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Hölder's Inequalities

Hölder's inequalities bound the integral of a product in terms of integrals of powers of its factors. On a measure space, if f in L^p and g in L^q with conjugate exponents 1<=p,q<=infty, then

||fg||_1<=||f||_p||g||_q.
(1)

Here L^p denotes an Lp-space. Conjugacy includes the endpoints p=1, q=infty and p=infty, q=1, where the infinity norm is the essential supremum (Axler 2026, pp. 196-197). For finite exponents, let

1/p+1/q=1
(2)

with p, q>1. For measurable functions with finite L^p and L^q norms, respectively, 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).
(3)

When both norms are positive, equality holds iff, almost everywhere,

|g(x)|=c|f(x)|^(p-1)
(4)

for a constant c>0 (Axler 2026, p. 199). If either norm is zero, equality holds automatically. For p=q=2, this is Schwarz's inequality applied to |f| and |g|; the usual bound on |intfg^_| also uses the triangle inequality for the integral.

Similarly, for real or complex a_k and b_k, 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).
(5)

If both vectors are nonzero, equality holds iff

|b_k|=c|a_k|^(p-1)
(6)

for some c>0 and every k. A zero vector gives equality automatically. For p=q=2, this is Cauchy's inequality applied to the absolute values of the entries.

See also

Cauchy's Inequality, Schwarz's Inequality

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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. "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.

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Hölder's Inequalities

Cite this as:

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

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