In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Clarkson's inequalities are results in the theory of L^p spaces

Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

. They give bounds for the L^p-norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

s of the sum and difference of two measurable function

Measurable function

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

s in L^p in terms of the L^p-norms of those functions individually.

Statement of the inequalities

Let (X, Σ, μ) be a measure space; let f, g : X → R be measurable functions in L^p. Then, for 2 ≤ p < +∞,


For 1 < p < 2,


where


i.e., q = p ⁄ (p − 1).

The case p ≥ 2 is somewhat easier to prove, being a simple application of the triangle inequality

Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....

 and the convexity

Convex function

In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

of