- Open Access
A relation between Hilbert and Carlson inequalities
© Azar; licensee Springer 2012
- Received: 13 February 2012
- Accepted: 7 November 2012
- Published: 28 November 2012
In this paper, we introduce some new inequalities with a best constant factor. As an application, we obtain a sharper form of Hilbert’s inequality. Some inequalities of Carlson type are also considered.
- Hilbert’s inequality
- Carlson’s inequality
- Cauchy’s inequality
- sharper form
the constant is sharp. Regarding these inequalities and their extensions, we refer the reader to the book .
In this paper, we introduce two new inequalities with a best constant factor which gives an upper estimate for the double series and the double integral , where is a double sequence of positive numbers and is a positive function on . As an application, we obtain a sharper form of the Hilbert inequality. Some examples of Carlson type inequalities are also considered. The proof of the inequalities depends on inequalities (1.7), (1.8) and Hardy’s idea in proving Carlson’s inequality.
here , , , and the constant is the best possible.
Set , , and consider the function . Since , we conclude that the minimum of this function attains for . Therefore, if we let and , we get (2.1).
Substituting (2.4) and (2.5) in (2.3), we obtain the contradiction . The theorem is proved. □
2.1 Some applications
- 1.If in (2.1), then we have the following form of Hilbert’s inequality:(2.6)
- 2.The more accurate Hilbert inequality is given as(2.9)
- 4.If we let and , , , we get
where , , , and the constant is the best possible.
Substituting the above estimates in (3.2) and then letting , we find . The theorem is proved. □
3.1 Some applications
- 4.Let , in (3.1), then we obtain the following Carlson type inequality:
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