Comments on: Means and inequalities http://www.johndcook.com/blog/2009/03/23/inequalities-means/ The blog of John D. Cook Sat, 11 Feb 2012 01:10:06 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: Solution to Renaissance problem — The Endeavour http://www.johndcook.com/blog/2009/03/23/inequalities-means/comment-page-1/#comment-118446 Solution to Renaissance problem — The Endeavour Wed, 30 Nov 2011 13:29:05 +0000 http://www.johndcook.com/blog/?p=1842#comment-118446 [...] Means and inequalities The middle size of the universe [...] [...] Means and inequalities The middle size of the universe [...]

]]> By: Giles Warrack http://www.johndcook.com/blog/2009/03/23/inequalities-means/comment-page-1/#comment-41247 Giles Warrack Mon, 05 Jul 2010 20:47:18 +0000 http://www.johndcook.com/blog/?p=1842#comment-41247 A really excellent source of these types of inequalities is Chapter 2 of Bela Bollobas's "Linear Analysis". It summarizes quite a lot of Hardy, Littlewood and Polya, but with rather more up-to-date notation. A really excellent source of these types of inequalities is Chapter 2 of Bela Bollobas’s “Linear Analysis”. It summarizes quite a lot of Hardy, Littlewood and Polya, but with rather more up-to-date notation.

]]> By: Mycroft Holmes http://www.johndcook.com/blog/2009/03/23/inequalities-means/comment-page-1/#comment-15179 Mycroft Holmes Tue, 31 Mar 2009 10:46:48 +0000 http://www.johndcook.com/blog/?p=1842#comment-15179 Another useful generalization is the concept of Chisini mean: Chisini was a less-known Italian mathematician. You can read the idea here: http://en.wikipedia.org/wiki/Chisini_mean Another useful generalization is the concept of Chisini mean: Chisini was a less-known Italian mathematician. You can read the idea here:

http://en.wikipedia.org/wiki/Chisini_mean

]]> By: Mark Reid http://www.johndcook.com/blog/2009/03/23/inequalities-means/comment-page-1/#comment-14917 Mark Reid Wed, 25 Mar 2009 03:36:45 +0000 http://www.johndcook.com/blog/?p=1842#comment-14917 By "generalised" I meant exactly those cases for when r < 1. The wikipedia article on <a href="http://en.wikipedia.org/wiki/Lp_space" rel="nofollow">L_p spaces</a> talks about these generalisations for 0 ≤ r < 1. I hadn't seen the case of r < 0 before, however. By “generalised” I meant exactly those cases for when r < 1. The wikipedia article on L_p spaces talks about these generalisations for 0 ≤ r < 1. I hadn’t seen the case of r < 0 before, however.

]]> By: John Moeller http://www.johndcook.com/blog/2009/03/23/inequalities-means/comment-page-1/#comment-14914 John Moeller Wed, 25 Mar 2009 03:04:09 +0000 http://www.johndcook.com/blog/?p=1842#comment-14914 Actually, all p-norms require an absolute value: $latex \left(\sum_{i=1}^n |x_i|^r\right)^{1/r}$ That's pretty cool about r = 0. I had to plot it to convince myself it was true. The limit comes in from both directions, too. Now I'm trying to prove it for fun. Actually, all p-norms require an absolute value: $latex \left(\sum_{i=1}^n |x_i|^r\right)^{1/r}$

That’s pretty cool about r = 0. I had to plot it to convince myself it was true. The limit comes in from both directions, too. Now I’m trying to prove it for fun.

]]> By: John http://www.johndcook.com/blog/2009/03/23/inequalities-means/comment-page-1/#comment-14880 John Tue, 24 Mar 2009 12:15:11 +0000 http://www.johndcook.com/blog/?p=1842#comment-14880 Mark, These means correspond to p-norms if r ≥ 1, but not for smaller values of r. The famous theorem I refer to is the geometric mean – arithmetic mean inequality. I agree about Steele's book. It's one of my favorites. Mark,

These means correspond to p-norms if r ≥ 1, but not for smaller values of r.

The famous theorem I refer to is the geometric mean – arithmetic mean inequality.

I agree about Steele’s book. It’s one of my favorites.

]]> By: Mark Reid http://www.johndcook.com/blog/2009/03/23/inequalities-means/comment-page-1/#comment-14878 Mark Reid Tue, 24 Mar 2009 10:21:41 +0000 http://www.johndcook.com/blog/?p=1842#comment-14878 These are just generalised p-norms, right? What is the "famous theorem" you refer to? Is it a consequence of Hölder's inequality? On the topic of inequalities, if you haven't got it already I strongly recommend Steele's <a href="http://www.maa.org/reviews/cauchyschwarz.html" rel="nofollow">The Cauchy-Schwarz Master Class</a>. It's a wonderfully readable tour through inequalities and their history. These are just generalised p-norms, right? What is the “famous theorem” you refer to? Is it a consequence of Hölder’s inequality?

On the topic of inequalities, if you haven’t got it already I strongly recommend Steele’s The Cauchy-Schwarz Master Class. It’s a wonderfully readable tour through inequalities and their history.

]]> By: Nicou http://www.johndcook.com/blog/2009/03/23/inequalities-means/comment-page-1/#comment-14870 Nicou Tue, 24 Mar 2009 06:59:40 +0000 http://www.johndcook.com/blog/?p=1842#comment-14870 Of all those the one I find more counter-intuitive is M0. This is a great post. Keep up with the good work. Of all those the one I find more counter-intuitive is M0.
This is a great post. Keep up with the good work.

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