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Minkowski's Inequalities

Minkowski's inequalities express the triangle inequality for Lp-spaces. On any measure space, for 1<=p<=infty and f,g in L^p,

||f+g||_p<=||f||_p+||g||_p.
(1)

The infinity norm is the essential supremum (Axler 2026, p. 199). For finite p>=1 and real or complex measurable functions with finite L^p norm, Minkowski's integral inequality states that

[int_a^b|f(x)+g(x)|^pdx]^(1/p)<=[int_a^b|f(x)|^pdx]^(1/p)+[int_a^b|g(x)|^pdx]^(1/p).
(2)

Similarly, if 1<=p<infty and a_k, b_k are real or complex, then Minkowski's sum inequality states that

[sum_(k=1)^n|a_k+b_k|^p]^(1/p) <=(sum_(k=1)^n|a_k|^p)^(1/p)+(sum_(k=1)^n|b_k|^p)^(1/p).
(3)

For 1<p<infty and two nonzero vectors, equality holds iff one is a positive real multiple of the other. The integral version has the corresponding almost everywhere condition. If either norm is zero, equality holds automatically. At p=1, equality need not require proportionality: for example, (1,0) and (0,1) give equality (Axler 2026, p. 201).

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.Axler, S. Measure, Integration & Real Analysis. Author-hosted revision, June 12, 2026. https://measure.axler.net/MIRA.pdf.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1092 and 1099, 2000.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. "Minkowski's Inequality" and "Minkowski's Inequality for Integrals." §2.11, 5.7, and 6.13 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 30-32, 123, and 146-150, 1988.Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: Dover, p. 118, 2000.Minkowski, H. Geometrie der Zahlen, Vol. 1. Leipzig, Germany: pp. 115-117, 1896.Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, p. 33, 1991.

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Minkowski's Inequalities

Cite this as:

Weisstein, Eric W. "Minkowski's Inequalities." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MinkowskisInequalities.html

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