Minkowski inequality

The proper Minkowski inequality: For real numbers $ x _ {i} , y _ {i} \geq 0 $, $ i = 1 \dots n $, and for $ p > 1 $,

$$ \tag{1 } \left ( \sum_{i=1} ^ { n } ( x _ {i} + y _ {i} ) \right ) ^ {1/p} \leq \ \left ( \sum_{i=1} ^ { n } x _ {i} ^ {p} \right ) ^ {1/p} + \left ( \sum_{i=1} ^ { n } y _ {i} ^ {p} \right ) ^ {1/p} . $$

This was derived by H. Minkowski . For $ p < 1 $, $ p \neq 0 $, the inequality is reversed (for $ p < 0 $ one must have $ x _ {i} , y _ {i} > 0 $). In each case equality holds if and only if the rows $ \{ x _ {i} \} $ and $ \{ y _ {i} \} $ are proportional. For $ p = 2 $ 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 $ x _ {ij} \geq 0 $ for $ i = 1 \dots n $ and $ j = 1 \dots m $ and let $ p > 1 $. Then

$$ \tag{2 } \left [ \sum_{i=1} ^ { n } \left ( \sum_{j=1}^ { m } x _ {ij} \right ) ^ {p} \right ] ^ {1/p} \leq \ \sum_{j=1}^ { m } \left ( \sum_{i=1} ^ { n } x _ {ij} ^ {p} \right ) ^ {1/p} . $$

The inequality is reversed for $ p < 1 $, $ p \neq 0 $, and for $ p < 0 $ it is assumed that $ x _ {ij} > 0 $. In each case equality holds if and only if the rows $ \{ x _ {i1} \} \dots \{ x _ {im} \} $ 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 ).

Inequalities

and (2) are homogeneous with respect to $ \sum $ and therefore have analogues for various means, for example, if $ M _ \phi ( x _ {i} ) = \phi^{-1} \{ \sum \phi ( x _ {i} ) \} $, where $ \phi ( t) = \mathop{\rm log} t $, then

$$ M _ \phi \left ( \frac{x _ {i} + y _ {i} }{2} \right ) \leq \frac{1}{2} M _ \phi ( x _ {i} ) + \frac{1}{2} M _ \phi ( y _ {i} ) ; $$

for more details see .

Minkowski's inequality for integrals is similar to

and also holds because of the homogeneity with respect to $ \int $. Let $ f , g $ be integrable functions in a domain $ X \subset \mathbf R ^ {n} $ with respect to the volume element $ d V $. Then for $ p > 1 $,

$$ \tag{3 } \left ( \int\limits _ { X } | f + g | ^ {p} d V \right ) ^ {1/p\ } \leq $$

$$ \leq \ \left ( \int\limits _ { X } | f | ^ {p} d V \right ) ^ {1/p} + \left ( \int\limits _ { X } | g | ^ {p} d V \right ) ^ {1/p} . $$

A generalization of (3) to more general functions can be obtained naturally. A further generalization is: If $ k > 1 $, then

$$ \left ( \int\limits \left ( \int\limits f ( x , y ) d y \right ) ^ {k} d x \right ) ^ {1/k} \leq \ \int\limits \left ( \int\limits f ^ { k } ( x , y ) d x \right ) ^ {1/k} d y , $$

where equality holds only if $ f ( x , y ) = \phi ( x) \psi ( y) $.

Other inequalities of Minkowski type:

a) for products: If $ x _ {i} , y _ {i} \geq 0 $, then

$$ \prod_{i=1}^ { n } ( x _ {i} + y _ {i} ) ^ {1/n} \geq \ \left ( \prod_{i=1}^ { n } x _ {i} \right ) ^ {1/n} + \left ( \prod_{i=1}^ { n } y _ {i} \right ) ^ {1/n} ; $$

b) Mahler's inequality: Let $ F ( x) $ be a generalized norm on $ E ^ {n} $ and $ G ( y) $ its polar function; then

$$ ( x , y ) \leq F ( x) G ( y) , $$

where $ ( \cdot , \cdot ) $ is the inner product;

c) for determinants: If $ A , B $ are non-negative Hermitian matrices over $ \mathbf C $, then

$$ ( \mathop{\rm det} ( A + B ) ) ^ {1/n} \geq \ ( \mathop{\rm det} A ) ^ {1/n} + ( \mathop{\rm det} B ) ^ {1/n} . $$

Finally, connected with the name of Minkowski are other inequalities; in particular, in convex analysis and number theory. For example, the Brunn–Minkowski theorem.

References

[1]	H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953)
[2]	G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[3]	E.F. Beckenbach, R. Bellman, "Inequalities" , Springer (1961)
[4]	M. Marcus, H. Minc, "Survey of matrix theory and matrix inequalities" , Allyn & Bacon (1964)

Comments

A generalized norm on $ E ^ {n} $ is a function $ F $ for which: 1) $ F ( x) > 0 $ for $ x \neq 0 $; 2) $ F ( t x ) = t F ( x) $ for $ t \geq 0 $; and 3) $ F ( x) + F ( y) \geq F ( x + y ) $. The polar form (or polar function) $ G $ of the generalized norm $ F $ is defined by:

$$ G ( y) = \max _ { x } \frac{( x , y ) }{F ( x) } , $$

where $ ( \cdot , \cdot ) $ is the inner product.

References

[a1]	P.S. Bullen, D.S. Mitrinović, P.M. Vasić, "Means and their inequalities" , Reidel (1988)