On a generalization of a Hilbert-type integral inequality in the whole plane with a hypergeometric function
Journal of Inequalities and Applications volume 2013, Article number: 189 (2013)
By introducing some parameters, we establish a generalization of the Hilbert-type integral inequality in the whole plane with the homogeneous kernel of degree and the best constant factor which involves the hypergeometric function.
If , and satisfy
In 2001, Yang gave an extension of (1) involving beta function as (see ):
where the constant factor () is the best possible.
If , , , , , satisfy
then we have
where the constant factors and are the best possible. Inequalities (3) and (4) are equivalent.
By introducing some parameters, we establish generalizations of inequalities (3) and (4) with the homogeneous kernel of degree and the best constant factor which involves the hypergeometric function.
2 Preliminary lemmas
In order to prove our assertions, we need the following lemmas.
Recall that the hypergeometric function is defined  by
where is the Pochhammer symbol defined by
It is known the series (5) converges for and diverges for . The hypergeometric function satisfies the integral representation
Lemma 2.1 (See )
Suppose that , , . Then we have
Lemma 2.2 Let , , and be real parameters such that . Define the weight functions and () as follows:
Then we have (), where
Proof For , setting , in the following two integrals, respectively, and using Lemma 2.1, we get
For , setting , in the following two integrals, respectively, and using Lemma 2.1, we get
By the same way, we still can find that (). The lemma is proved. □
Lemma 2.3 Let p and q be conjugate parameters with , and let , , , and , and be a nonnegative measurable function in , then we have
Proof By Lemma 2.2 and Hölder’s inequality , we have
Then by the Fubini theorem, it follows that
The lemma is proved. □
3 Main results
Theorem 3.1 Let p and q be conjugate parameters with , and let , , , and , and , satisfy and . Then
where the constant factors and are the best possible and is defined in Lemma 2.2. Inequalities (8) and (9) are equivalent.
Proof If (7) takes the form of the equality for a , then there exist constants A and B such that they are not all zero, and
Hence, there exists a constant K such that
We suppose (otherwise ). Then a.e. in , which contradicts the fact that . Hence, (7) takes the form of a strict inequality, so does (6), and we have (9).
By Hölder’s inequality , we have
By (9), we have (8). On the other hand, suppose that (8) is valid. Set
then it follows . By (6), we have . If , then (9) is obviously valid. If , then by (8), we obtain
Hence, we have (9), which is equivalent to (8).
For , define functions , as follows:
Taking , by the Fubini theorem, we obtain
In view of the above results, if the constant factor in (8) is not the best possible, then there exists a positive number with such that
By the Fatou lemma and (11), we have
which contradicts the fact that . Hence, the constant factor in (8) is the best possible.
If the constant factor in (9) is not the best possible, then by (10), we may get a contradiction that the constant factor in (8) is not the best possible. Thus the theorem is proved. □
Remark 1 Setting in Theorem 3.1, we have (3) and (4).
Remark 2 Setting in Theorem 3.1, we have the following particular results:
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The authors declare that they have no competing interests.
Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.
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Adiyasuren, V., Batbold, T. On a generalization of a Hilbert-type integral inequality in the whole plane with a hypergeometric function. J Inequal Appl 2013, 189 (2013). https://doi.org/10.1186/1029-242X-2013-189
- Hilbert’s inequality
- Hölder’s inequality
- homogeneous kernel
- weight function
- equivalent form