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On reverse Hilbert-type inequalities
Journal of Inequalities and Applications volume 2014, Article number: 198 (2014)
Abstract
By introducing two pairs of conjugate exponents and estimating the weight coefficients, we establish reverse versions of Hilbert-type inequalities, as described by Jin (J. Math. Anal. Appl. 340:932-942, 2008), and we prove that the constant factors are the best possible. As applications, some particular results are considered.
1 Introduction
If both and , such that and , then we have (see [1])
where the constant factor π has the best possible value. Inequality (1.1) is the well-known Hilbert inequality, introduced in 1925; inequality (1.1) has been generalized by Hardy as follows.
If , , and both and , such that and , then we have
where the constant factor is the best possible. Inequality (1.2) is the well-known Hardy-Hilbert inequality, which is important in analysis and applications (see [2]). In recent years, many results with generalizations of this type of inequality have been obtained (see [3]).
Under the same conditions as in (1.2), there are some Hilbert-type inequalities that are similar to (1.2), which also have been studied and generalized by some mathematicians.
Recently, by studying a Hilbert-type operator, Jin [4] obtained a new bilinear operator inequality with the norm, and he provided some new Hilbert-type inequalities with the best constant factor. First, we repeat the results of [4].
Definition 1.1 Let be the set of functions satisfying the following conditions.
Let , , , , suppose that is continuous in and satisfies:
-
(1)
, where .
-
(2)
For and , the function () is decreasing in .
For small enough, , and can be written as
where is independent of x, (), is a positive constant independent of x, and ().
We have Jin’s result as follows.
Theorem 1.1 If , , , , and , , such that and , then we have
Here the constant factors and are the best possible. Inequality (1.3) is equivalent to (1.4).
If and in Theorem 1.1, then Theorem 1.1 reduces to Yang’s result [5] as follows.
Theorem 1.2 If , , , and both and , such that and , then we have
where the constant factors and are the best possible. Inequality (1.3) is equivalent to (1.4).
In this paper, by introducing some parameters, we establish a reverse version of the inequality (1.3). As applications, some particular results are considered.
2 Some lemmas
Definition 2.1 Let be the set of functions satisfying the following conditions:
Let , , , , suppose that is continuous in and satisfies:
-
(1)
, .
-
(2)
For and , the function () is decreasing in .
For small enough, for , can be described as
where is independent of x, and (), is a positive constant independent of x, and ().
-
(3)
There exists a positive constant such that
Lemma 2.2 If , , , , , and the weight coefficients and are defined as
then we have
Proof By the assumption of the lemma, because () is decreasing, then we find
However, we find
It is easy to show that the above inequalities take the form of a strict inequality. Hence, we have . Similarly, we can obtain . The lemma is proved. □
Lemma 2.3 If , , , , and , for small enough, we have
Proof For , by Definition 2.1, we have
The lemma is proved. □
3 Main results
Theorem 3.1 If , , , , and , , such that and , then we have
where the constant factor and are the best possible. Inequality (3.1) is equivalent to (3.2).
Proof By Hölder’s inequality, we have (see [6])
Then, by (2.3), in view of and , we have (3.1).
For , setting and , we find
By virtue of (2.4), we have
If the constant factor in (3.1) is not the best possible factor, then there exists a positive number K (with ), such that (3.1) is still valid if the constant factor is replaced by K. In particular, by (3.4) and (3.5), we have
that is,
For , it follows that , which contradicts the fact that . Hence, the constant factor in (3.1) is the best possible.
Setting as
by (3.1), we have
Hence, we obtain
By (3.1), both (3.6) and (3.7) take the form of a strict inequality, and we have (3.2).
However, if (3.2) is valid, by Hölder’s inequality, we find
Then, by (3.2), we have (3.1). Hence (3.2) and (3.1) are equivalent.
If the constant factor in (3.2) is not the best possible, by using (3.8), we find the contradiction that the constant factor in (3.1) is not the best possible. The theorem is completed. □
4 Some particular results
-
(1)
Setting
for , and for fixed , we find (see [4])
and ();
Hence, . Similarly, we obtain . For , , and fixed , the function
is decreasing in . Hence, . By Theorem 3.1, we have the following.
Corollary 4.1 If , , , , and both such that and , then we have
where the constant factors
are the best possible. Inequality (4.1) is equivalent to (4.2).
In particular, (a) for , , and , we have
-
(b)
For and , we have
(4.5)(4.6) -
(2)
Let
For and , we find (see [4])
and ();
Hence, . Similarly, we can obtain . For , , and , the function
is decreasing in . Hence . By Theorem 3.1, we have the following corollary.
Corollary 4.2 If , , , , , and both such that and , then we have
where the constant factors
are the best possible. Inequality (4.7) is equivalent to (4.8).
In particular, (a) for , , and , we have
-
(b)
For and , we have
(4.11)(4.12) -
(3)
Let
for and , then we find (see [4])
and ()
Hence, . Similarly, we can obtain . For , , and , the function
is decreasing in . Hence, . By Theorem 3.1, we have the following corollary.
Corollary 4.3 If , , , , , and both , such that and , then we have
Here the constant factors and are the best possible. Inequality (4.13) is equivalent to (4.14).
In particular, (a) for , , and , we have
-
(b)
For and , we have
(4.17)(4.18)
References
Hardy GH, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge; 1952.
Mitrinovic DS, Pecaric JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston; 1991.
Yang BC, Rassias TM: On the way of weight coefficient and research for the Hilbert-type inequalities. Math. Inequal. Appl. 2003,6(4):625–658.
Jin JJ: On Hilbert’s type inequalities. J. Math. Anal. Appl. 2008, 340: 932–942. 10.1016/j.jmaa.2007.09.036
Yang BC: On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl. 2007, 325: 529–541. 10.1016/j.jmaa.2006.02.006
Kuang JC: Applied Inequalities. Shandong Science and Technology Press, Jinan; 2004.
Acknowledgements
The work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (No. 2013JK1139), China Postdoctoral Science Foundation (No. 2013M542370), NNSFC (No. 11326161), key projects of Science and Technology Research of the Henan Education Department (No. 14A110011). The authors deeply appreciate the support.
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Xu, B., Wang, XH., Wei, W. et al. On reverse Hilbert-type inequalities. J Inequal Appl 2014, 198 (2014). https://doi.org/10.1186/1029-242X-2014-198
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DOI: https://doi.org/10.1186/1029-242X-2014-198