The proper Minkowski inequality: For real numbers , , and for ,
This was derived by H. Minkowski . For , , the inequality is reversed (for one must have ). In each case equality holds if and only if the rows and are proportional. For Minkowski's inequality is called the triangle inequality. Minkowski's inequality can be generalized in various ways (also called Minkowski inequalities). Below some of them are listed.
Minkowski's inequality for sums. Let for and and let . Then
The inequality is reversed for , , and for it is assumed that . In each case equality holds if and only if the rows are proportional. There are also generalizations of
for multiple and infinite sums. However, when limit processes are used special attention is required in formulating the case of possible equality (see ).
and (2) are homogeneous with respect to and therefore have analogues for various means, for example, if , where , then
for more details see .
Minkowski's inequality for integrals is similar to
and also holds because of the homogeneity with respect to . Let be integrable functions in a domain with respect to the volume element . Then for ,
A generalization of (3) to more general functions can be obtained naturally. A further generalization is: If , then
where equality holds only if .
Other inequalities of Minkowski type:
a) for products: If , then
b) Mahler's inequality: Let be a generalized norm on and its polar function; then
where is the inner product;
c) for determinants: If are non-negative Hermitian matrices over , then
Finally, connected with the name of Minkowski are other inequalities; in particular, in convex analysis and number theory. For example, the Brunn–Minkowski theorem.
|||H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953)|
|||G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)|
|||E.F. Beckenbach, R. Bellman, "Inequalities" , Springer (1961)|
|||M. Marcus, H. Minc, "Survey of matrix theory and matrix inequalities" , Allyn & Bacon (1964)|
A generalized norm on is a function for which: 1) for ; 2) for ; and 3) . The polar form (or polar function) of the generalized norm is defined by:
where is the inner product.
|[a1]||P.S. Bullen, D.S. Mitrinović, P.M. Vasić, "Means and their inequalities" , Reidel (1988)|
Minkowski inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minkowski_inequality&oldid=17387