Hölder inequality (integrals)

∫|fg| ≤ (∫|f|^p)^(1/p)(∫|g|^q)^(1/q) for conjugate exponents

Hölder inequality (integrals)

Proposition (Hölder inequality for integrals). Let $f,g:[a,b]\to\mathbb{R}$ be measurable functions with $f\in L^p[a,b]$ and $g\in L^q[a,b]$ , where $p>1$ , $q>1$ , and $1/p+1/q=1$ . Then

\int_a^b |f(x)g(x)|\,dx \le \left(\int_a^b |f(x)|^p\,dx\right)^{1/p} \left(\int_a^b |g(x)|^q\,dx\right)^{1/q}.

Context. This is the continuous analogue of Hölder's inequality for sums and is foundational for $L^p$ estimates and duality.

Proof sketch. Assume the right-hand side is finite and normalize so the $L^p$ and $L^q$ norms are 1. Apply the pointwise inequality from weighted AM–GM (equivalently Young’s inequality) to obtain $|f(x)g(x)|\le |f(x)|^p/p+|g(x)|^q/q$ almost everywhere, then integrate and rescale.