Supervised neural learning for the predator-prey delay differential system of Holling form-III

Naret Ruttanaprommarin; Zulqurnain Sabir; Salem Ben Said; Muhammad Asif Zahoor Raja; Saira Bhatti; Wajaree Weera; Thongchai Botmart; Naret Ruttanaprommarin; Zulqurnain Sabir; Salem Ben Said; Muhammad Asif Zahoor Raja; Saira Bhatti; Wajaree Weera; Thongchai Botmart

doi:10.3934/math.20221101

AIMS Mathematics

2022, Volume 7, Issue 11: 20126-20142. doi: 10.3934/math.20221101

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Research article

Supervised neural learning for the predator-prey delay differential system of Holling form-III

1.
Department of Science and Mathematics, Faculty of Industry and Technology, Rajamangala University of Technology Isan Sakonnakhon Campus, Sakonnakhon 47160, Thailand
2.
Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, UAE
3.
Department of Mathematics, Hazara University, Mansehra, Pakistan
4.
Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan
5.
Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad Pakistan
6.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 05 July 2022 Revised: 22 August 2022 Accepted: 30 August 2022 Published: 13 September 2022
MSC : 92B20, 34A34

The purpose of this work is to present the stochastic computing study based on the artificial neural networks (ANNs) along with the scaled conjugate gradient (SCG), ANNs-SCG for solving the predator-prey delay differential system of Holling form-III. The mathematical form of the predator-prey delay differential system of Holling form-III is categorized into prey class, predator category and the recent past effects. Three variations of the predator-prey delay differential system of Holling form-III have been numerical stimulated by using the stochastic ANNs-SCG procedure. The selection of the data to solve the predator-prey delay differential system of Holling form-III is provided as 13%, 12% and 75% for testing, training, and substantiation together with 15 neurons. The correctness and exactness of the stochastic ANNs-SCG method is provided by using the comparison of the obtained and data-based reference solutions. The constancy, authentication, soundness, competence, and precision of the stochastic ANNs-SCG technique is performed through the analysis of the correlation measures, state transitions (STs), regression analysis, correlation, error histograms (EHs) and MSE.

Keywords:

Citation: Naret Ruttanaprommarin, Zulqurnain Sabir, Salem Ben Said, Muhammad Asif Zahoor Raja, Saira Bhatti, Wajaree Weera, Thongchai Botmart. Supervised neural learning for the predator-prey delay differential system of Holling form-III[J]. AIMS Mathematics, 2022, 7(11): 20126-20142. doi: 10.3934/math.20221101

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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.

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