On Opial’s Type Integral Inequalities

Abstract

In the article we establish some new Opial’s type inequalities involving higher order partial derivatives. These new inequalities, in special cases, yield Agarwal-Pang’s, Pachpatte’s and Das’s type inequalities.

1. Introduction and Statement of Results

In 1960, Opial [1] established the following integral inequality:

Theorem 1.

Suppose $x\in C^1[0,a]$ satisfies $x\left(0\right)=x\left(a\right)=0$ and $x\left(t\right)>0$ for all $t\in (0,a).$ Then the integral inequality holds

$${\int }_0^a\left|x\left(t\right)x^{\prime }\left(t\right)\right|dt\le \frac{a}{4}{\int }_0^a{\left(x^{\prime }\left(t\right)\right)}^{2}dt,$$

(1)

where this constant $\frac{a}{4}$ is best possible.

The first natural extension of Opial’s inequality (1) involving the higher order derivatives $x^{\left(n\right)}\left(s\right)(n\ge 1)$ instead of $x^{\prime }\left(s\right)$ is embodied in the following:

Theorem 2 ([2]).

Let $x\left(t\right)\in C^{\left(n\right)}[0,a]$ be such that $x^{\left(i\right)}\left(0\right)=0,0\le i\le n-1(n\ge 1)$. Then the following inequality holds

$${\int }_0^a\left|x\left(t\right)x^{\left(n\right)}\left(t\right)\right|dt\le \frac{1}{2}{a}^n{\int }_0^a{\left|x^{\left(n\right)}\left(t\right)\right|}^{2}dt.$$

(2)

A sharp version of (2) is the following:

Theorem 3 ([3]).

Let $x\left(t\right)\in C^{(n-1)}[0,a]$ be such that $x^{\left(i\right)}\left(0\right)=0,0\le i\le n-1(n\ge 1)$. Further, let $x^{(n-1)}\left(t\right)$ be absolutely continuous, and ${\int }_0^a{\left|(x^{\left(n\right)}\left(t\right)\right|}^{2}dt<\infty .$ Then the following inequality holds

$${\int }_0^a\left|x\left(t\right)x^{\left(n\right)}\left(t\right)\right|dt\le \frac{1}{2n!}{\left(\frac{n}{2n-1}\right)}^{1/2}{a}^n{\int }_0^a{\left|x^{\left(n\right)}\left(t\right)\right|}^{2}dt.$$

(3)

A more general version of (3) was established in [4] as follows:

Theorem 4.

Let $r_k,0\le k\le n-1$ and ℓ be non-negative real numbers such that $\ell \sigma \ge 1$, where $\sigma ={\sum }_{k=0}^{n-1}{r}_k$, and let $p\left(t\right)$ be a non-negative continuous function on $[0,h]$. Further, let $x\left(t\right)\in C^{(n-1)}[0,h]$ be such that $x^{\left(i\right)}\left(0\right)=x^{\left(i\right)}\left(h\right)=0,0\le i\le n-1$, and let $x^{(n-1)}\left(t\right)$ be absolutely continuous. Then the following inequality holds

$${\int }_0^{h}p\left(t\right){\left({\left|{\displaystyle \prod _{\kappa =0}^n}{x}^{\left(\kappa \right)}\left(t\right)\right|}^{r_k}\right)}^{\ell }dt\le \frac{1}{2\sigma }\left({\int }_0^h{\left[t(h-t)\right]}^{(\ell \sigma -1)/2}p\left(t\right)dt\right){\displaystyle \sum _{k=0}^{n-1}}{r}_k{\left|x^{(k+1)}\left(t\right)\right|}^{\ell \sigma }dt.$$

(4)

Opial’s inequality and its generalizations, extensions and discretizations play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations [2,5,6,7,8]. The inequality (1) has received considerable attention and a large number of papers dealing with new proofs, extensions, generalizations, variants and discrete analogues of Opial’s inequality have appeared in the literature [1,3,4,7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. For an extensive survey on these inequalities, see [2,8].

The first aim of the present paper is to establish a new Opial’s type inequality involving higher order partial derivatives which is a generalization of inequality (4).

Theorem 5.

Let $r_{k_1,\dots ,k_n},0\le k_i\le n_i-1,1\le i\le n$ and ℓ be non-negative real numbers such that $\ell \sigma \ge 1$, where $\sigma ={\sum }_{k_1=0}^{n_1-1}\cdots {\sum }_{k_n=0}^{n_n-1}{r}_{k_1,\dots ,k_n}$, and let $p(s_1,\dots ,s_n)$ be a non-negative continuous function on $[0,a_1]\times \cdots \times [0,a_n]$. Let $x(s_1,\dots ,s_n)$ be a continuous function on $[0,a_1]\times \cdots \times [0,a_n]$ such that $\frac{{\partial }^{k_i}}{\partial s_i^{k_i}}x(s_1,\dots ,s_i,\dots ,s_n){|}_{s_i=0\mathit{and}{s}_i=a_i}=0,1\le i\le n$ and $0\le k_i\le n_i-1$. Further, let $\frac{{\partial }^{n_i}}{\partial s_i^{n_i}}\left(\frac{{\partial }^{n_i-1}}{\partial s_i^{n_i-1}}x(s_1,\dots ,s_i,\dots ,s_n)\right)$ and $\frac{{\partial }^{n_i-1}}{\partial s_i^{n_i-1}}\left(\frac{{\partial }^{n_i}}{\partial s_i^{n_i}}x(s_1,\dots ,s_i,\dots ,s_n)\right)$ be absolutely continuous on $[0,a_1]\times \cdots \times [0,a_n]$. Then

$$\begin{array}{c}{\int }_0^{a_1}\cdots {\int }_0^{a_n}p(s_1,\dots ,s_n){\left({\displaystyle \prod _{k_1=0}^{n_1-1}\cdots \prod _{k_n=0}^{n_n-1}}{\left|\frac{{\partial }^{k_1+\cdots +k_n}}{\partial s_1^{k_1}\cdots \partial s_n^{k_n}}x(s_1,\dots ,s_n)\right|}^{r_{k_1,\dots ,k_n}}\right)}^{\ell }d{s}_1\cdots d{s}_n\\ \le \frac{1}{2\sigma }\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left[{\displaystyle \prod _{i=1}^n}{s}_i(a_i-s_i)\right]}^{(\ell \sigma -1)/2}p(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n\right)\times \\ \times {\displaystyle \sum _{k_1=0}^{n_1-1}\cdots \sum _{k_n=0}^{n_n-1}}{r}_{k_1,\dots ,k_n}{\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{(k_1+1)+\cdots +(k_n+1)}}{\partial s_1^{k_1+1}\cdots \partial s_n^{k_n+1}}x(s_1,\dots ,s_n)\right|}^{\ell \sigma }d{s}_1\cdots d{s}_n.\end{array}$$

(5)

Remark 1.

Let $x(s_1,\dots ,s_n)$ and $p(s_1,\dots ,s_n)$ reduce to $x\left(t\right)$ and $p\left(t\right)$, respectively, and with suitable modifications, then (5) changes to (4). A special case of Theorem 5 was proved in [31].

A result involving two functions and their higher order derivatives is embodied in [18]:

Theorem 6.

For $j=1,2$ let $x_j\left(t\right)\in C^{n-1}[0,a]$ be such that $x_j^{\left(i\right)}\left(0\right)=0,0\le i\le n-1$. Further, let $x_j^{(n-1)}$ be absolutely continuous, and ${\int }_0^a{\left|x_j^{\left(n\right)}\left(t\right)\right|}^{2}dt<\infty$. Then

$${\int }_0^a\left\{\left|x_2\left(t\right)x_1^{\left(n\right)}\left(t\right)\right|+\left|x_1\left(t\right)x_2^{\left(n\right)}\left(t\right)\right|\right\}dt\le \frac{1}{2n!}{\left(\frac{n}{2n-1}\right)}^{1/2}{a}^n{\int }_0^a\left({\left|x_1^{\left(n\right)}\left(t\right)\right|}^2+{\left|x_2^{\left(n\right)}\left(t\right)\right|}^2\right)dt,$$

(6)

The second aim of the present paper is to establish a new Opial’s type inequality involving higher order partial derivatives which is a generalization of inequality (6).

Theorem 7.

For $j=1,2,$ let $x_j(s_1,\dots ,s_n)$ be a continuous function on $[0,a_1]\times \cdots \times [0,a_n]$ such that $\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_{i-1}+{\kappa }_{i+1}+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_{i-1}^{{\kappa }_{i-1}}\partial s_{i+1}^{{\kappa }_{i+1}}\cdots \partial s_1^{{\kappa }_n}}{x}_j(s_1,\dots ,s_{i-1},s_i,s_{i+1},\dots ,s_n){|}_{s_i=0\mathit{and}{s}_i=a_i}=0,0\le {\kappa }_i\le n_i-1$, and $i=1,\dots ,n$. Further, let $\frac{{\partial }^{n_i}}{\partial s_i^{n_i}}\left(\frac{{\partial }^{n_i-1}}{\partial s_i^{n_i-1}}{x}_j(s_1,\dots ,s_i,\dots ,s_n)\right)$ and $\frac{{\partial }^{n_i-1}}{\partial s_i^{n_i-1}}\left(\frac{{\partial }^{n_i}}{\partial s_i^{n_i}}{x}_j(s_1,\dots ,s_i,\dots ,s_n)\right)$ be absolutely continuous on $[0,a_1]\times \cdots \times [0,a_n]$, and ${\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_j(s_1,\dots ,s_n)\right|}^{2}d{s}_1\cdots d{s}_n$, exist. Then

$$\begin{array}{c}{\int }_0^{a_1}\cdots {\int }_0^{a_n}\{\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_1(s_1,\dots ,s_n)\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|\\ +\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_2(s_1,\dots ,s_n)\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|\}d{s}_1\cdots d{s}_n\\ \le \sqrt{2}D(n_1,\dots ,n_n,{\kappa }_1,\dots ,{\kappa }_n){\displaystyle \prod _{i=1}^n}{a}_i^{n_i-{\kappa }_i}{\int }_0^{a_1}\cdots {\int }_0^{a_n}\{{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|}^2\\ +{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|}^2\}d{s}_1\cdots d{s}_n,\end{array}$$

(7)

where

$$D(n_1,\dots ,n_n,{\kappa }_1,\dots ,{\kappa }_n)=\frac{1}{4{\prod }_{i=1}^n(n_i-{\kappa }_i)!}{\left(\frac{{\prod }_{i=1}^n(n_i-{\kappa }_i)}{{\prod }_{i=1}^n(2n_i-2{\kappa }_i-1)}\right)}^{1/2}.$$

Remark 2.

Taking for ${\kappa }_1=\cdots ={\kappa }_n=0$ and $n_1=\cdots =n_n=1$ in (7) and for $j=1,2$, let $x_j(s_1,\dots ,s_n)$ reduce to $x_j\left(t\right)$, respectively and with suitable modifications, then (7) changes to (6). A special case of Theorem 7 was proved in [31].

2. Proofs of Main Results

In order to prove Theorem 5, we need the following lemma.

Lemma 1.

Let $\lambda \ge 1$ be a real number, and let $p(s_1,\dots ,s_n)$ be a nonnegative and continuous functions on $[0,a_1]\times \cdots \times [0,a_n]$. Further, let $x(s_1,\dots ,s_n)$ be an absolutely continuous function on $[0,a_1]\times \cdots \times [0,a_n]$, with $x(s_1,\dots ,s_{i-1},0,s_{i+1},\dots ,s_n)=0,$ $x(s_1,\dots ,s_{i-1},a_i,s_{i+1},\dots ,s_n)=0$ and $i=1,\dots ,n$. Then

$$\begin{array}{c}{\int }_0^{a_1}\cdots {\int }_0^{a_n}p(s_1,\dots ,s_n){\left|x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n\\ \le \frac{1}{2}\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left[{\displaystyle \prod _{i=1}^n}{s}_i(a_i-s_i)\right]}^{(\lambda -1)/2}p(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n\right)\\ \times {\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^n}{\partial s_1\dots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }dsdt.\end{array}$$

(8)

Proof. 

From the hypotheses, we have

$$x(s_1,\dots ,s_n)={\int }_0^{s_1}\cdots {\int }_0^{s_n}\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n.$$

By Hölder’s inequality with indices $\lambda$ and $\lambda /(\lambda -1)$, it follows that

$$\begin{array}{c}|x(s_1,\dots ,s_n){|}^{\lambda /2}\le {\left[{\left({\int }_0^{s_1}\cdots {\int }_0^{s_n}\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|d{s}_1\cdots d{s}_n\right)}^{\lambda }\right]}^{1/2}\\ \le {\left({\displaystyle \prod _{i=1}^n}{s}_i\right)}^{(\lambda -1)/2}{\left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n\right)}^{1/2}.\end{array}$$

(9)

Similarly

$$|x(s_1,\dots ,s_n){|}^{\lambda /2}\le {\left({\displaystyle \prod _{i=1}^n}(a_i-s_i)\right)}^{(\lambda -1)/2}{\left({\int }_{s_1}^{a_1}\cdots {\int }_{s_n}^{a_n}{\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n\right)}^{1/2}.$$

(10)

Now a multiplication of (9) and (10), and by the elementary inequality $2\sqrt{\alpha \beta }\le \alpha +\beta ,\alpha \ge 0,\beta \ge 0$ gives

$$\begin{array}{c}|x(s_1,\dots ,s_n){|}^{\lambda }\le {\left[{\displaystyle \prod _{i=1}^n}{s}_i(a_i-s_i)\right]}^{(\lambda -1)/2}{\left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n\right)}^{1/2}\\ \times {\left({\int }_{s_1}^{a_1}\cdots {\int }_{s_n}^{a_n}{\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n\right)}^{1/2}\\ \le \frac{1}{2}{\left[{\displaystyle \prod _{i=1}^n}{s}_i(a_i-s_i)\right]}^{(\lambda -1)/2}\{{\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n\\ +{\int }_{s_1}^{a_1}\cdots {\int }_{s_n}^{a_n}{\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n\}\\ =\frac{1}{2}{\left[{\displaystyle \prod _{i=1}^n}{s}_i(a_i-s_i)\right]}^{(\lambda -1)/2}{\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n.\end{array}$$

(11)

Multiplying the both sides of (11) by $p(s_1,\dots ,s_n)$ and integrating both sides over $s_i$ from 0 to $a_i,i=1,\dots ,n$ respectively, we obtain

$$\begin{array}{c}{\int }_0^{a_1}\cdots {\int }_0^{a_n}p(s_1,\dots ,s_n){\left|x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n\\ \le \frac{1}{2}{\int }_0^{a_1}\cdots {\int }_0^{a_n}({\left[{\displaystyle \prod _{i=1}^n}{s}_i(a_i-s_i)\right]}^{(\lambda -1)/2}p(s_1,\dots ,s_n)\times \\ \times {\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n)d{s}_1\cdots d{s}_n\\ =\frac{1}{2}\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left[{\displaystyle \prod _{i=1}^n}{s}_i(a_i-s_i)\right]}^{(\lambda -1)/2}p(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n\right)\\ \times {\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}x(s_1,\dots ,s_n)\right|}^{\lambda }d{s}_1\cdots d{s}_n.\end{array}$$

 □

Proof of Theorem 5.

We recall that for the real numbers ${\alpha }_{k_1,\dots ,k_n},r_{k_1,\dots ,k_n}\ge 0,0\le k_i\le n_i-1,1\le i\le n$, and any $\ell \ge 1$, the following elementary inequality holds

$${\displaystyle \prod _{k_1=0}^{n_1-1}\cdots \prod _{k_n=0}^{n_n-1}}{\alpha }_{k_1,\dots ,k_n}^{r_{k_1,\dots ,k_n}}\le {\displaystyle \sum _{k_1=0}^{n_1-1}\cdots \sum _{k_n=0}^{n_n-1}}{r}_{k_1,\dots ,k_n}{\alpha }_{k_1,\dots ,k_n}\le {\left(\sum _{k_1=0}^{n_1-1}\cdots \sum _{k_n=0}^{n_n-1}{r}_{k_1,\dots ,k_n}{\alpha }_{k_1,\dots ,k_n}^{\ell }\right)}^{1/\ell }.$$

(12)

From the inequality (12), we have

$$\begin{array}{c}{\left({\displaystyle \prod _{k_1=0}^{n_1-1}\cdots \prod _{k_n=0}^{n_n-1}}{\left|\frac{{\partial }^{k_1+\cdots +k_n}}{\partial s_1^{k_1}\cdots \partial s_n^{k_n}}x(s_1,\dots ,s_n)\right|}^{r_{k_1,\dots ,k_n}}\right)}^{\ell }\\ ={\left({\displaystyle \prod _{k_1=0}^{n_1-1}\cdots \prod _{k_n=0}^{n_n-1}}{\left|\frac{{\partial }^{k_1+\cdots +k_n}}{\partial s_1^{k_1}\cdots \partial s_n^{k_n}}x(s_1,\dots ,s_n)\right|}^{r_{k_1,\dots ,k_n}/\sigma }\right)}^{\ell \sigma }\\ \le {\left({\displaystyle \sum _{k_1=0}^{n_1-1}\cdots \sum _{k_n=0}^{n_n-1}}\frac{r_{k_1,\dots ,k_n}}{\sigma }\left|\frac{{\partial }^{k_1+\cdots +k_n}}{\partial s_1^{k_1}\cdots \partial s_n^{k_n}}x(s_1,\dots ,s_n)\right|\right)}^{\ell \sigma }\\ \le {\displaystyle \sum _{k_1=0}^{n_1-1}\cdots \sum _{k_n=0}^{n_n-1}}\frac{r_{k_1,\dots ,k_n}}{\sigma }{\left|\frac{{\partial }^{k_1+\cdots +k_n}}{\partial s_1^{k_1}\cdots \partial s_n^{k_n}}x(s_1,\dots ,s_n)\right|}^{\ell \sigma }.\end{array}$$

(13)

Multiplying (13) by $p(s_1,\dots ,s_n)$, integrating the two sides of (13) over $s_i$ from 0 to $a_i,i,\dots ,n$, respectively, and then applying the Lemma 1 to the right side again, we observe

$$\begin{array}{c}{\int }_0^{a_1}\cdots {\int }_0^{a_n}p(s_1,\dots ,s_n){\left({\displaystyle \prod _{k_1=0}^{n_1-1}\cdots \prod _{k_n=0}^{n_n-1}}{\left|\frac{{\partial }^{k_1+\cdots +k_n}}{\partial s_1^{k_1}\cdots \partial s_n^{k_n}}x(s_1,\dots ,s_n)\right|}^{r_{k_1,\dots ,k_n}}\right)}^{\ell }d{s}_1\cdots d{s}_n\\ \le \frac{1}{\sigma }{\displaystyle \sum _{k_1=0}^{n_1-1}\cdots \sum _{k_n=0}^{n_n-1}}{r}_{k_1,\dots ,k_n}{\int }_0^{a_1}\cdots {\int }_0^{a_n}p(s_1,\dots ,s_n){\left|\frac{{\partial }^{k_1+\cdots +k_n}}{\partial s_1^{k_1}\cdots \partial s_n^{k_n}}x(s_1,\dots ,s_n)\right|}^{\ell \sigma }d{s}_1\cdots d{s}_n\\ \le \frac{1}{\sigma }{\displaystyle \sum _{k_1=0}^{n_1-1}\cdots \sum _{k_n=0}^{n_n-1}}\frac{r_{k_1,\dots ,k_n}}{2}\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left[{\displaystyle \prod _{i=1}^n}{s}_i(a_i-s_i)\right]}^{(\ell \sigma -1)/2}p(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n\right)\times \\ \times {\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{(k_1+1)+\cdots +(k_n+1)}}{\partial s_1^{k_1+1}\cdots \partial s_n^{k_n+1}}x(s_1,\dots ,s_n)\right|}^{\ell \sigma }d{s}_1\cdots d{s}_n\\ =\frac{1}{2\sigma }\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left[{\displaystyle \prod _{i=1}^n}{s}_i(a_i-s_i)\right]}^{(\ell \sigma -1)/2}p(s_1,\dots ,s_n)d{s}_1\cdots d{s}_n\right)\times \\ \times {\displaystyle \sum _{k_1=0}^{n_1-1}\cdots \sum _{k_n=0}^{n_n-1}}{r}_{k_1,\dots ,k_n}{\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{(k_1+1)+\cdots +(k_n+1)}}{\partial s_1^{k_1+1}\cdots \partial s_n^{k_n+1}}x(s_1,\dots ,s_n)\right|}^{\ell \sigma }d{s}_1\cdots d{s}_n.\end{array}$$

This completes the proof of Theorem 5.

Proof of Theorem 7.

From the hypotheses of the Theorem 7, we have for $0\le {\kappa }_i\le n_i-1,i=1,\dots ,n$

$$\begin{array}{c}\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_1(s_1,\dots ,s_n)\right|\le \frac{1}{{\prod }_{i=1}^n(n_i-{\kappa }_i-1)!}\\ \times {\int }_0^{s_1}\cdots {\int }_0^{s_n}{\displaystyle \prod _{i=1}^n}{(s_i-{\sigma }_i)}^{n_i-{\kappa }_i-1}\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_1({\sigma }_1,\dots ,{\sigma }_n)\right|d{\sigma }_1\cdots d{\sigma }_n.\end{array}$$

(14)

Multiplying (14) by $\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|$ and using Cauchy-Schwarz inequality, we obtain

$$\begin{array}{c}\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_1(s_1,\dots ,s_n)\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|\\ \le \frac{1}{{\prod }_{i=1}^n(n_i-{\kappa }_i-1)!}\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|{\left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\displaystyle \prod _{i=1}^n}{(s_i-{\sigma }_i)}^{2(n_i-{\kappa }_i-1)}d{\sigma }_1\cdots d{\sigma }_n\right)}^{1/2}\\ \times {\left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_1({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)}^{1/2}\\ =\frac{{\prod }_{i=1}^n{s}_i^{n_i-{\kappa }_i-1/2}}{{\prod }_{i=1}^n(n_i-{\kappa }_i-1)!{\left[(2n_i-2{\kappa }_i-1)\right]}^{1/2}}\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|\\ \times {\left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_1({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)}^{1/2}.\end{array}$$

(15)

Integrating the two sides of (15) over $s_i$ from 0 to $a_i,i=1,\dots ,n$, respectively and then applying the Cauchy-Schwarz inequality to the right side again, we observe

$$\begin{array}{c}{\int }_0^{a_1}\cdots {\int }_0^{a_n}\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_1(s_1,\dots ,s_n)\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|d{s}_1\cdots d{s}_n\\ \le \frac{1}{{\prod }_{i=1}^n(n_i-{\kappa }_i-1)!{\left[(2n_i-2{\kappa }_i-1)\right]}^{1/2}}{\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\displaystyle \prod _{i=1}^n}{s}_i^{2n_i-2{\kappa }_i-1}d{s}_1\cdots d{s}_n\right)}^{1/2}\\ \times \{{\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|}^2\\ \times \left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_1({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)d{s}_1\cdots d{s}_n{\}}^{1/2}.\end{array}$$

(16)

Similarly

$$\begin{array}{c}{\int }_0^{a_1}\cdots {\int }_0^{a_n}\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_2(s_1,\dots ,s_n)\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|d{s}_1\cdots d{s}_n\\ \le \frac{1}{{\prod }_{i=1}^n(n_i-{\kappa }_i-1)!{\left[(2n_i-2{\kappa }_i-1)\right]}^{1/2}}{\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\displaystyle \prod _{i=1}^n}{s}_i^{2n_i-2{\kappa }_i-1}d{s}_1\cdots d{s}_n\right)}^{1/2}\\ \times \{{\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|}^2\\ \times \left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_2({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)d{s}_1\cdots d{s}_n{\}}^{1/2}.\end{array}$$

(17)

Taking the sum of the two sides of (16) and (17), and in view of the elementary inequality ${\alpha }^{1/2}+{\beta }^{1/2}\le {\left[2(\alpha +\beta )\right]}^{1/2},\alpha \ge 0,\beta \ge 0,$ we have

$$\begin{array}{c}{\int }_0^{a_1}\cdots {\int }_0^{a_n}\{\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_1(s_1,\dots ,s_n)\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|\\ +\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_2(s_1,\dots ,s_n)\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|\}d{s}_1\cdots d{s}_n\\ \le \frac{1}{{\prod }_{i=1}^n(n_i-{\kappa }_i-1)!{\left[(2n_i-2{\kappa }_i-1)\right]}^{1/2}}{\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\displaystyle \prod _{i=1}^n}{s}_i^{2n_i-2{\kappa }_i-1}d{s}_1\cdots d{s}_n\right)}^{1/2}\\ \times \{2{\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|}^2\\ \times \left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_1({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)d{s}_1\cdots d{s}_n\\ +2{\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|}^2\\ \times \left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_2({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)d{s}_1\cdots d{s}_n{\}}^{1/2}.\end{array}$$

(18)

On the other hand, note the derivation rule of integral upper bound function and the derivation rule of product function, we have

$$\begin{array}{c}\frac{{\partial }^n}{\partial s_1\cdots \partial s_n}[\left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_1({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)\\ \times \left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_2({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)]\\ ={\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|}^2\left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_2({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)\\ +{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|}^2\left({\int }_0^{s_1}\cdots {\int }_0^{s_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial {\sigma }_1^{n_1}\cdots \partial {\sigma }_n^{n_n}}{x}_1({\sigma }_1,\dots ,{\sigma }_n)\right|}^{2}d{\sigma }_1\cdots d{\sigma }_n\right)\end{array}$$

(19)

and

$$\begin{array}{l}{\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\displaystyle \prod _{i=1}^n}{s}_i^{2n_i-2{\kappa }_i-1}d{s}_1\cdots d{s}_n\right)}^{1/2}\\ ={\left({\int }_0^{a_1}{s}_1^{2n_1-2k_1-1}d{s}_1·{\int }_0^{a_2}{s}_2^{2n_2-2k_2-1}d{s}_2·\cdots ·{\int }_0^{a_n}{s}_n^{2n_n-2k_n-1}d{s}_n\right)}^{1/2}\\ =\frac{{\prod }_{i=1}^n{a}_i^{n_i-{\kappa }_i}}{2{\left[{\prod }_{i=1}^n(n_i-{\kappa }_i)\right]}^{1/2}}.\end{array}$$

(20)

From (18)–(20) and in view the elementary inequality ${\left(\alpha \beta \right)}^{1/2}\le \frac{1}{2}(\alpha +\beta ),\alpha \ge 0,\beta \ge 0$, we have

$$\begin{array}{c}{\int }_0^{a_1}\cdots {\int }_0^{a_n}\{\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_1(s_1,\dots ,s_n)\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|\\ +\left|\frac{{\partial }^{{\kappa }_1+\cdots +{\kappa }_n}}{\partial s_1^{{\kappa }_1}\cdots \partial s_n^{{\kappa }_n}}{x}_2(s_1,\dots ,s_n)\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|\}d{s}_1\cdots d{s}_n\\ \le 2D(n_1,\dots ,n_n,{\kappa }_1,\dots ,{\kappa }_n){\displaystyle \prod _{i=1}^n}{a}_i^{n_i-{\kappa }_i}[2\left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|}^{2}d{s}_1\cdots d{s}_n\right)\\ \times \left({\int }_0^{a_1}\cdots {\int }_0^{a_n}{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|}^{2}d{s}_1\cdots d{s}_n\right){]}^{1/2}\\ \le \sqrt{2}D(n_1,\dots ,n_n,{\kappa }_1,\dots ,{\kappa }_n){\displaystyle \prod _{i=1}^n}{a}_i^{n_i-{\kappa }_i}{\int }_0^{a_1}\cdots {\int }_0^{a_n}\{{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_1(s_1,\dots ,s_n)\right|}^2\\ +{\left|\frac{{\partial }^{n_1+\cdots +n_n}}{\partial s_1^{n_1}\cdots \partial s_n^{n_n}}{x}_2(s_1,\dots ,s_n)\right|}^2\}d{s}_1\cdots d{s}_n,\end{array}$$

where

$$D(n_1,\dots ,n_n,{\kappa }_1,\dots ,{\kappa }_n)=\frac{1}{4{\prod }_{i=1}^n(n_i-{\kappa }_i)!}{\left(\frac{{\prod }_{i=1}^n(n_i-{\kappa }_i)}{{\prod }_{i=1}^n(2n_i-2{\kappa }_i-1)}\right)}^{1/2},i-1,\dots ,n.$$