Asymptotic analysis of high frequency modes for thin elastic plates

1.   Introduction

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

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

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

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

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]$)$:

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

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

satisfying

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

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

(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]):

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)

Let us denote

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:

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

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

Remark 1.

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

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

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

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

where,

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

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

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

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

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

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

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

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

Taking

In (2.3), we have

which implies

Using (2.6) in (2.5), we get

Application of Leibnitz's Theorem on (1.9) gives

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

Or, equivalently

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

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

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

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

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

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

By taking

reduces to the inequality

where

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

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

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

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

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

where

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

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

given by

where

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

given by

where

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

then

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

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

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

From (3.3), we have

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

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

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

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

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

$\Phi _{2}(\rho)$ 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.

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

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

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

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

and

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

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

given by

and

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

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

given by

and

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.