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On a New Hilbert-Hardy-Type Integral Operator and Applications
Journal of Inequalities and Applications volume 2010, Article number: 812636 (2010)
Abstract
By applying the way of weight functions and a Hardy's integral inequality, a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained. As applications, a new Hilbert-Hardy-type inequality similar to Hilbert-type integral inequality is given, and two equivalent inequalities with the best constant factors as well as some particular examples are considered.
1. Introduction
In 1934, Hardy published the following theorem (cf. [1, Theorem 319]).
Theorem A.
If is a homogeneous function of degree
in
,
,
, and
, then for
,
, and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ1_HTML.gif)
where the constant factor is the best possible.
Hardy [2] also published the following Hardy's integral inequality.
Theorem B.
If ,
,
, and
;
,
, then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ2_HTML.gif)
where the constant factor is the best possible (cf. [1, Theorem  330]).
In 2009, Yang [3] published the following theorem.
Theorem C.
If ,
,
,
is a homogeneous function of degree
in
, and for any
,
, then for
,
,
,
,
, and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ3_HTML.gif)
where the constant factor is the best possible.
For,
, (1.3) reduces to (1.1). We name of (1.1) and (1.3) Hilbert-type integral inequalities. Inequalities (1.1), (1.2) and (1.3) are important in analysis and its applications (cf. [4–6]).
Setting ,
,
,
, by applying (1.2) (for
), Das and Sahoo gave a new integral inequality similar to Pachpatte's inequality (cf. [7, 8]) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ4_HTML.gif)
where the constant factor is the best possible (cf. [9]). Sulaiman [10] also considered a Hilbert-Hardy-type integral inequality similar to (1.4) with the kernel
,
,
. But he cannot show that the constant factor in the new inequality is the best possible.
In this paper, by applying the way of weight functions and inequality (1.2) for , a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained. As applications, a new Hilbert-Hardy-type inequality similar to (1.3) is given, and two equivalent inequalities with a best constant factor as well as some particular examples are considered.
2. A Lemma and Two Equivalent Inequalities
Lemma 2.1.
If ,
is a nonnegative homogeneous function of degree
in
with
, and for any
,
, then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ5_HTML.gif)
Proof.
Setting , we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ6_HTML.gif)
There exists , satisfying
and
. Since we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ7_HTML.gif)
there exists , such that
, and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ8_HTML.gif)
The lemma is proved.
Theorem 2.2.
If ,
,
,
is a homogeneous function of degree
in
, and for any
,
, then for
,
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_IEq70_HTML.gif)
, and , one has the following equivalent inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ11_HTML.gif)
Proof.
Setting the weight functions and
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ12_HTML.gif)
then by Lemma 2.1, we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ13_HTML.gif)
By Hölder's inequality (cf. [11]) and (2.8), (2.9), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ14_HTML.gif)
Then by Fubini theorem (cf. [12]), it follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ15_HTML.gif)
Since ,
, then by (1.2) (for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ16_HTML.gif)
Hence by (2.11), we have (2.7). Still by Hölder's inequality, we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ17_HTML.gif)
Then by (2.7), we have (2.6).
On the other-hand, supposing that (2.6) is valid, by (2.11) and (1.2) (for , it follows
. If
, then (2.7) is naturally valid; if
, setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ18_HTML.gif)
then by (2.6), we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ19_HTML.gif)
Hence, we have (2.7), which is equivalent to (2.6).
3. A Hilbert-Hardy-Type Integral Operator and Applications
Setting a real function space as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ20_HTML.gif)
for ,
, define an integral operator
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ21_HTML.gif)
Then, by (2.7), , and
is bounded with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ22_HTML.gif)
Theorem 3.1.
Let the assumptions of Theorem 2.2 be fulfilled, and additionally setting . Then one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ23_HTML.gif)
where the constant factor is the best possible. Moreover the constant factor in (2.6) and (2.7) is the best possible and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ24_HTML.gif)
Proof.
Since , by (1.2), for
, it follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ25_HTML.gif)
Then, by (2.6), we have (3.4).
For , setting
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ26_HTML.gif)
then for ,
, we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ27_HTML.gif)
where ,
, and
are indicated as follows;
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ28_HTML.gif)
If there exists a positive constant , such that (3.4) is still valid as we replace
by
, then in particular, we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ29_HTML.gif)
By (3.8) and (3.10), we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ30_HTML.gif)
Since by Fubini theorem, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ31_HTML.gif)
then for in (3.10), by Lemma 2.1, we obtain
. Hence
, and then
is the best value of (3.4).
We conclude that the constant factor in (2.6) is the best possible, otherwise we can get a contradiction by (1.2) that the constant factor in (3.4) is not the best possible. By the same way, if the constant factor in (2.7) is not the best possible, then by (2.13), we can get a contradiction that the constant factor in (2.6) is not the best possible. Therefore in view of (3.3), we have (3.5).
Corollary 3.2.
For ,
,
,
,
, in (2.6), (2.7) and (3.4), one has the following basic Hilbert-Hardy-type integral inequalities with the best constant factors:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ32_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ33_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ34_HTML.gif)
where , and (3.13) is equivalent to (3.14).
Example 3.3.
For ,
,
,
, and
in(3.4),
(a)if ,
,
and
, then we obtain the following integral inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ35_HTML.gif)
(b)if,
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ36_HTML.gif)
(c)if , then we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F812636/MediaObjects/13660_2010_Article_2259_Equ37_HTML.gif)
where the constant factors in the above inequalities are the best possible.
References
Hardy GH, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.
Hardy GH: Note on some points in the integral calculus (LXIV). Messenger of Math 1928, 57: 12–16.
Yang B: A survey of the study of Hilbert-type inequalities with parameters. Advances in Math 2009, 38(3):257–268.
Mintrinovic DS, Pecaric JE, Kink AM: Inequalities Involving Functions and their Integrals and Derivertives. Kluwer Academic Publishers, Boston, Mass, USA; 1991.
Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science, Beijin, China; 2009.
Yang B: On the norm of a Hilbert's type linear operator and applications. Journal of Mathematical Analysis and Applications 2007, 325(1):529–541. 10.1016/j.jmaa.2006.02.006
Pachpatte BG: On some new inequalities similar to Hilbert's inequality. Journal of Mathematical Analysis and Applications 1998, 226(3):166–179.
Pachpatte BG: Inequalities similar to certain extensions of Hilbert's inequality. Journal of Mathematical Analysis and Applications 2000, 243(2):217–227. 10.1006/jmaa.1999.6646
Das N, Sahoo S: New inequalities similar to Hardy-Hilbert's inequality. Turkish Journal of Mathematics 2009, 33: 1–13.
Sulaiman WT: On three inequalities similar to Hardy-Hilbert's integral inequality. Acta Mathematica Universitatis Comenianae 2007, 76(2):273–278.
Kuang J: Applied Inequalities. Shangdong Science Technic, Jinan, China; 2004.
Kuang J: Introduction to Real Analysis. Hunan Education, Changsha, China; 1996.
Acknowledgments
This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (no. 05Z026) and the Guangdong Natural Science Foundation (no. 7004344).
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Liu, X., Yang, B. On a New Hilbert-Hardy-Type Integral Operator and Applications. J Inequal Appl 2010, 812636 (2010). https://doi.org/10.1155/2010/812636
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DOI: https://doi.org/10.1155/2010/812636