TOPICS
Search

Hölder's Inequalities

Let

1/p+1/q=1
(1)

with p, q>1. Then Hölder's inequality for integrals states that

int_a^b|f(x)g(x)|dx<=[int_a^b|f(x)|^pdx]^(1/p)[int_a^b|g(x)|^qdx]^(1/q),
(2)

with equality when

|g(x)|=c|f(x)|^(p-1).
(3)

If p=q=2, this inequality becomes Schwarz's inequality.

Similarly, Hölder's inequality for sums states that

sum_(k=1)^n|a_kb_k|<=(sum_(k=1)^n|a_k|^p)^(1/p)(sum_(k=1)^n|b_k|^q)^(1/q),
(4)

with equality when

|b_k|=c|a_k|^(p-1).
(5)

If p=q=2, this becomes Cauchy's inequality.

See also

Cauchy's Inequality, Schwarz's Inequality

Explore with Wolfram|Alpha

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. "Hölder's Inequality and Its Extensions." §2.7 and 2.8 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 21-26, 1988.Hölder, O. "Über einen Mittelwertsatz." Göttingen Nachr., 38-47, 1889.Riesz, F. "Untersuchungen über Systeme integrierbarer Funktionen." Math. Ann. 69, 456, 1910.Riesz, F. "Su alcune disuguaglianze." Boll. Un. Mat. It. 7, 77-79, 1928.Rogers, L. J. "An Extension of a Certain Theorem in Inequalities." Messenger Math. 17, 145-150, 1888.Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 32-33, 1991.

Referenced on Wolfram|Alpha

Hölder's Inequalities

Cite this as:

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

Subject classifications