A NEW EXTENSION ON THE HARDY-HILBERT INEQUALITY

Abstract

A new Hardy-Hilbert type integral inequality for double series with weights can be established by introducing a parameter ${\lambda}$ (with ${\lambda}>1-\frac{2}{pq}$) and a weight function of the form $x^{1-\frac{2}{r}}$ (with $r$ > 1). And the constant factors of new inequalities established are proved to be the best possible. In particular, for case $r$ = 2, a new Hilbert type inequality is obtained. As applications, an equivalent form is considered.

Keywords

• Hardy-Hilbert type inequality;
• double series;
• Euler-Maclaurin summation formula;
• weight function

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References

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