If 
,
then 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).](/images/equations/MinkowskisInequalities/NumberedEquation1.svg) |
Similarly, if 
and 
,
,
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).](/images/equations/MinkowskisInequalities/NumberedEquation2.svg) |
Equality holds iff the sequences 
, 
, ... and 
, 
, ... are proportional.
Minkowski's Inequalities
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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.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.Referenced on Wolfram|Alpha
Minkowski's InequalitiesCite this as:
Weisstein, Eric W. "Minkowski's Inequalities." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MinkowskisInequalities.html