A new Log-Lomax distribution, properties, stock price, and heart attack predictions using machine learning techniques

Aliyu Ismail Ishaq; Abdullahi Ubale Usman; Hana N. Alqifari; Amani Almohaimeed; Hanita Daud; Sani Isah Abba; Ahmad Abubakar Suleiman; Aliyu Ismail Ishaq; Abdullahi Ubale Usman; Hana N. Alqifari; Amani Almohaimeed; Hanita Daud; Sani Isah Abba; Ahmad Abubakar Suleiman

doi:10.3934/math.2025575

AIMS Mathematics

2025, Volume 10, Issue 5: 12761-12807. doi: 10.3934/math.2025575

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Research article Special Issues

A new Log-Lomax distribution, properties, stock price, and heart attack predictions using machine learning techniques

1.
Department of Statistics, Ahmadu Bello University, Zaria, Nigeria
2.
Department of Statistics, Aliko Dangote University of Science and Technology, Wudil, Nigeria
3.
Department of Statistics and Operation Research, College of Science, Qassim University, Saudi Arabia
4.
Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, 32610 Seri Iskandar, Malaysia
5.
Department of Civil Engineering, Prince Mohammad Bin Fahd University, Al Khobar, Saudi Arabia

Received: 17 February 2025 Revised: 24 April 2025 Accepted: 14 May 2025 Published: 30 May 2025
MSC : 60E05, 62F10, 62H12

In this study, we introduced the log-Lomax distribution, a more versatile probabilistic model for capturing various statistical properties. The study was divided into two sections: Modeling stock price exchange rates with the proposed log-Lomax distribution and incorporating log-Lomax features into machine learning models for prediction. In the modeling section, we introduced the log-Lomax distribution, which employed a logarithmic transformation with an exponent parameter. The model was left- and right-skewed, monotonic, inverted, and bathtub-shaped. Some properties were obtained, and several parameter estimation techniques were evaluated using a simulation study. The model was applied to two Nigerian stock exchange rate datasets: Naira-to-Euro and Naira-to-Riyal, as well as the Worcester heart attack patient dataset. The prediction section used insights from modeling methods and machine learning workflows to improve accuracy and reduce overfitting. The predictions were evaluated in two ways: With raw data and features derived from the log-Lomax model. Employing log-Lomax model features, Random Forest, and XGBoost achieved 99.87% accuracy in the Euro dataset, respectively. Random Forest and XGBoost had accuracy rates of 98.67% and 99.33% on the Riyal dataset, respectively, and 91.25% and 88.75% on the heart attack dataset. Random Forests and XGBoost are the preferred models, as they consistently provide the best prediction performance and stability mix across datasets.

Keywords:

Citation: Aliyu Ismail Ishaq, Abdullahi Ubale Usman, Hana N. Alqifari, Amani Almohaimeed, Hanita Daud, Sani Isah Abba, Ahmad Abubakar Suleiman. A new Log-Lomax distribution, properties, stock price, and heart attack predictions using machine learning techniques[J]. AIMS Mathematics, 2025, 10(5): 12761-12807. doi: 10.3934/math.2025575

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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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