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Kantorovich Inequality


Suppose x_1<x_2<...<x_n are given positive numbers. Let lambda_1, ..., lambda_n>=0 and sum_(j=1)^(n)lambda_j=1. Then

 (sum_(j=1)^nlambda_jx_j)(sum_(j=1)^nlambda_jx_j^(-1))<=A^2G^(-2),
(1)

where

A=1/2(x_1+x_n)
(2)
G=sqrt(x_1x_n)
(3)

are the arithmetic and geometric mean, respectively, of the first and last numbers. The Kantorovich inequality is central to the study of convergence properties of descent methods in optimization (Luenberger 1984).


See also

Arithmetic Mean, Geometric Mean

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References

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.

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Kantorovich Inequality

Cite this as:

Weisstein, Eric W. "Kantorovich Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KantorovichInequality.html

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