Uniform bound of the highest-order energy for three dimensional inhomogeneous incompressible elastodynamics

1.   Introduction and preliminaries

Let $I\subseteq\mathbb{R}$ be an interval. Then a real-valued function $\mathcal{X}:I\rightarrow\mathbb{R}$ is said to be convex if the inequality

holds for all $x, y\in I$ and $\mu\in[0, 1]$.

It is well-known that convexity has wild applications in pure and applied mathematics ^{[1,2,3,4,5,6,7,8]}. In particular, many remarkable inequalities can be found in the literature ^{[9,10,11,12,13,14,15,16,17,18,19,20]} via the convexity theory. Recently, the generalizations, extensions and variants for the convexity have attracted the attentions of many researchers ^{[21,22,23,24,25]}.

İşcan ^[26] introduced the class of reciprocal convex functions as follows.

A real-vauled function $\mathcal{X}:I\subseteq(0, \infty)\rightarrow\mathbb{R}$ is said to be reciprocal convex if the inequality

holds for all $x, y\in I$ and $\mu\in[0, 1]$.

In ^[27], Noor et al. introduced and discussed the class of reciprocal ${\rho}$-convex functions. Later, Noor et al. ^[28] extended the class of reciprocal convex functions on coordinates and introduced the class of $2D$ reciprocal convex functions.

Let $\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)$. Then a real-valued function $\mathcal{X}: \Omega\rightarrow\mathbb{R}$ is said to be $2D$ reciprocal convex if the inequality

holds for all $x, y \in [a, b]$, $u, w \in [c, d]$ and ${\mu}, {\lambda} \in [0, 1]$.

Very recently, Awan et al. ^[29] gave the definition of approximately reciprocal ${\rho}$-convex functions depending on a metric function.

It is well-known that the classical Hermite-Hadamard inequality ^{[30,31,32,33,34,35]} is one of the most famous and important inequalities in convexity theory, which can be stated as follows.

The double inequality

holds for all $a, b\in I$ with $a\neq b$ if $f: I\rightarrow\mathbb{R}$ is a convex function.

In the past half century, many researchers have devoted themselves to the generalizations, improvements and variants of the Hermite-Hadamard inequality. For example, Dragomir ^[36] obtained a two dimensional version of the Hermite-Hadamard inequality using the coordinated convex functions, Budak et al. ^[37] provided a two dimensional extension of the Hermite-Hadamard inequality by use of coordinated trigonometrically $\rho$-convex functions, İşcan ^[26] derived a new variant of the Hermite-Hadamard inequality by using the class of reciprocal convex functions, Noor et al. ^[27] obtained a generalized version of the Hermite-Hadamard inequality via the reciprocal $\rho$-convex functions, and Noor et al. ^[28] establshed a $2D$ version of the Hermite-Hadamard inequality using $2D$ reciprocal convex functions.

The main purpose of the article is to introduce the $2D$ approximately reciprocal $\rho$-convex functions, discuss how this class of functions unifies several other unrelated classes of reciprocal convex functions by considering some suitable choices of the given function $\Delta(\cdot, \cdot)$ and the real function $\rho(\cdot)$, derive several new refinements of the Hermite-Hadamard like inequalities involving $2D$ approximately reciprocal ${\rho}$-convex functions, and discuss the special cases of the main obtained results.

2.   Definition of $2D$ approximately reciprocal ${\rho}$-convex function and its special cases

In this section, we provide the definition of the class of $2D$ approximately reciprocal ${\rho}$-convex functions, and discuss its special cases.

Definition 2.1. Let $\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)$. Then a real-valued function $\mathcal{X}: \Omega\rightarrow\mathbb{R}$ is said to be a $2D$ approximately reciprocal ${\rho}$-convex function if the inequality

holds for $x, y \in [a, b]$, $u, w \in [c, d]$ and ${\mu}, {\lambda} \in [0, 1]$.

Next, We discuss some special cases of Definition 2.1.

Ⅰ. If we take $\Delta(x, y) = \epsilon(\| x^{-1}-y^{-1} \|)^{\gamma}$ and $\Delta(u, w) = \epsilon(\| u^{-1}-w^{-1} \|)^{\gamma}$ for some $\epsilon\in\mathbb{R}$ and $\gamma > 1$ in Definition 2.1, then we have a new definition of "$\gamma$-paraharmonic ${\rho}$-convex function of higher order".

Definition 2.2. Let $\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)$. Then a real-valued function $\mathcal{X}: \Omega\rightarrow\mathbb{R}$ is said to be a $2D$ $\gamma$-paraharmonic ${\rho}$-convex function of higher order if the inequality

takes place for all $x, y \in [a, b]$, $u, w \in [c, d]$ and ${\mu}, {\lambda} \in [0, 1]$.

Ⅱ. If we take $\Delta(x, y) = \epsilon(\| x^{-1}-y^{-1} \|)$ and $\Delta(u, w) = \epsilon(\|u^{-1}-w^{-1}\|)$ for some $\epsilon\in\mathbb{R}$ in Definition 2.1, then we obtain a new definition of "$\epsilon$-paraharmonic ${\rho}$-convex function".

Definition 2.3. Let $\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)$. Then a function $\mathcal{X}:\Omega\rightarrow\mathbb{R}$ is said to be a $2D$ $\epsilon$-paraharmonic ${\rho}$-convex function if

whenever $x, y \in [a, b]$, $u, w \in [c, d]$ and ${\mu}, {\lambda} \in [0, 1]$.

Ⅲ. If we take

and

in Definition 2.1, then we get a new definition of $2D$ reciprocal strong ${\rho}$-convex function of higher order.

Definition 2.4. Let $\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)$. Then a real-valued function $\mathcal{X}: \Omega\rightarrow\mathbb{R}$ is said to be a $2D$ reciprocal strong ${\rho}$-convex function of higher order if the inequality

is valid for all $x, y \in [a, b]$, $u, w \in [c, d]$ and ${\mu}, {\lambda} \in [0, 1]$.

Ⅳ. If we take $\sigma = 2$ in Definition 2.4, then

and we have the definition of $2D$ reciprocal strong ${\rho}$-convex function.

Definition 2.5. Let $\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)$. Then a real-valued function $\mathcal{X}: \Omega\rightarrow\mathbb{R}$ is said to be a $2D$ reciprocal strong ${\rho}$-convex function if the inequality

holds for $x, y \in [a, b]$, $u, w \in [c, d]$ and ${\mu}, {\lambda} \in [0, 1]$.

Ⅴ. If we take $\Delta(x, y) = \mu {\mu}(1-{\mu})\left(\frac{1}{x}-\frac{1}{y}\right)^{2}$ and $\Delta(u, w) = \mu {\lambda}(1-{\lambda})\left(\frac{1}{u}-\frac{1}{w}\right)^{2}$ for some $\mu > 0$ in Definition 2.1, then we obtain the definition of $2D$ reciprocal relaxed ${\rho}$-convex function.

Definition 2.6. Let $\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)$. Then a real-valued function $\mathcal{X}: \Omega\rightarrow\mathbb{R}$ is said to be a $2D$ reciprocal relaxed ${\rho}$-convex function if

whenever $x, y \in [a, b]$, $u, w \in [c, d]$ and ${\mu}, {\lambda} \in [0, 1]$.

Ⅵ. If we take $\Delta(x, y) = -{\mu}(1-{\mu})\left(\frac{xy}{x-y}\right)^{2}$ and $\Delta(u, w) = -{\lambda}(1-{\lambda})\left(\frac{uw}{u-w}\right)^{2}$ in Definition 2.1, then we have a new definition of $2D$ strongly $F$ reciprocal ${\rho}$-convex function.

Definition 2.7. Let $\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)$. Then a real-valued function $\mathcal{X}: \Omega\rightarrow\mathbb{R}$ is said to be a $2D$ strongly $F$ reciprocal ${\rho}$-convex function if the inequality

holds for all $x, y \in [a, b]$, $u, w \in [c, d]$ and ${\mu}, {\lambda}\in [0, 1]$.

Remark 2.8. It is pertinent to mention here that we can recapture other new classes of reciprocal convexity from Definition 2.1 by considering suitable choices of function $\rho(\cdot)$. For example, if we take $\rho(\mu) = \mu^{s}$ and $\rho(\lambda) = \lambda^{s}$ in Definition 2.1, then we have the class of Breckner type $2D$ approximately reciprocal $s$-convex functions; if we take $\rho(\mu) = \mu^{-s}$ and $\rho(\lambda) = \lambda^{-s}$ in Definition 2.1, then we get the class of Godunova-Levin type $2D$ approximately reciprocal $s$-convex functions; if we take $\rho(\mu) = 1$ and $\rho(\lambda) = 1$ in Definition 2.1, then we obtain the class of $2D$ approximately reciprocal $P$-convex functions. Moreover, if we choose suitable function $\Delta(\cdot, \cdot)$ in these discussed classes, then we also can get new refinements of reciprocal convexity, we left the details to the interested readers.

3.   New variant of the Hermite-Hadamard inequality

In this section, we derive a new variant of the Hermite-Hadamard inequality using the class of $2D$ approximately reciprocal ${\rho}$-convex functions.

Theorem 3.1. Let $\mathcal{X}: \Omega = [a, b]\times[c, d]\rightarrow\mathbb{R}$ be an integrable $2D$ approximately reciprocal ${\rho}$-convex function. Then we have the Hermite-Hadamard type inequality as follows

Proof. It follows from the $2D$ approximately reciprocal ${\rho}$-convexity of $\mathcal{X}$ that

Integrating above inequality with respect to $({\mu}, {\lambda})$ on $[0, 1]\times[0, 1]$ leads to

Similarly, we have

Integrating both sides of the above inequality with respect to $({\mu}, {\lambda})$ on $[0, 1]\times[0, 1]$, we get

This completes the proof.

4.   Applications of Theorem 3.1

In this section, we present some applications of Theorem 3.1.

Ⅰ. If ${\rho}({\mu}) = {\mu}$ and ${\rho}({\lambda}) = {\lambda}$, then Theorem 3.1 leads to Corollary 4.1.

Corollary 4.1. Let $\mathcal{X}: \Omega = [a, b]\times[c, d]\rightarrow\mathbb{R}$ be an integrable $2D$ approximately reciprocal convex function. Then one has

Ⅱ. If ${\rho}({\mu}) = {\mu}^{s}$ and ${\rho}({\lambda}) = {\lambda}^{s}$, then Theorem 3.1 becomes Corollary 4.2.

Corollary 4.2. Let $\mathcal{X}: \Omega = [a, b]\times[c, d]\rightarrow\mathbb{R}$ be an integrable Breckner type $2D$ approximately reciprocal $s$-convex function. Then

Ⅲ. If ${\rho}({\mu}) = {\mu}^{-s}$ and ${\rho}({\lambda}) = {\lambda}^{-s}$, then Theorem 3.1 reduces to Corollary 4.3.

Corollary 4.3. Let $\mathcal{X}: \Omega = [a, b]\times[c, d]\rightarrow\mathbb{R}$ be an integrable Godunova-Levin type $2D$ approximately reciprocal $s$-convex function. Then we get

Ⅳ. If ${\rho}({\mu}) = {\rho}({\lambda}) = 1$, then Theorem 3.1 leads to Corollary 4.4.

Corollary 4.4. Let $\mathcal{X}: \Omega = [a, b]\times[c, d]\rightarrow\mathbb{R}$ be an integrable $2D$ approximately reciprocal $P$-convex function. Then one has

Ⅴ. If we take

and

for some $\mu > 0$, then Theorem 3.1 reduces to Corollary 4.5.

Corollary 4.5. Let $\mathcal{X}: \Omega = [a, b]\times[c, d]\rightarrow\mathbb{R}$ be an integrable $2D$ reciprocal strong ${\rho}$-convex function of higher order. Then we obtain the inequality

Ⅵ. If we take $\sigma = 2$. Then Corollary 4.5 becomes Corollary 4.6.

Corollary 4.6. Let $\mathcal{X}: \Omega = [a, b]\times[c, d]\rightarrow\mathbb{R}$ be an integrable $2D$ reciprocal strong ${\rho}$-convex function. Then one has

5.   Bounds pertaining to trapezium like inequality using partial differentiable $2D$ approximately reciprocal $\rho$-convex functions

In this section, we present some bounds pertaining to trapezium like inequality using partial differentiable $2D$ approximately reciprocal $\rho$-convex functions. The following auxiliary result will play significant role in our Theorem 5.2.

Lemma 5.1. (See ^[28]) Let $\mathcal{X}: \Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)\rightarrow\mathbb{R}$ be a partial differential function on $\Omega$ such that $\frac{\partial^{2}\mathcal{X}}{\partial {\mu} \partial {\lambda}}\in L_{1}(\Omega)$. Then

where

In order to obtain our results we need the gamma function $\Gamma$ ^[38,39], beta function $B$ ^[40] and Gaussian hypergeometric functions $_{2}\mathscr{F}_{1}$ ^[41,42], which are defined by

and

respectively.

Theorem 5.2 Let $p, q > 1$ with $1/p+1/q = 1$, and $\mathcal{X}:\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)\rightarrow\mathbb{R}$ be a partial differentiable function on $\Omega$ such that $\frac{\partial^{2}\mathcal{X}}{\partial {\mu} \partial {\lambda}}\in L_{1}(\Omega)$ and $\left|\frac{\partial^{2}\mathcal{X}}{\partial {\lambda} \partial {\mu}}\right|^{q}$ is a $2D$ approximately reciprocal ${\rho}$-convex function. Then we have

where

and

Proof. It follows from Lemma 5.1, Hölder inequality and the $2D$ approximately reciprocal ${\rho}$-convexity of $\left|\frac{\partial^{2}\mathcal{X}}{\partial{\lambda}\partial{\mu}}\right|^{q}$ that

This completes the proof.

Ⅰ. If we take ${\rho}({\mu}) = {\mu}$ and ${\rho}({\lambda}) = {\lambda}$, then Theorem 5.2 leads to Corollary 5.3.

Corollary 5.3. Let $p, q > 1$ with $1/p+1/q = 1$, and $\mathcal{X}:\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)\rightarrow\mathbb{R}$ be a partial differentiable function on $\Omega$ such that $\frac{\partial^{2}\mathcal{X}}{\partial {\mu} \partial {\lambda}}\in L_{1}(\Omega)$ and $\left|\frac{\partial^{2}\mathcal{X}}{\partial {\lambda} \partial {\mu}}\right|^{q}$ is a $2D$ approximately reciprocal convex function. Then one has

where

and $\varphi_{5}(a, b, c, d; \Omega)$ and $\varphi_{6}(a, b, c, d; \Omega)$ are given in Theorem 5.2.

Ⅱ. Let ${\rho}({\mu}) = {\mu}^{s}$ and ${\rho}({\lambda}) = {\lambda}^{s}$. Then Theorem 5.2 reduces to Corollary 5.4.

Corollary 5.4. Let $p, q > 1$ with $1/p+1/q = 1$, and $\mathcal{X}:\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)\rightarrow\mathbb{R}$ be a partial differentiable function on $\Omega$ such that $\frac{\partial^{2}\mathcal{X}}{\partial {\mu} \partial {\lambda}}\in L_{1}(\Omega)$ and $\left|\frac{\partial^{2}\mathcal{X}}{\partial {\lambda} \partial {\mu}}\right|^{q}$ is a Breckner type $2D$ approximately reciprocal $s$-convex function. Then

where

and $\varphi_{5}(a, b, c, d; \Omega)$ and $\varphi_{6}(a, b, c, d; \Omega)$ are given in Theorem 5.2.

Ⅲ. If we take ${\rho}({\mu}) = {\mu}^{-s}$ and ${\rho}({\lambda}) = {\lambda}^{-s}$, then Theorem 5.2 becomes Corollary 5.5.

Corollary 5.5. Let $p, q > 1$ with $1/p+1/q = 1$, and $\mathcal{X}:\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)\rightarrow\mathbb{R}$ be a partial differentiable function on $\Omega$ such that $\frac{\partial^{2}\mathcal{X}}{\partial {\mu} \partial {\lambda}}\in L_{1}(\Omega)$ and $\left|\frac{\partial^{2}\mathcal{X}}{\partial {\lambda} \partial {\mu}}\right|^{q}$ is a Godunova-Levin type $2D$ approximately reciprocal $s$-convex function. Then we obtain

where

and $\varphi_{5}(a, b, c, d; \Omega)$ and $\varphi_{6}(a, b, c, d; \Omega)$ are given in Theorem 5.2.

Ⅳ. Let ${\rho}({\mu}) = {\rho}({\lambda}) = 1$. Then Theorem 5.2 leads to Corollary 5.6.

Corollary 5.6. Let $p, q > 1$ with $1/p+1/q = 1$, and $\mathcal{X}:\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)\rightarrow\mathbb{R}$ be a partial differentiable function on $\Omega$ such that $\frac{\partial^{2}\mathcal{X}}{\partial {\mu} \partial {\lambda}}\in L_{1}(\Omega)$ and $\left|\frac{\partial^{2}\mathcal{X}}{\partial {\lambda} \partial {\mu}}\right|^{q}$ is a $2D$ approximately reciprocal $P$-convex function. Then we have

where

Ⅴ. Let

and

for some $\mu > 0$. Then Theorem 5.2 reduces to Corollary 5.7.

Corollary 5.7. Let $p, q > 1$ with $1/p+1/q = 1$, and $\mathcal{X}:\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)\rightarrow\mathbb{R}$ be a partial differentiable function on $\Omega$ such that $\frac{\partial^{2}\mathcal{X}}{\partial {\mu} \partial {\lambda}}\in L_{1}(\Omega)$ and $\left|\frac{\partial^{2}\mathcal{X}}{\partial {\lambda} \partial {\mu}}\right|^{q}$ is a $2D$ reciprocal strong ${\rho}$-convex function of higher order. Then one has

where $\varphi_{1}(a, b, c, d:\Omega)$, $\varphi_{2}(a, b, c, d:\Omega)$, $\varphi_{3}(a, b, c, d:\Omega)$, $\varphi_{4}(a, b, c, d:\Omega)$ are given in Theorem 5.2, and

Ⅵ. If we take $\sigma = 2$, then Corollary 5.7 becomes Corollary 5.8.

Corollary 5.8. Let $p, q > 1$ with $1/p+1/q = 1$, and $\mathcal{X}:\Omega = [a, b]\times[c, d]\subseteq(0, \infty)\times(0, \infty)\rightarrow\mathbb{R}$ be a partial differentiable function on $\Omega$ such that $\frac{\partial^{2}\mathcal{X}}{\partial {\mu} \partial {\lambda}}\in L_{1}(\Omega)$ and $\left|\frac{\partial^{2}\mathcal{X}}{\partial {\lambda} \partial {\mu}}\right|^{q}$ is a $2D$ reciprocal strong ${\rho}$-convex function. Then

where $\varphi_{1}(a, b, c, d:\Omega)$, $\varphi_{2}(a, b, c, d:\Omega)$, $\varphi_{3}(a, b, c, d:\Omega)$, $\varphi_{4}(a, b, c, d:\Omega)$ are given in Theorem 5.2, and

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.

The research was supported by the National Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485, 11971142, 11871202).

Conflict of interest

The authors declare that they have no competing interests.