On Gardner–Hartenstine’s inequality

Abstract

In the paper, we give a new generalization of Gardner–Hartenstine inequality and establish its integral form. As applications, we combine an important inequality and give some broader improvements.

1 Introduction

In [1], Gardner and Hartenstine established an interesting inequality. This inequality is crucial in their proof (as it was in [2]).

Theorem A

For $x_{0}, y_{0}>0$ and reals $x_{i}$, $y _{i}$, $i = 1,\ldots ,n$, we have

$$ \frac{ (\sum_{i=1}^{n}(x_{i}+y_{i})^{2} )^{(n-1)/2}}{(x _{0}+y_{0})^{n-2}} \leq \frac{ (\sum_{i=1}^{n}x_{i}^{2} ) ^{(n-1)/2}}{x_{0}^{n-2}}+ \frac{ (\sum_{i=1}^{n}y_{i}^{2} ) ^{(n-1)/2}}{y_{0}^{n-2}}, $$

(1.1)

with equality if and only if either $x_{i}=y_{i}=0$ for $i=1,2,\ldots ,n$ or $x_{i}=\alpha y_{i}$ for $i=0,1,\ldots ,n$, and some $\alpha >0$.

The first aim of this paper is to give a new generalization of the Gardner–Hartenstine inequality (1.1). Our result is given in the following theorem.

Theorem 1.1

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $x_{00}, y_{00}>0$ and reals $x_{ij}$, $y_{ij}$, $i=1,2,\ldots ,n$, $j=1,2,\ldots ,m$, then

$$ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}(x_{ij}+y_{ij})^{r} ) ^{1/r}}{(x_{00}+y_{00})^{1/q} } \biggr)^{p}\leq \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} )^{1/r}}{x_{00}^{1/q}} \biggr) ^{p}+ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} ) ^{1/r}}{y_{00}^{1/q}} \biggr)^{p}, $$

(1.2)

with equality if and only if either $x_{ij}=y_{ij}=0$ for $i=1,\ldots ,n$, $j=1,\ldots ,m$ or $x_{ij}=\alpha y_{ij}$ for $i=0,1,\ldots ,n$, $j=0,1, \ldots ,m$, and some $\alpha >0$.

Remark 1.2

Let $x_{ij}$ and $y_{ij}$ change $x_{i}$ and $y_{i}$, respectively, with appropriate transformation, and putting $m=1$, $r=2$, $p=n-1$, and $q=(n-1)/(n-2)$ in (1.2), (1.2) becomes (1.1).

Another aim of this paper is to give an integral form of (1.2). Our result is given in the following theorem.

Theorem 1.3

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $u(x,y), v(x,y)>0$ and $f(x,y)$, $g(x,y)$ are continuous functions on $[a,b]\times [c,d]$, then

$$ \begin{aligned}[b] &\biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}(f(x,y)+g(x,y))^{r}\,dx \,dy ) ^{1/r}}{(u(x,y)+v(x,y))^{1/q} } \biggr)^{p}\\ &\quad \leq \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy )^{1/r}}{u(x,y)^{1/q}} \biggr) ^{p}+ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{1/r}}{v(x,y)^{1/q}} \biggr)^{p}, \end{aligned} $$

(1.3)

with equality if and only if either $(\|f(x,y)\|_{r}^{r},\| g(x,y) \|_{r}^{r} )=\alpha (\|u(x,y)\|_{r}^{r},\|v(x,y)\|_{r}^{r} )$ for some $\alpha >0$ or $\|f(x,y)\|_{r}^{r}=\|g(x,y)\|_{r}^{r}=0$.

Let $f(x,y)$ and $g(x,y)$ change $f(x)$ and $g(x)$, respectively, with appropriate transformation, and putting $r=2$, $p=n-1$ and $q=(n-1)/(n-2)$ in (1.3), (1.3) becomes the following result.

Corollary 1.4

If $u(x), v(x)>0$ and $f(x)$, $g(x)$ are continuous functions on $[a,b]$, then

$$ \frac{ (\int _{a}^{b}(f(x)+g(x))^{2}\,dx )^{(n-1)/2}}{(u(x)+v(x))^{n-2} }\leq \frac{ (\int _{a}^{b}f(x)^{2}\,dx )^{(n-1)/2}}{u(x)^{n-2}}+ \frac{ (\int _{a}^{b}g(x)^{2}\,dx )^{(n-1)/2}}{v(x)^{n-2}}, $$

(1.4)

with equality if and only if either $\|f(x)\|_{r}^{r}=\|g(x)\|_{r} ^{r}=0$ or $(\|f(x)\|_{r}^{r},\| g(x)\|_{r}^{r} )=\alpha (\|u(x)\|_{r}^{r}, \|v(x)\|_{r}^{r} )$ for some $\alpha >0$.

This is just an integral form of (1.1) established by Gardner and Hartenstine [1].

As applications, we combine another important inequality and give some broader improvements. Our results are given in the following theorems.

Theorem 1.5

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $x_{00}, y_{00}, a_{00}, b_{00}>0$ and reals $x_{ij}$, $y_{ij}$, $a_{ij}$, $b_{ij}$, $i=1,2,\ldots ,n$, $j=1,2,\ldots ,m$, then

$$\begin{aligned}& \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}[(x_{ij}+y_{ij})^{r}+(a_{ij}+b _{ij})^{r}] )^{p}}{ [ (x_{00}+y_{00} )^{r}+ (a_{00}+b_{00} )^{r} ]^{p/q}} \\& \quad \leq \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} )^{p/r}}{x_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} )^{p/r}}{y_{00} ^{p/q}} \biggr)^{r} \\& \qquad {}+ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}a_{ij}^{r} ) ^{p/r}}{a_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}b_{ij} ^{r} )^{p/r}}{b_{00}^{p/q}} \biggr)^{r} \end{aligned}$$

(1.5)

with equality if and only if either $x_{ij}=y_{ij}=0$ and $a_{ij}=b _{ij}=0$ for $i=1,\ldots ,n$ and $j=1,\ldots ,m$ or $x_{ij}=\alpha y _{ij}$ and $a_{ij}=\beta b_{ij}$ for $i=0,1,\ldots ,n$ and $j=0,1, \ldots ,m$ and some $\alpha , \beta >0$, and

$$\begin{aligned}& \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} )^{p/r}}{x _{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} ) ^{p/r}}{y_{00}^{p/q}} \biggr): \biggl( \frac{ (\sum_{j=1}^{m}\sum_{i=1} ^{n}a_{ij}^{r} )^{p/r}}{a_{00}^{p/q}}+ \frac{ (\sum_{j=1} ^{m}\sum_{i=1}^{n}b_{ij}^{r} )^{p/r}}{b_{00}^{p/q}} \biggr) \\& \quad =(x_{00}+y_{00}):(a_{00}+b_{00}). \end{aligned}$$

Theorem 1.6

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $u(x,y), v(x,y), u'(x,y), v'(x,y)>0$ and $f(x,y)$, $g(x,y)$, $f'(x,y)$, $g'(x,y)$ are continuous functions on $[a,b]\times [c,d]$, then

$$\begin{aligned}& \frac{ (\int _{a}^{b}\int _{c}^{d}[(f(x,y)+g(x,y))^{r}+(f'(x,y)+g'(x,y))^{r}]\,dx \,dy ) ^{p}}{ [ (u(x,y)+v(x,y) )^{r}+ (u'(x,y)+v'(x,y) ) ^{r} ]^{p/q}} \\& \quad \leq \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d} f(x,y)^{r}\,dx \,dy ) ^{p/r}}{u(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{p/r}}{v(x,y)^{p/q}} \biggr)^{r} \\& \qquad {} + \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f'(x,y)^{r}\,dx \,dy ) ^{p/r}}{u'(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g'(x,y)^{r}\,dx \,dy ) ^{p/r}}{v'(x,y)^{p/q}} \biggr)^{r} \end{aligned}$$

(1.6)

with equality if and only if either $f(x,y)=g(x,y)=0$ and $f'(x,y)=g'(x,y)=0$ or $(f(x,y), g(x,y))=\alpha (u(x,y),v(x,y))$ and $(f'(x,y),g'(x,y))=\beta (u'(x,y),v'(x,y))$ and some $\alpha , \beta >0$, and

$$\begin{aligned}& \biggl( \frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy )^{p/r}}{u(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy )^{p/r}}{v(x,y)^{p/q}} \biggr) \\& \qquad{} : \biggl( \frac{ (\int _{a}^{b}\int _{c}^{d}f'(x,y)^{r}\,dx \,dy ) ^{p/r}}{u'(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g'(x,y)^{r} ) ^{p/r}}{v'(x,y)^{p/q}} \biggr) \\& \quad =\bigl(u(x,y)+v(x,y)\bigr):\bigl(u'(x,y)+v'(x,y) \bigr). \end{aligned}$$

2 Generalizations

Our main results are given in the following theorems.

Theorem 2.1

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $x_{00}, y_{00}>0$ and reals $x_{ij}$, $y_{ij}$, $i=1,2,\ldots ,n$, $j=1,2,\ldots ,m$, then

$$ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}(x_{ij}+y_{ij})^{r} ) ^{1/r}}{(x_{00}+y_{00})^{1/q} } \biggr)^{p}\leq \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} )^{1/r}}{x_{00}^{1/q}} \biggr) ^{p}+ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} ) ^{1/r}}{y_{00}^{1/q}} \biggr)^{p}, $$

(2.1)

with equality if and only if either $x_{ij}=y_{ij}=0$ for $i=1,\ldots ,n$, $j=1,\ldots ,m$ or $x_{ij}=\alpha y_{ij}$ for $i=0,1,\ldots ,n$, $j=0,1, \ldots ,m$, and some $\alpha >0$.

Proof

From Minkowski’s and Hölder’s inequalities, we obtain

$$\begin{aligned} \Biggl(\sum_{j=1}^{m}\sum _{i=1}^{n}(x_{ij}+y_{ij})^{r} \Biggr)^{1/r} \leq & \Biggl(\sum_{j=1}^{m} \sum_{i=1}^{n}x_{ij}^{r} \Biggr)^{1/r}+ \Biggl(\sum_{j=1}^{m} \sum_{i=1}^{n}y_{ij}^{r} \Biggr)^{1/r} \\ =& \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} ) ^{1/r}}{x_{00}^{1/q}} \biggr)x_{00}^{1/q} + \biggl( \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} )^{1/r}}{y_{00}^{1/q}} \biggr)y _{00}^{1/q} \\ \leq & \biggl\{ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij} ^{r} )^{1/r}}{x_{0}^{1/q}} \biggr)^{p}+ \biggl( \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} )^{1/r}}{y_{00}^{1/q}} \biggr) ^{p} \biggr\} ^{1/p} \\ &{}\times \bigl(\bigl(x_{00}^{1/q}\bigr)^{q}+ \bigl(y_{00}^{1/q}\bigr)^{q} \bigr)^{1/q} \\ =& \biggl\{ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} ) ^{p/r}}{x_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ijj} ^{r} )^{p/r}}{y_{00}^{p/q}} \biggr\} ^{1/p} (x_{00}+y_{00} ) ^{1/q}. \end{aligned}$$

Rearranging, (2.1) follows.

The following is a discussion of the conditions for this equal sign to hold. Suppose that equality holds in (2.1). Then equality holds in Minkowski’s inequality, which implies that $x_{ij}=\alpha y_{ij}$ for $i=1,\ldots ,n$ and $j=1,\ldots ,m$ and some $\alpha \geq 0$. Equality also holds in Hölder’s inequality, implying that there are constants β and γ with $\beta ^{2}+\gamma ^{2}>0$ such that

$$ \beta \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} ) ^{1/r}}{x_{00}^{1/q}} \biggr)^{p} =\gamma \bigl(x_{00}^{1/q}\bigr)^{q}, $$

or equivalently

$$ \beta \Biggl(\sum_{j=1}^{m}\sum _{i=1}^{n}x_{ij}^{r} \Biggr)^{p/r}= \gamma x_{00}^{p}, $$

and the same equation with $y_{ij}$ instead of $x_{ij}$, $i=0,1,\ldots ,n$; $j=0,1,\ldots ,m$. Therefore

$$\begin{aligned} \gamma x_{00}^{p} =&\beta \Biggl(\sum _{j=1}^{m}\sum_{i=1}^{n}x_{ij} ^{r} \Biggr)^{p/r} \\ =&\beta \Biggl(\sum_{j=1}^{m}\sum _{i=1}^{n}(\alpha y_{ij})^{r} \Biggr) ^{p/r} \\ =&\gamma (\alpha y_{00})^{p}. \end{aligned}$$

Obviously, if $\gamma =0$, then $x_{ij}=y_{ij}=0$ for $i=1,\ldots ,n$; $j=1,\ldots ,m$. If $\gamma \neq 0$, then $\alpha >0$ and $x_{ij}= \alpha y_{ij}$ for $i=0,1,\ldots ,n$ and $j=0,1,\ldots ,m$.

This proof is complete. □

Let $x_{ij}$ become $x_{i}$ with appropriate transformation, and $m=1$, (2.2) reduces to the following result.

Corollary 2.2

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $x_{0}, y_{0}>0$ and reals $x_{i}$, $y_{i}$, $i=1,2,\ldots ,n$, then

$$ \biggl(\frac{ (\sum_{i=1}^{n}(x_{i}+y_{i})^{r} )^{1/r}}{(x _{0}+y_{0})^{1/q} } \biggr)^{p}\leq \biggl(\frac{ (\sum_{i=1} ^{n}x_{i}^{r} )^{1/r}}{x_{0}^{1/q}} \biggr)^{p}+ \biggl(\frac{ (\sum_{i=1}^{n}y_{i}^{r} )^{1/r}}{y_{0}^{1/q}} \biggr) ^{p}, $$

with equality if and only if either $x_{i}=y_{i}=0$ for $i=1,\ldots ,n$ or $x_{i}=\alpha y_{i}$ for $i=0,1,\ldots ,n$, for some $\alpha >0$.

Theorem 2.3

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $u(x,y), v(x,y)>0$ and $f(x,y)$, $g(x,y)$ are continuous functions on $[a,b]\times [c,d]$, then

$$ \begin{aligned}[b] &\biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}(f(x,y)+g(x,y))^{r}\,dx \,dy ) ^{1/r}}{(u(x,y)+v(x,y))^{1/q} } \biggr)^{p}\\ &\quad \leq \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy )^{1/r}}{u(x,y)^{1/q}} \biggr) ^{p}+ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{1/r}}{v(x,y)^{1/q}} \biggr)^{p}, \end{aligned} $$

(2.2)

with equality if and only if either $(\|f(x,y)\|_{r}^{r},\| g(x,y) \|_{r}^{r} )=\alpha (\|u(x,y)\|_{r}^{r},\|v(x,y)\|_{r}^{r} )$ for some $\alpha >0$ or $\|f(x,y)\|_{r}^{r}=\|g(x,y)\|_{r}^{r}=0$.

Proof

From Minkowski’s and Hölder’s integral inequalities, we obtain

$$\begin{aligned}& \biggl( \int _{a}^{b} \int _{c}^{d}\bigl(f(x,y)+g(x,y) \bigr)^{r}\,dx \,dy \biggr)^{1/r} \\& \quad \leq \biggl( \int _{a}^{b} \int _{c}^{d}f(x,y)^{r}\,dx \,dy \biggr)^{1/r}+ \biggl( \int _{a}^{b} \int _{c}^{d}g(x,y)^{r}\,dx \,dy \biggr)^{1/r} \\& \quad = \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy ) ^{1/r}}{u(x,y)^{1/q}} \biggr)u(x,y)^{1/q} \\& \qquad {}+ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r} \,dx \,dy ) ^{1/r}}{v(x,y)^{1/q}} \biggr)v(x,y)^{1/q} \\& \quad \leq \biggl\{ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy ) ^{1/r}}{u(x,y)^{1/q}} \biggr)^{p} \\& \qquad {}+ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{1/r}}{v(x,y)^{1/q}} \biggr)^{p} \biggr\} ^{1/p} \\& \qquad {}\times \bigl(\bigl(u(x,y) ^{1/q}\bigr)^{q}+ \bigl(v(x,y)^{1/q}\bigr)^{q} \bigr)^{1/q} \\& \quad = \biggl\{ \frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy ) ^{p/r}}{u(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{p/r}}{v(x,y)^{p/q}} \biggr\} ^{1/p} \\& \qquad {}\times \bigl(u(x,y)+v(x,y) \bigr)^{1/q}. \end{aligned}$$

Rearranging, (2.2) follows.

The following is a discussion of the conditions for this equal sign to hold. Suppose that equality holds in (2.2). Then equality holds in Minkowski’s inequality, which implies that $f(x,y)=\alpha g(x,y)$ and some $\alpha \geq 0$. Equality also holds in Hölder’s inequality, implying that there are constants β and γ with $\beta ^{2}+\gamma ^{2}>0$ such that

$$ \beta \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx ) ^{1/r}}{u(x,y)^{1/q}} \biggr)^{p}=\gamma \bigl(u(x,y)^{1/q}\bigr)^{q}, $$

or equivalently

$$ \beta \biggl( \int _{a}^{b} \int _{c}^{d}f(x,y)^{r}\,dx \,dy \biggr)^{p/r}= \gamma u(x,y)^{p}, $$

and the same equation with $g(x,y)$ instead of $f(x,y)$. Therefore

$$\begin{aligned} \gamma u(x,y)^{p} =&\beta \biggl( \int _{a}^{b} \int _{c}^{d}f(x,y)^{r}\,dx \,dy \biggr) ^{p/r} \\ =&\beta \biggl( \int _{a}^{b} \int _{c}^{d}\bigl(\alpha g(x,y) \bigr)^{r}\,dx \,dy \biggr) ^{p/r} \\ =&\gamma \bigl(\alpha v(x,y)\bigr)^{p}. \end{aligned}$$

Obviously, if $\gamma =0$, then $\|f(x,y)\|_{r}^{r}=\|g(x,y)\|_{r} ^{r}=0$. If $\gamma \neq 0$, then $\alpha >0$ and $(\|f(x,y)\| _{r}^{r}, \| g(x,y)\|_{r}^{r} )=\alpha (\|u(x,y)\|_{r}^{r},\|v(x,y) \|_{r}^{r} )$.

This proof is complete. □

Let $f(x,y)$ become $f(x)$ with appropriate transformation, (2.2) reduces to the following result.

Corollary 2.4

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $u(x), v(x)>0$ and $f(x)$, $g(x)$ are continuous functions on $[a,b]$, then

$$ \biggl(\frac{ (\int _{a}^{b}(f(x)+g(x))^{r}\,dx )^{1/r}}{(u(x)+v(x))^{1/q} } \biggr)^{p}\leq \biggl(\frac{ (\int _{a}^{b}f(x)^{r}\,dx ) ^{1/r}}{u(x)^{1/q}} \biggr)^{p}+ \biggl(\frac{ (\int _{a}^{b}g(x)^{r}\,dx ) ^{1/r}}{v(x)^{1/q}} \biggr)^{p}, $$

(2.3)

with equality if and only if either $\|f(x)\|_{r}^{r}=\|g(x)\|_{r} ^{r}=0$ or $(\|f(x)\|_{r}^{r},\| g(x)\|_{r}^{r} )=\alpha (\|u(x)\|_{r}^{r}, \| v(x)\|_{r}^{r} )$ for some $\alpha >0$.

3 Improvements

We need the following lemma to prove our main results.

Lemma 3.1

([3] p.39)

If $a_{i}\geq 0$, $b_{i}>0$, $i=1, \ldots ,m$, and $\sum_{i=1}^{m}\alpha _{i}=1$, then

$$ \Biggl(\prod_{i=1}^{m}(a_{i}+b_{i}) \Biggr)^{\alpha _{i}}\geq \Biggl(\prod_{i=1}^{m}a_{i} \Biggr)^{\alpha _{i}} + \Biggl(\prod_{i=1}^{m}b_{i} \Biggr) ^{\alpha _{i}}, $$

(3.1)

with equality if and only if $a_{1}/b_{1}=\cdots =a_{m}/b_{m}$.

Theorem 3.2

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $x_{00}, y_{00}, a_{00}, b_{00}>0$ and reals $x_{ij}$, $y_{ij}$, $a_{ij}$, $b_{ij}$, $i=1,2,\ldots ,n$, $j=1,2,\ldots ,m$, then

$$\begin{aligned}& \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}[(x_{ij}+y_{ij})^{r}+(a_{ij}+b _{ij})^{r}] )^{p}}{ [ (x_{00}+y_{00} )^{r}+ (a_{00}+b_{00} )^{r} ]^{p/q}} \\& \quad \leq \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} )^{p/r}}{x_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} )^{p/r}}{y_{00} ^{p/q}} \biggr)^{r} \\& \qquad {}+ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}a_{ij}^{r} ) ^{p/r}}{a_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}b_{ij} ^{r} )^{p/r}}{b_{00}^{p/q}} \biggr)^{r} \end{aligned}$$

(3.2)

with equality if and only if either $x_{ij}=y_{ij}=0$ and $a_{ij}=b _{ij}=0$ for $i=1,\ldots ,n$ and $j=1,\ldots ,m$ or $x_{ij}=\alpha y _{ij}$ and $a_{ij}=\beta b_{ij}$ for $i=0,1,\ldots ,n$ and $j=0,1, \ldots ,m$ and some $\alpha , \beta >0$, and

$$\begin{aligned}& \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} )^{p/r}}{x _{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} ) ^{p/r}}{y_{00}^{p/q}} \biggr): \biggl( \frac{ (\sum_{j=1}^{m}\sum_{i=1} ^{n}a_{ij}^{r} )^{p/r}}{a_{00}^{p/q}}+ \frac{ (\sum_{j=1} ^{m}\sum_{i=1}^{n}b_{ij}^{r} )^{p/r}}{b_{00}^{p/q}} \biggr) \\& \quad =(x_{00}+y_{00}):(a_{00}+b_{00}). \end{aligned}$$

Proof

From (2.1), we have

$$ \begin{aligned}[b] &\Biggl(\sum_{j=1}^{m}\sum _{i=1}^{n}(x_{ij}+y_{ij})^{r} \Biggr)^{1/r} \\ &\quad \leq \biggl\{ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} ) ^{p/r}}{x_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij} ^{r} )^{p/r}}{y_{00}^{p/q}} \biggr\} ^{1/p} (x_{00}+y_{00} ) ^{1/q}, \end{aligned} $$

(3.3)

with equality if and only if either $x_{ij}=y_{ij}=0$ for $i=1,\ldots ,n$ and $j=1,\ldots ,m$ or $x_{ij}=\alpha y_{ij}$ for $i=0,1,\ldots ,n$ and $j=0,1,\ldots ,m$ and some $\alpha >0$, and

$$ \begin{aligned}[b] &\Biggl(\sum_{j=1}^{m}\sum _{i=1}^{n}(a_{ij}+b_{ij})^{r} \Biggr)^{1/r} \\ &\quad \leq \biggl\{ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}a_{ij}^{r} ) ^{p/r}}{a_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}b_{ij} ^{r} )^{p/r}}{b_{00}^{p/q}} \biggr\} ^{1/p} (a_{00}+b_{00} ) ^{1/q}, \end{aligned} $$

(3.4)

with equality if and only if either $a_{ij}=b_{ij}=0$ for $i=1,\ldots ,n$ and $j=1,\ldots ,m$ or $a_{ij}=\alpha b_{ij}$ for $i=0,1,\ldots ,n$ and $j=0,1,\ldots ,m$ and some $\alpha >0$.

From (3.1), (3.3), and (3.4), we obtain

$$\begin{aligned}& \sum_{j=1}^{m}\sum _{i=1}^{n}\bigl[(x_{ij}+y_{ij})^{r}+(a_{ij}+b_{ij})^{r} \bigr] \\& \quad \leq \biggl\{ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij} ^{r} )^{p/r}}{x_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} )^{p/r}}{y_{00}^{p/q}} \biggr)^{r} \biggr\} ^{1/p} \bigl( (x_{00}+y_{00} )^{r} \bigr)^{1/q} \\& \qquad {}+ \biggl\{ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}a_{ij}^{r} ) ^{p/r}}{a_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}b_{ij} ^{r} )^{p/r}}{b_{00}^{p/q}} \biggr)^{r} \biggr\} ^{1/p} \bigl( (a _{00}+b_{00} )^{r} \bigr)^{1/q} \\& \quad \leq \biggl\{ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij} ^{r} )^{p/r}}{x_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} )^{p/r}}{y_{00}^{p/q}} \biggr)^{r} \\& \qquad {}+ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}a_{ij}^{r} ) ^{p/r}}{a_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}b_{ij} ^{r} )^{p/r}}{b_{00}^{p/q}} \biggr)^{r} \biggr\} ^{1/p} \bigl[ (x _{00}+y_{00} )^{r}+ (a_{00}+b_{00} )^{r} \bigr]^{1/q}. \end{aligned}$$

Hence

$$\begin{aligned} \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}[(x_{ij}+y_{ij})^{r}+(a_{ij}+b _{ij})^{r}] )^{p}}{ [ (x_{00}+y_{00} )^{r}+ (a_{00}+b_{00} )^{r} ]^{p/q}} \leq & \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}x_{ij}^{r} )^{p/r}}{x_{00} ^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}y_{ij}^{r} ) ^{p/r}}{y_{00}^{p/q}} \biggr)^{r} \\ &{}+ \biggl(\frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}a_{ij}^{r} ) ^{p/r}}{a_{00}^{p/q}}+ \frac{ (\sum_{j=1}^{m}\sum_{i=1}^{n}b_{ij} ^{r} )^{p/r}}{b_{00}^{p/q}} \biggr)^{r}. \end{aligned}$$

From the equality conditions of (3.3), (3.4), and (3.1), we easily get the equality in (3.2). □

Remark 3.3

Let $a_{ij}=b_{ij}=0$, (3.2) becomes a similar form of (2.1). Putting $x_{ij}=a_{ij}$, $y_{ij}=b_{ij}$ in (3.2), where $i=0,1,\ldots,n$ and $j=0,1,\ldots,m$, (3.2) reduces to (2.1).

Theorem 3.4

For $p>1$, $\frac{1}{p}+\frac{1}{q}=1$ and $r>1$. If $u(x,y), v(x,y), u'(x,y), v'(x,y)>0$ and $f(x,y)$, $g(x,y)$, $f'(x,y)$, $g'(x,y)$ are continuous functions on $[a,b]\times [c,d]$, then

$$\begin{aligned}& \frac{ (\int _{a}^{b}\int _{c}^{d}[(f(x,y)+g(x,y))^{r}+(f'(x,y)+g'(x,y))^{r}]\,dx \,dy ) ^{p}}{ [ (u(x,y)+v(x,y) )^{r}+ (u'(x,y)+v'(x,y) ) ^{r} ]^{p/q}} \\& \quad \leq \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d} f(x,y)^{r}\,dx \,dy ) ^{p/r}}{u(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{p/r}}{v(x,y)^{p/q}} \biggr)^{r} \\& \qquad{} + \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f'(x,y)^{r}\,dx \,dy ) ^{p/r}}{u'(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g'(x,y)^{r}\,dx \,dy ) ^{p/r}}{v'(x,y)^{p/q}} \biggr)^{r} \end{aligned}$$

(3.5)

with equality if and only if either $f(x,y)=g(x,y)=0$ and $f'(x,y)=g'(x,y)=0$ or $(f(x,y), g(x,y))=\alpha (u(x,y),v(x,y))$ and $(f'(x,y), g'(x,y))=\beta (u'(x,y),v'(x,y))$ and some $\alpha , \beta >0$, and

$$\begin{aligned}& \biggl( \frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy )^{p/r}}{u(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy )^{p/r}}{v(x,y)^{p/q}} \biggr)\\& \qquad {}: \biggl( \frac{ (\int _{a}^{b}\int _{c}^{d}f'(x,y)^{r}\,dx \,dy ) ^{p/r}}{u'(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g'(x,y)^{r} ) ^{p/r}}{v'(x,y)^{p/q}} \biggr) \\& \quad =\bigl(u(x,y)+v(x,y)\bigr):\bigl(u'(x,y)+v'(x,y) \bigr). \end{aligned}$$

Proof

From (2.1), we have

$$\begin{aligned} &\biggl( \int _{a}^{b} \int _{c}^{d}\bigl(f(x,y)+g(x,y) \bigr)^{r}\,dx \,dy \biggr)^{1/r} \\ &\quad \leq \biggl\{ \frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy ) ^{p/r}}{u(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{p/r}}{v(x,y)^{p/q}} \biggr\} ^{1/p} \\ &\qquad {}\times \bigl(u(x,y)+v(x,y) \bigr)^{1/q}, \end{aligned}$$

(3.6)

with equality if and only if either $f(x,y)=g(x,y)=0$ or $(f(x,y),g(x,y))= \alpha (u(x,y),v(x,y))$ for some $\alpha >0$. And

$$\begin{aligned} &\biggl( \int _{a}^{b} \int _{c}^{d}\bigl(f'(x,y)+g'(x,y) \bigr)^{r}\,dx \,dy \biggr)^{1/r} \\ &\quad \leq \biggl\{ \frac{ (\int _{a}^{b}\int _{c}^{d}f'(x,y)^{r}\,dx \,dy ) ^{p/r}}{u'(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g'(x,y)^{r}\,dx \,dy ) ^{p/r}}{v'(x,y)^{p/q}} \biggr\} ^{1/p} \\ &\qquad {}\times \bigl(u'(x,y)+v'(x,y) \bigr)^{1/q}, \end{aligned}$$

(3.7)

with equality if and only if either $f'(x,y)=g'(x,y)=0$ or $(f'(x,y),g'(x,y))= \beta (u'(x,y),v'(x,y))$ and for some $\beta >0$,

From (3.1), (3.6), and (3.7), we obtain

$$\begin{aligned}& \int _{a}^{b} \int _{c}^{d}\bigl[\bigl(f(x,y)+g(x,y) \bigr)^{r}+\bigl(f'(x,y)+g'(x,y) \bigr)^{r}\bigr]\,dx \,dy \\& \quad \leq \biggl\{ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy ) ^{p/r}}{u(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{p/r}}{v(x,y)^{p/q}} \biggr)^{r} \biggr\} ^{1/p} \bigl( \bigl(u(x,y)+v(x,y) \bigr) ^{r} \bigr)^{1/q} \\& \qquad {}+ \biggl\{ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f'(x,y)^{r}\,dx \,dy ) ^{p/r}}{u'(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g'(x,y)^{r}\,dx \,dy ) ^{p/r}}{v'(x,y)^{p/q}} \biggr)^{r} \biggr\} ^{1/p}\\& \qquad {}\times \bigl( \bigl(u'(x,y)+v'(x,y) \bigr) ^{r} \bigr)^{1/q} \\& \quad \leq \biggl\{ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f(x,y)^{r}\,dx \,dy ) ^{p/r}}{u(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{p/r}}{v(x,y)^{p/q}} \biggr)^{r} \\& \qquad {}+ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f'(x,y)^{r}\,dx \,dy ) ^{p/r}}{u'(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g'(x,y)^{r}\,dx \,dy ) ^{p/r}}{v'(x,y)^{p/q}} \biggr)^{r} \biggr\} ^{1/p} \\& \qquad {}\times \bigl[ \bigl(u(x,y)+v(x,y) \bigr)^{r}+ \bigl(u'(x,y)+v'(x,y) \bigr) ^{r} \bigr]^{1/q}. \end{aligned}$$

Hence

$$\begin{aligned}& \frac{ (\int _{a}^{b}\int _{c}^{d}[(f(x,y)+g(x,y))^{r}+(f'(x,y)+g'(x,y))^{r}]\,dx \,dy ) ^{p}}{ [ (u(x,y)+v(x,y) )^{r}+ (u'(x,y)+v'(x,y) ) ^{r} ]^{p/q}} \\& \quad \leq \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d} f(x,y)^{r}\,dx \,dy ) ^{p/r}}{u(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g(x,y)^{r}\,dx \,dy ) ^{p/r}}{v(x,y)^{p/q}} \biggr)^{r} \\& \qquad {}+ \biggl(\frac{ (\int _{a}^{b}\int _{c}^{d}f'(x,y)^{r}\,dx \,dy ) ^{p/r}}{u'(x,y)^{p/q}}+ \frac{ (\int _{a}^{b}\int _{c}^{d}g'(x,y)^{r}\,dx \,dy ) ^{p/r}}{v'(x,y)^{p/q}} \biggr)^{r}. \end{aligned}$$

From the equality conditions of (3.6), (3.7), and (3.1), we easily get the equality in (3.2). □

Remark 3.5

Let $f'(x,y)=g'(x,y)=0$, (3.3) becomes a similar form of (2.2). Putting $f(x,y)=f'(x,y)$, $g(x,y)=g'(x,y)$, $u(x,y)=u'(x,y)$ and $v(x,y)=v'(x,y)$ in (3.5), (3.5) reduces to (2.2).