On a generalization of a Hilbert-type integral inequality in the whole plane with a hypergeometric function
© Adiyasuren and Batbold; licensee Springer 2013
Received: 17 May 2012
Accepted: 16 January 2013
Published: 19 April 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.
where the constant factor () is the best possible.
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.
Lemma 2.1 (See )
By the same way, we still can find that (). The lemma is proved. □
The lemma is proved. □
3 Main results
where the constant factors and are the best possible and is defined in Lemma 2.2. Inequalities (8) and (9) are equivalent.
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).
Hence, we have (9), which is equivalent to (8).
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).
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