On the Hubert Inequality

  • Gao Mingzhe

    Xiangxi Education College, Hunan, China


It is shown that the Hilbert inequality for double series can be improved by introducing the positive real number 1π2(s2(b)a2+s2(b)b2)\frac{1}{\pi^2} (\frac{s^2(b)}{\|a\|^2} + \frac{s^2(b)}{\|b\|^2}) where s(x)=n=1xnns(x) = \sum^{\infty}_{n=1} \frac{x_n}{n} and x2=n=1xn2(x=a,b)\|x\|^2 = \sum^{\infty}_{n=1} x^2_n (x = a, b). The coefficient π\pi of the classical Hilbert inequality is proved not to be the best possible if a\|a\| or b\|b\| is finite. A similar result for the Hubert integral inequality is also proved.

Cite this article

Gao Mingzhe, On the Hubert Inequality . Z. Anal. Anwend. 18 (1999), no. 4, pp. 1117–1122

DOI 10.4171/ZAA/932