- Open Access
On reverse Hilbert-type inequalities
© Xu et al.; licensee Springer. 2014
- Received: 8 November 2013
- Accepted: 6 May 2014
- Published: 20 May 2014
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.
- reverse Hilbert-type inequality
- weight coefficient
- best constant factor
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.
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 ). In recent years, many results with generalizations of this type of inequality have been obtained (see ).
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  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 .
Definition 1.1 Let be the set of functions satisfying the following conditions.
, where .
For and , the function () is decreasing in .
We have Jin’s result as follows.
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  as follows.
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.
Definition 2.1 Let be the set of functions satisfying the following conditions:
For and , the function () is decreasing in .
- (3)There exists a positive constant such that
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. □
The lemma is proved. □
where the constant factor and are the best possible. Inequality (3.1) is equivalent to (3.2).
Then, by (2.3), in view of and , we have (3.1).
For , it follows that , which contradicts the fact that . Hence, the constant factor in (3.1) is the best possible.
By (3.1), both (3.6) and (3.7) take the form of a strict inequality, and we have (3.2).
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. □
is decreasing in . Hence, . By Theorem 3.1, we have the following.
are the best possible. Inequality (4.1) is equivalent to (4.2).
- (b)For and , we have(4.5)(4.6)
is decreasing in . Hence . By Theorem 3.1, we have the following corollary.
are the best possible. Inequality (4.7) is equivalent to (4.8).
- (b)For and , we have(4.11)(4.12)
is decreasing in . Hence, . By Theorem 3.1, we have the following corollary.
Here the constant factors and are the best possible. Inequality (4.13) is equivalent to (4.14).
- (b)For and , we have(4.17)(4.18)
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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