Talk:Hölder's inequality

message: Following some discussion on Talk:Minkowski inequality I wonder if the inequality can still be proved if we allow one of \|f\|_p and \|g\|_q to be infinite. Or perhaps we can even allow both f and g to have infinite norm? --MarSch 13:29, 2 October 2007 (UTC)
replies: text: Okay, yeah, R.e.b. is right when he says it holds trivially. Still it would be nice to mention for completeness and because the proof of the Minkowski inequality uses this. --MarSch 16:59, 2 October 2007 (UTC)
thread_title:  does it hold in case of infinite norm? 

message: I have removed the inequality which seems trivial and has nothing to to with Holder's.
* One can also apply the inequality to a set of a set of numbers aij, where i is in the interval [1,m] and j is in the interval [1,n]. In this case,
:\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}
Basically, for a_{ij}\geq 0 and m>1, because of the convexity of m-th root, we have
:\prod_i \sum_j (\cdot) \geq \sum_j\prod_i(\cdot)\geq \left[\sum_j\left(\prod_i(\cdot)\right)^{1/m}\right]^m
Or I am missing something? (Igny (talk) 19:05, 30 January 2008 (UTC)
replies: text: \sum_{j=1}^{n}\prod_{i=1}^m a_{ij}\leq
\prod_{i=1}^{m}\left(\sum_{j=1}^{n} a_{ij}^{p_i}\right)^{1/p_i}
which follows from generalized Holder's inequality. (Igny (talk) 02:56, 1 February 2008 (UTC)
thread_title:  Removed inequality 

message: I added this "converse inequality" that was not exactly given by the article, and that is useful for the duality of Lp spaces.
Bdmy (talk) 23:22, 2 November 2008 (UTC)
thread_title:  Converse inequality, extremal equality? 

message: I think in that case, we should use the conjugate of g.  —Preceding unsigned comment added by 201.223.189.127 (talk) 06:41, 16 December 2008 (UTC)
replies: text: No. Why adding a useless "bar" in these inequalities? You may apply the given inequality to \overline g, if you need to. Bdmy (talk) 12:25, 16 December 2008 (UTC)
thread_title:  What about f and g are complex valued functions? 

message: I added this section to the page.  The distinction is almost trivial, but it thoroughly confused me when I first saw the inequality, since some people use Wikipedia's convention and others use the inner product version, and nobody seems to mention the difference.  Saying that Cauchy-Schwarz follows with p=q=2 doesn't help (since this seems to hint that ||fg||1=|(f,g)| ).  I imagine that this is what confused the person (201.223.189.127) who started the section just above this one on the talk page.  If someone thinks that this should be integrated into the article in a different way then fair enough, but I feel strongly that there should at least be a mention of it. Quietbritishjim (talk) 13:57, 2 May 2009 (UTC)
thread_title:  Inner product form 

message: The page mentions that Holder's inequality is used to prove Minkowski's inequality. This is true for the proof given on the Minkowski inequality Wikipedia page, but it doesn't seem necessary at all; as best I can tell, the Minkowski Inequality follows easily from the convexity of t^p on the interval [0,\infty) for 1 \leq p < \infty. Royden's Real Analysis (a standard textbook) seems to agree with me, and in fact it proves Minkowski before it proves Holder. Is there some subtlety I'm missing? Obscureeconomist (talk) 03:00, 6 September 2010 (UTC)
thread_title:  Used to prove Minkowski's Inequality? 

message: *The generalization of Holder's inequality(58.187.37.209 (talk) 19:17, 11 December 2010 (UTC)
thread_title:  See also 

message: Page 55 of Terrance Tao's Topics in Random Matrix Theory (freely available for download), he gives the noncommutative Holder's inequality |tr(AB)| <= ||A||_Sp ||B||_Sq for 1/p + 1/q =1 and A,B hermitian and Sp the p-Schatten norm.  This should probably be included under generalizations  — Preceding unsigned comment added by 68.7.130.63 (talk) 08:54, 3 February 2016 (UTC)
thread_title:  Noncommutative Holder's Inequality 

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thread_title:  External links modified 

message: An editor has identified a potential problem with the redirect :Hölder conjugates and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 May 11#Hölder conjugates until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  1234qwer1234qwer4 16:14, 11 May 2022 (UTC)
thread_title:  "Hölder conjugates" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: The article has a lot of proofs. I've added some more proof environments to make it easier to see what's prose and what's proofs. Thomasda (talk) 19:30, 31 October 2022 (UTC)
thread_title:  Proof sections 

message: I added a third proof that is more constructive and shows how one "discovers" a proof. It explains where and why the Holder exponent comes in. 129.137.96.17 (talk) 18:39, 22 April 2024 (UTC)
thread_title:  Explaining the conjugate