On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena

Fouad Mohammad Salama; Faisal Fairag; Fouad Mohammad Salama; Faisal Fairag

doi:10.3934/math.2024020

AIMS Mathematics

2024, Volume 9, Issue 1: 340-370. doi: 10.3934/math.2024020

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Research article

On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena

Fouad Mohammad Salama ^,,
Faisal Fairag

Department of Mathematics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

Received: 09 August 2023 Revised: 29 September 2023 Accepted: 12 October 2023 Published: 29 November 2023
MSC : 35R11, 65M06, 65M12

In recent years, the application of variable-order (VO) fractional differential equations for describing complex physical phenomena ranging from biology, hydrology, mechanics and viscoelasticity to fluid dynamics has become one of the most hot topics in the context of scientific modeling. An interesting aspect of VO operators is their capability to address the behavior of scientific and engineering systems with time and spatially varying properties. The VO fractional diffusion equation is a fundamental model that allows transitions among sub-diffusive, diffusive and super-diffusive behaviors without altering the underlying governing equations. In this paper, we considered the two-dimensional fractional diffusion equation with the Caputo time VO derivative, which is essential for describing anomalous diffusion in real-world complex systems. A new Crank-Nicolson (C-N) difference scheme and an efficient explicit decoupled group (EDG) method were proposed to solve the problem under consideration. The proposed EDG method is based on a skewed difference scheme in conjunction with a grouping procedure of the solution grid points. Special attention was devoted to investigating the stability and convergence of the proposed methods. Three numerical examples with known exact analytical solutions were provided to illustrate our considerations. The proposed methods were shown to be stable and convergent theoretically as well as numerically. In addition, a comparative study was done between the EDG method and the C-N difference scheme. It was found that the proposed methods are accurate in simulating the considered problem, while the EDG method is superior to the C-N difference method in terms of Central Processing Unit (CPU) timing, verifying the efficiency of the former method in solving the VO problem.

Keywords:

Citation: Fouad Mohammad Salama, Faisal Fairag. On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena[J]. AIMS Mathematics, 2024, 9(1): 340-370. doi: 10.3934/math.2024020

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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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