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Holder's Inequality

   Also found in: Wikipedia 0.02 sec.
Hölder’s Inequality 

For finite sums, it is

│a1b1 + … + anbn │

≤(│a1 │ P + … + │ an │ p)1/p (│ b1 │ q + … + │ bn │ q + … + │ bn │ q) 1/q

and for integrals,

│∫g(x)h(x) dx │ ≤[ ∫ │g(x)│ pdx]1/p[∫ │h(x)│q dx]1/q

where p > 1 and 1/p + 1/q = 1. Hölder’s inequality was established by the German mathematician O. L. Hölder in 1889 and is one of the most commonly used in mathematical analysis. For p = q = 2 it is transformed for finite sums into Cauchy’s inequality and for integrals, into Buniakovskii’s inequality.
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[v+1]]) by (16), and applying Holder's inequality we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as m [right arrow] [infinity], by the hypotheses of the Theorem and Lemma 1.
A new note on the increasing sequences by Bor, Huseyin / Tamsui Oxford Journal of Mathematical Sciences
Then, by (1) we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Applying first Abel's transformation and then using Lemma 2, we have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in order to complete the proof of the Theorem, by (3) it is sufficient to show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Whenever k > 1, we can apply Holder's inequality with indices k and k', where 1/k + 1/k' = 1.
On a new application of power increasing sequences/astmeliselt ... by Bor, Huseyin / Proceedings of the Estonian Academy of Sciences
First, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Holder's inequality implies for any p > 3 that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.
Error bounds in stochastic-geometric normal approximation by Penrose, Mathew D.; Rosoman, Tom / DMTCS Proceedings
 
 
 
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