Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space L p (μ), and also to establish that L q (μ) is the dual space of L p (μ) for p ∈ [1, ∞). Hölder's inequality (in a slightly different form) was first found by Leonard James Rogers . Se mer In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of L spaces. The numbers p and q … Se mer Statement Assume that 1 ≤ p < ∞ and let q denote the Hölder conjugate. Then for every f ∈ L (μ), where max indicates that there actually is a g maximizing the … Se mer Two functions Assume that p ∈ (1, ∞) and that the measure space (S, Σ, μ) satisfies μ(S) > 0. Then for all … Se mer It was observed by Aczél and Beckenbach that Hölder's inequality can be put in a more symmetric form, at the price of introducing an extra vector (or function): Let Se mer Conventions The brief statement of Hölder's inequality uses some conventions. • In the definition of Hölder conjugates, 1/∞ means zero. Se mer For the following cases assume that p and q are in the open interval (1,∞) with 1/p + 1/q = 1. Counting measure For the n-dimensional Se mer Statement Assume that r ∈ (0, ∞] and p1, ..., pn ∈ (0, ∞] such that $${\displaystyle \sum _{k=1}^{n}{\frac {1}{p_{k}}}={\frac {1}{r}}}$$ where 1/∞ is interpreted as 0 in this equation. Then for all … Se mer Nettet14. feb. 2016 · Cauchy Schwarz inequality can be generalized as follows: \begin{equation}\label{d} x^\top y \leq \ x\ \ y\ _{\star}, \forall x,y \in \mathbb{R^{n}} …

$p-$ Norm for Integrals $\\int_{a}^b$ and Hölder and Minkowski- inequality

Nettet1 Answer Sorted by: 1 Let C be a cone and C ∗ = { y: x, y ≥ 0 ∀ x ∈ C } its dual cone. If a point y satisfies x, y ≥ 0 for all extreme rays of C, then it satisfies this inequality for all rays of C. Therefore, we can restrict attention to the extreme rays of C. Each of these rays determines a half-plane { y: x, y ≥ 0 }. NettetIn mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequalitybetween integralsand an indispensable tool for the study of Lpspaces. Hölder's inequality — Let (S, Σ, μ)be a measure spaceand let p, q∈[1, ∞]with 1/p+ 1/q= 1. ‖fg‖1≤‖f‖p‖g‖q.{\displaystyle \ fg\ _{1}\leq \ f\ _{p}\ g\ _{q}.} slaying the badger

When does the equality hold in the Holder inequality?

Nettet$\begingroup$ It is not obvious how your consideration of three vectors relates to the statement of Holder's inequality (in Euclidean spaces) which involves two vectors and not three $\endgroup$ – Martin Geller Nettet1. jan. 1991 · The well known Hölder inequality involves the inner product of vectors measured by Minkowski norms. In this paper, another step of extension is taken so that a Hölder type inequality may apply to general, paired non-Euclidean norms. We restrict the discussion to finite dimensional spaces. Nettet27. mar. 2015 · The Hölder inequality generalizes the Cauchy-Schwarz inequality to arbitrary 1 ≤ p ≤ ∞ : f, g ≤ ‖ f ‖ p ‖ g ‖ q where q is the number satisfying 1 / p + 1 / q = 1, so p = q q − 1 and q = p p − 1. This immediately gives us your desired inequality, x, y ≤ ‖ x ‖ q / ( q − 1) ‖ y ‖ q slaying the badger book review