Young's inequality for products

Young's inequality for products bounds a product by a weighted sum of powers: for conjugate exponents p and q, nonnegative a and b satisfy ab ≤ a^p/p + b^q/q.

Hölder's inequality: A bound on sums or integrals of products using conjugate exponents. Applying the scalar bound term by term yields the finite-sum form of Hölder's inequality.
Hölder space: A function space whose elements satisfy a specified Hölder continuity condition. Young's inequality helps estimate products of functions with complementary integrability.
Hölder conjugate: An exponent q paired with p by 1/p + 1/q = 1. The reciprocal relation determines the exponents in the product bound.
Cauchy–Schwarz inequality: An inner-product inequality bounding the magnitude of an inner product by the product of vector norms. It bounds an inner product, whereas Young's inequality bounds a scalar product by powers.
William Henry Young: An English mathematician known for work in real analysis and measure theory. The inequality bears Young's name and is associated with his work on analysis.
Convex conjugate: A function derived from a convex function by taking the supremum of linear functions minus that function. Young's inequality is the defining inequality between a convex function and its conjugate.
Sobolev inequality: An inequality controlling a function norm by norms of its derivatives. Product estimates using Young's inequality often absorb derivative terms in Sobolev arguments.
Convex function: A function whose value along each line segment lies below the segment joining endpoint values. Convexity underlies both the power-function proof and the Fenchel–Young generalization.
Minkowski inequality: A triangle inequality for norms, including the norms of sums of functions. Minkowski controls sums in a norm; Young controls products by separate power terms.
Real analysis: The branch of mathematics studying real numbers, limits, functions, and integration. Young's inequality became a standard tool in the development of rigorous analysis.

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Rearrangement inequalityCompared with: It controls individual products through powers, not the effect of changing their pairing.