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.
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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
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
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
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
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
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
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
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
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
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
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
or equivalently
and the same equation with \(y_{ij}\) instead of \(x_{ij}\), \(i=0,1,\ldots ,n\); \(j=0,1,\ldots ,m\). Therefore
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
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
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
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
or equivalently
and the same equation with \(g(x,y)\) instead of \(f(x,y)\). Therefore
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
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
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
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
Proof
From (2.1), we have
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
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
Hence
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
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
Proof
From (2.1), we have
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
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
Hence
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).
References
Gardner, R.J., Hartenstine, D.: Capacities, surface area, and radial sums. Adv. Math. 221, 601–626 (2009)
Bandle, C., Marcus, M.: Radial averaging transformations and generalized capacities. Math. Z. 145, 11–17 (1975)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Univ. Press, Cambridge (1934)
Acknowledgements
The first author expresses his gratitude to professor W. Li for their valuable help.
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The first author’s research is supported by the Natural Science Foundation of China (11371334, 10971205). The second author’s research is partially supported by a HKU Seed Grant for Basic Research.
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C-JZ and W-SC jointly contributed to the main results. All authors read and approved the final manuscript.
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Zhao, CJ., Cheung, WS. On Gardner–Hartenstine’s inequality. J Inequal Appl 2019, 161 (2019). https://doi.org/10.1186/s13660-019-2109-4
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DOI: https://doi.org/10.1186/s13660-019-2109-4