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In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality relating Lp spaces: let S be a measure space, let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, let f be in Lp(S) and g be in Lq(S). Then fg is in L1(S) and
- <math>\|fg\|_1 \le \|f\|_p \|g\|_q.<math>
By choosing S to be the set {1,...,n} with the counting measure, we obtain as a special case the inequality
- <math>\sum_{k=1}^n |x_k y_k| \leq \left( \sum_{k=1}^n |x_k|^p \right)^{1/p} \left( \sum_{k=1}^n |y_k|^q \right)^{1/q}<math>
valid for all real (or complex) numbers x1,...,xn, y1,...,yn. By choosing S to be the natural numbers with the counting measure, one obtains a similar inequality for infinite series.
For p = q = 2 results a special case of the Cauchy-Schwarz inequality.
Hölder's inequality is used to prove the generalization of the triangle inequality in the space Lp, the Minkowski inequality, and also to establish that Lp is dual to Lq.
Generalizations
Hölder's inequality can be generalized to lessen requirements on its two parameters. While p ≥ 1 and q ≥ 1, with 1/p + 1/q = 1, one has in terms of any two positive numbers, r > 0 and s > 0:
- <math> \| f ~ g \|_{\frac{1}{1/r + 1/s}} \le \| f \|_{r} ~ \| g \|_{s}, <math>
provided only that the integrability conditions can be generalized as well, namely that f is in Lr(S) and g is in Ls(S).
The latter inequality can further be expressed as a sequence of two separately stronger inequalities (for any two positive real numbers, r and s):
- <math> \| f ~ g \|_{\frac{1}{1/r + 1/s}} \le \| f \|_{\frac{2}{1/r + 1/s}} ~ \| g \|_{\frac{2}{1/r + 1/s}}, <math>
that is, an instance of the Cauchy-Bunyakovski-Schwarz inequality, and
- <math> \| f \|_{\frac{2}{1/r + 1/s}} ~ \| g \|_{\frac{2}{1/r + 1/s}} \le \| f \|_{r} ~ \| g \|_{s}. <math>de:Hlder-Ungleichung