为一个
为一个 |
(1)
|
和
 |
(2)
|
并取 
。 对于求和，
 |
(3)
|
(除非所有 
)， 以及对于积分，
![int_0^infty[(F(x))/x]^pdx<(p/(p-1))^pint_0^infty[f(x)]^pdx](/images/equations/HardysInequality/NumberedEquation4.svg) |
(4)
|
(除非 
恒等于 0)。
哈代不等式
另请参阅
卡尔曼不等式使用 Wolfram|Alpha 探索
参考文献
Broadbent, T. A. A. "A Proof of Hardy's Convergence Theorem." J. London Math. Soc. 3, 232-243, 1928.Elliot, E. B. "A Simple Exposition of Some Recently Proved Facts as to Convergency." J. London Math. Soc. 1, 93-96, 1926.Grandjot, K. "On Some Identities Relating to Hardy's Convergence Theorem." J. London Math. Soc. 3, 114-117, 1928.Hardy, G. H. "Note on a Theorem of Hilbert." Math. Z. 6, 314-317, 1920.Hardy, G. H. "Notes on Some Points in the Integral Calculus. LX." Messenger Math. 54, 150-156, 1925.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. "Hardy's Inequality." §9.8 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 239-243, 1988.Kaluza, T. and Szegö, G. "Über Reihen mit lauter positiven Gliedern." J. London Math. Soc. 2, 266-272, 1927.Knopp, K. "Über Reihen mit positiven Gliedern." J. London Math. Soc. 3, 205-211, 1928.Landau, E. "A Note on a Theorem Concerning Series of Positive Terms." J. London Math. Soc. 1, 38-39, 1926.Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. New York: Kluwer, 1991.Opic, B. and Kufner, A. Hardy-Type Inequalities. Essex, England: Longman, 1990.在 Wolfram|Alpha 中被引用
哈代不等式请引用为
Weisstein, Eric W. "哈代不等式。" 来自 MathWorld-- Wolfram Web 资源。 https://mathworld.net.cn/HardysInequality.html