Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.Recall the Jensen's inequality for the convex function x p {displaystyle x^{p}} (it is convex because obviously p ≥ 1 {displaystyle pgeq 1} ):By Hölder's inequality, the integrals are well defined and, for 1 ≤ p ≤ ∞,We use Hölder's inequality and mathematical induction. For n = 1, the result is obvious. Let us now pass from n − 1 to n. Without loss of generality assume that p1 ≤ … ≤ pn.Note that p andDefine the random variables In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces. The numbers p and q above are said to be Hölder conjugates of each other. The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if ||fg||1 is infinite, the right-hand side also being infinite in that case. Conversely, if f  is in Lp(μ) and g is in Lq(μ), then the pointwise product fg is in L1(μ). Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space Lp(μ), and also to establish that Lq(μ) is the dual space of Lp(μ) for p ∈ [1, ∞). Hölder's inequality was first found by Leonard James Rogers (Rogers (1888)), and discovered independently by Hölder (1889).