假设 
是给定的 正 数。令 
, ..., 
和 
。则
 |
(1)
|
其中
 |  |  |
(2)
|
 |  |  |
(3)
|
分别是第一个和最后一个数的算术平均值和几何平均值。Kantorovich 不等式对于研究优化中下降方法的收敛性至关重要 (Luenberger 1984)。
Kantorovich 不等式
另请参阅
算术平均值, 几何平均值使用 Wolfram|Alpha 探索
参考文献
Bauer, F. L. "A Further Generalization of the Kantorovich Inequality." Numer. Math. 3, 117-119, 1961.Greub, W. and Rheinboldt, W. "On a Generalization of an Inequality of L. V. Kantorovich." Proc. Amer. Math. Soc. 10, 407-413, 1959.Henrici, P. "Two Remarks of the Kantorovich Inequality." Amer. Math. Monthly 68, 904-906, 1961.Kantorovič, L. V. "Functional Analysis and Applied Mathematics" [Russian]. Uspekhi Mat. Nauk 3, 89-185, 1948.Luenberger, D. G. Linear and Nonlinear Programming, 2nd ed. Reading, MA: Addison-Wesley, pp. 217-219, 1984.Newman, M. "Kantorovich's Inequality." J. Res. National Bur. Standards 64B, 33-34, 1960.Pólya, G. and Szegö, G. Aufgaben und Lehrsätze der Analysis. Berlin: Springer-Verlag, 1925.Pták, V. "The Kantorovich Inequality." Amer. Math. Monthly 102, 820-821, 1995.Schopf, A. H. "On the Kantorovich Inequality." Numer. Math. 2, 344-346, 1960.Strang, W. G. "On the Kantorovich Inequality." Proc. Amer. Math. Soc. 11, 468, 1960.在 Wolfram|Alpha 中被引用
Kantorovich 不等式请引用为
Weisstein, Eric W. "Kantorovich 不等式。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/KantorovichInequality.html