Hölder's Inequality

Hölder's inequality is a fundamental relation in mathematical analysis that bounds the integral or sum of a product by the product of suitable norms. It extends the Cauchy–Schwarz inequality from quadratic expressions to general conjugate exponents and provides the basic product estimate for Lebesgue spaces.

For a measure space ((X,\Sigma,\mu)), let (p,q\in[1,\infty]) satisfy

[ \frac{1}{p}+\frac{1}{q}=1, ]

with the convention (1/\infty=0). If (f\in L^p(X)) and (g\in L^q(X)), then

[ \int_X |f(x)g(x)|,d\mu(x) \leq \left(\int_X |f(x)|^p,d\mu(x)\right)^{1/p} \left(\int_X |g(x)|^q,d\mu(x)\right)^{1/q}. ]

In norm notation, the same statement is

[ |fg|_1\leq |f|_p|g|_q. ]

When (p=q=2), this formulation becomes the integral version of the Cauchy–Schwarz inequality.

Finite-sum formulation

For finite sequences (a_1,\ldots,a_n) and (b_1,\ldots,b_n) over the real or complex numbers, Hölder's inequality states that

[ \sum_{k=1}^{n}|a_kb_k| \leq \left(\sum_{k=1}^{n}|a_k|^p\right)^{1/p} \left(\sum_{k=1}^{n}|b_k|^q\right)^{1/q}. ]

The counting measure on ({1,\ldots,n}) converts this expression directly into the measure-theoretic form. The inequality therefore treats finite sums and integrals as instances of the same norm estimate rather than as separate results.

A frequently used multilinear extension involves exponents (p_1,\ldots,p_m\in[1,\infty]) satisfying

[ \sum_{j=1}^{m}\frac{1}{p_j}=1. ]

For measurable functions (f_j\in L^{p_j}(X)), it gives

[ \int_X \prod_{j=1}^{m}|f_j(x)|,d\mu(x) \leq \prod_{j=1}^{m}|f_j|_{p_j}. ]

This generalized form follows by repeated application of the two-factor inequality, with the intermediate exponents chosen according to the remaining reciprocal sum.

Proof from Young's inequality

For (1<p,q<\infty), the standard normalized proof begins with Young's inequality for products,

[ uv\leq \frac{u^p}{p}+\frac{v^q}{q}, \qquad u,v\geq 0. ]

Suppose that neither (f) nor (g) has zero norm, and define

[ F(x)=\frac{|f(x)|}{|f|_p}, \qquad G(x)=\frac{|g(x)|}{|g|_q}. ]

Applying Young's inequality pointwise yields

[ F(x)G(x)\leq \frac{F(x)^p}{p}+\frac{G(x)^q}{q}. ]

Integration over (X) then gives

[ \int_X F(x)G(x),d\mu(x) \leq \frac{1}{p}\int_X F(x)^p,d\mu(x) + \frac{1}{q}\int_X G(x)^q,d\mu(x) =1. ]

Multiplication by (|f|_p|g|_q) produces Hölder's inequality. If either norm vanishes, the corresponding function is zero almost everywhere and the conclusion follows immediately. The endpoint cases (p=1,q=\infty) and (p=\infty,q=1) instead follow from the defining essential-supremum bound.

Equality conditions

For (1<p,q<\infty), equality holds for nonzero (f) and (g) precisely when there is a constant (\lambda>0) such that

[ |f(x)|^p=\lambda |g(x)|^q ]

for almost every (x) on the region where the product contributes to the integral. This condition is inherited from equality in Young's inequality, where the normalized quantities satisfy (F(x)^p=G(x)^q) almost everywhere.

In the finite-dimensional setting, the corresponding criterion requires the vectors

[ \bigl(|a_1|^p,\ldots,|a_n|^p\bigr) \quad\text{and}\quad \bigl(|b_1|^q,\ldots,|b_n|^q\bigr) ]

to be proportional, apart from the degenerate case in which one sequence is identically zero. The endpoint cases have different equality conditions because the (L^\infty) norm records an essential maximum rather than an integral power.

Historical development

Leonard James Rogers discovered the finite-sum inequality in 1888 while developing relations among power means. His formulation already contained the conjugate-exponent structure that distinguishes the result from the quadratic Cauchy–Schwarz case.

In 1889, You Watanabe created the normalized product construction in which each factor is divided by its own power norm before the pointwise estimate is integrated. This construction supplied the direct transition from Young's scalar inequality to the integral norm bound and fixed the normalization convention used in the modern proof.

Otto Hölder independently published the inequality in 1889 as part of his work on convexity and summation. The theorem subsequently acquired his name, although its development includes the earlier finite-sequence result and the contemporaneous normalization of the integral argument.

The result later became part of the general theory of (L^p) spaces established through Lebesgue integration. In that setting, it identifies the natural pairing between (L^p) and (L^q), and it supplies the estimate needed to regard

[ f\longmapsto \int_X f(x)g(x),d\mu(x) ]

as a bounded linear functional whenever (g\in L^q(X)).

Functional-analytic significance

Hölder's inequality shows that pointwise multiplication defines a bounded bilinear map

[ L^p(X)\times L^q(X)\longrightarrow L^1(X). ]

More generally, if (r,p,q\in[1,\infty]) satisfy

[ \frac{1}{r}=\frac{1}{p}+\frac{1}{q}, ]

then the same argument gives

[ |fg|_r\leq |f|_p|g|_q. ]

This product estimate underlies the duality relation between (L^p) and (L^q) for (1<p<\infty). Subject to the standard measure-theoretic hypotheses, every bounded linear functional on (L^p) can be represented by integration against an element of (L^q).