Hölder's inequality

Hölder's inequality bounds the sum or integral of products by the product of corresponding norms with conjugate exponents. It generalizes the Cauchy–Schwarz inequality.

Conjugate exponents: Numbers p and q greater than one are conjugate when 1/p + 1/q = 1. Their reciprocal relation determines which norms can bound a product.
Lᵖ space: A space of measurable functions whose absolute values have finite p-th power integral, with suitable modifications for p = infinity. Hölder bounds pairings between functions in conjugate Lᵖ spaces.
Minkowski inequality: The triangle inequality for Lᵖ norms and related sequence norms. Its standard proof applies Hölder's inequality to expand a power of a sum.
Generalized Hölder inequality: An extension bounding integrals or sums of multiple factors using several exponents whose reciprocals sum to one. It applies the same conjugate-exponent principle to more than two factors.
Hölder's inequality for sums: A discrete form bounding the sum of paired products by the product of sequence norms. It states the inequality directly for finite or infinite sequences.
p-norm: A measure of vector or function size defined by aggregating the p-th powers of absolute values. The inequality compares the product pairing against two such size measures.
Convolution inequality: An inequality bounding the norm of a convolution by norms of its input functions. Hölder estimates inside the convolution integral help establish norm bounds.
Reverse Hölder inequality: A family of estimates giving lower bounds on averages of powers under additional conditions. Despite its name, it reverses the usual bound only under extra hypotheses.
Hölder's inequality for integrals: An integral form bounding the integral of a product by the product of function norms. It extends the same exponent rule from sums to integrals.
Lebesgue integral: An integral defined by approximating measurable functions with simple functions and measuring their level sets. The integral version requires a framework for integrating products of measurable functions.

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