jmaa.2024.129005 doi: 10.1016/j.jmaa.2024.129005
[18]	W. Ko, K. Ryu, A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: Ⅱ stationary pattern formation, J. Math. Anal. Appl., 543 (2025), 128947. https://doi.org/10.1016/j.jmaa.2024.128947 doi: 10.1016/j.jmaa.2024.128947
[19]	Y. Kuang, Delay differential equations, In: Encyclopedia of theoretical ecology, Boston: Academic Press, 1993,163–166. https://doi.org/10.1525/9780520951785-032
[20]	Z. Liu, R. Yuan, Stability and bifurcation in a delayed predator–prey system with Beddington–DeAngelis functional response, J. Math. Anal. Appl., 296 (2004), 521–537. https://doi.org/10.1016/j.jmaa.2004.04.051 doi: 10.1016/j.jmaa.2004.04.051
[21]	Z. Liu, R. Yuan, The effect of diffusion for a predator-prey system with nonmonotonic functional response, Int. J. Bifurcat. Chaos, 14 (2004), 4309–4316. https://doi.org/10.1142/S0218127404011867 doi: 10.1142/S0218127404011867
[22]	S. Liu, E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM J. Appl. Math., 66 (2006), 1101–1129. https://doi.org/10.1137/050630003 doi: 10.1137/050630003
[23]	C. Nie, D. Jin, R. Yang, Hopf bifurcation analysis in a delayed diffusive predator-prey system with nonlocal competition and generalist predator, AIMS Math., 7 (2022), 13344–13360. https://doi.org/10.3934/math.2022737 doi: 10.3934/math.2022737
[24]	L. Přibylová, L. Berec, Predator interference and stability of predator-prey dynamics, J. Math. Biol., 71 (2015), 301–323. https://doi.org/10.1007/s00285-014-0820-9 doi: 10.1007/s00285-014-0820-9
[25]	S. Ruan, D. Xiao, Gobal analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445–1472. https://doi.org/10.1137/S0036139999361896 doi: 10.1137/S0036139999361896
[26]	K. Ryu, W. Ko, On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation, Discrete. Cont. Dyn. Sys. B, 27 (2022), 6679–6709. https://doi.org/10.3934/dcdsb.2022015 doi: 10.3934/dcdsb.2022015
[27]	K. Ryu, W. Ko, Asymptotic behavior of positive solutions to a predator–prey elliptic system with strong hunting cooperation in predators, Physica A, 531 (2019), 121726. https://doi.org/10.1016/j.physa.2019.121726 doi: 10.1016/j.physa.2019.121726
[28]	K. Ryu, W. Ko, M. Haque, Bifurcation analysis in a predator–prey system with a functional response increasing in both predator and prey densities, Nonlinear Dyn., 94 (2018), 1639–1656. https://doi.org/10.1007/s11071-018-4446-0 doi: 10.1007/s11071-018-4446-0
[29]	D. Sen, S. Ghorai, M. Banerjee, Allee effect in prey versus hunting cooperation on predator – enhancement of stable coexistence, Int. J. Bifurcat. Chaos, 29 (2019), 1950081. https://doi.org/10.1142/S0218127419500810 doi: 10.1142/S0218127419500810
[30]	Q. Shi, Y. Song, Spatially nonhomogeneous periodic patterns in a delayed predator-prey model with predator-taxis diffusion, Appl. Math. Lett., 131 (2022), 108062. https://doi.org/10.1016/j.aml.2022.108062 doi: 10.1016/j.aml.2022.108062
[31]	H. Shu, X. Hu, L. Wang, J. Watmough, Delay induced stability switch, multitype bistability and chaos in an intraguild predation model, J. Math. Biol., 71 (2015), 1269–1298. https://doi.org/10.1007/s00285-015-0857-4 doi: 10.1007/s00285-015-0857-4
[32]	D. Song, C. Li, Y. Song, Stability and cross-diffusion-driven instability in a diffusive predator-prey system with hunting cooperation functional response, Nonlinear Anal.-Real, 54 (2020), 103106. https://doi.org/10.1016/j.nonrwa.2020.103106 doi: 10.1016/j.nonrwa.2020.103106
[33]	Y. Song, J. Shi, H. Wang, Spatiotemporal dynamics of a diffusive consumer-resource model with explicit spatial memory, Stud. Appl. Math., 148 (2021), 373–395. https://doi.org/10.1111/sapm.12443 doi: 10.1111/sapm.12443
[34]	Z. Sun, W. Jiang, Bifurcation and spatial patterns driven by predator-taxis in a predator-prey system with Beddington-DeAngelis functional response, Discrete. Cont. Dyn. Sys. B, 29 (2024), 4043–4070. https://doi.org/10.3934/dcdsb.2024034 doi: 10.3934/dcdsb.2024034
[35]	S. K. Verma, B. Kumar, Bifurcation and pattern formation in a prey–predator model with cooperative hunting, Eur. Phys. J. Plus, 139 (2024), 734. https://doi.org/10.1140/epjp/s13360-024-05543-y doi: 10.1140/epjp/s13360-024-05543-y
[36]	F. Wang, R. Yang, Y. Xie, J. Zhao, Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator, AIMS Math., 8 (2023), 17719–17743. https://doi.org/10.3934/math.2023905 doi: 10.3934/math.2023905
[37]	C. Wang, S. Yuan, H. Wang, Spatiotemporal patterns of a diffusive prey-predator model with spatial memory and pregnancy period in an intimidatory environment, J. Math. Biol., 84 (2022), 12. https://doi.org/10.1007/s00285-022-01716-4 doi: 10.1007/s00285-022-01716-4
[38]	J. Wang, S. Wu, J. Shi, Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis, Discrete. Cont. Dyn. Sys. B, 26 (2021), 1273–1289. https://doi.org/10.3934/dcdsb.2020162 doi: 10.3934/dcdsb.2020162
[39]	X. Wang, S. Li, Y. Dai, K. Wu, Dynamics in a slow-fast Leslie-Gower predator-prey model with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 545 (2025), 129177. https://doi.org/10.1016/j.jmaa.2024.129177 doi: 10.1016/j.jmaa.2024.129177
[40]	S. Wu, J. Shi, B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differ. Equations, 260 (2016), 5847–5874. https://doi.org/10.1016/j.jde.2015.12.024 doi: 10.1016/j.jde.2015.12.024
[41]	D. Xiao, S. Ruan, Multiple bifurcations in a delayed predator–prey system with nonmonotonic functional response, J. Differ. Equations, 176 (2001), 494–510. https://doi.org/10.1006/jdeq.2000.3982 doi: 10.1006/jdeq.2000.3982
[42]	J.-F. Zhang, W.-T. Li, X.-P. Yan, Multiple bifurcations in a delayed predator–prey diffusion system with a functional response, Nonlinear Anal.-Real, 11 (2010), 2708–2725. https://doi.org/10.1016/j.nonrwa.2009.09.019 doi: 10.1016/j.nonrwa.2009.09.019
[43]	T. Zhang, H. Zang, Delay-induced Turing instability in reaction-diffusion equations, Phys. Rev. E, 90 (2014), 052908. https://doi.org/10.1103/PhysRevE.90.052908 doi: 10.1103/PhysRevE.90.052908

This article has been cited by:

1.	Syed Ghoos Ali Shah, Saqib Hussain, Akhter Rasheed, Zahid Shareef, Maslina Darus, Fanglei Wang, Application of Quasisubordination to Certain Classes of Meromorphic Functions, 2020, 2020, 2314-8888, 1, 10.1155/2020/4581926
2.	Syed Ghoos Ali Shah, Saima Noor, Saqib Hussain, Asifa Tasleem, Akhter Rasheed, Maslina Darus, Rashad Asharabi, Analytic Functions Related with Starlikeness, 2021, 2021, 1563-5147, 1, 10.1155/2021/9924434
3.	Syed Ghoos Ali Shah, Saqib Hussain, Saima Noor, Maslina Darus, Ibrar Ahmad, Teodor Bulboaca, Multivalent Functions Related with an Integral Operator, 2021, 2021, 1687-0425, 1, 10.1155/2021/5882343
4.	Syed Ghoos Ali Shah, Shahbaz Khan, Saqib Hussain, Maslina Darus, $q$ -Noor integral operator associated with starlike functions and $q$ -conic domains, 2022, 7, 2473-6988, 10842, 10.3934/math.2022606
5.	Neelam Khan, Muhammad Arif, Maslina Darus, Abdellatif Ben Makhlouf, Majorization Properties for Certain Subclasses of Meromorphic Function of Complex Order, 2022, 2022, 1099-0526, 1, 10.1155/2022/2385739
6.	Ibrar Ahmad, Syed Ghoos Ali Shah, Saqib Hussain, Maslina Darus, Babar Ahmad, Firdous A. Shah, Fekete-Szegö Functional for Bi-univalent Functions Related with Gegenbauer Polynomials, 2022, 2022, 2314-4785, 1, 10.1155/2022/2705203
7.	F. Müge SAKAR, Syed Ghoos Ali SHAH, Saqib HUSSAİN, Akhter RASHEED, Muhammad NAEEM, q-Meromorphic closed-to-convex functions related with Janowski function, 2022, 71, 1303-5991, 25, 10.31801/cfsuasmas.883970
8.	Syed Ghoos Ali Shah, Sa’ud Al-Sa’di, Saqib Hussain, Asifa Tasleem, Akhter Rasheed, Imran Zulfiqar Cheema, Maslina Darus, Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination, 2023, 56, 2391-4661, 10.1515/dema-2022-0232
9.	Abdul Basir, Muhammad Adil Khan, Hidayat Ullah, Yahya Almalki, Saowaluck Chasreechai, Thanin Sitthiwirattham, Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory, 2023, 12, 2075-1680, 885, 10.3390/axioms12090885
10.	Shatha S. Alhily, Alina Alb Lupas, Certain Class of Close-to-Convex Univalent Functions, 2023, 15, 2073-8994, 1789, 10.3390/sym15091789

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)