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        general >> wanted >> convex functions
        
(Message started by: trusure on Nov 2^nd, 2009, 7:39pm)

Title: convex functions
Post by trusure on Nov 2^nd, 2009, 7:39pm

I'm trying to prove the folowing form of Jensen's Inequality for convex functions:

" a function f is convex iff
f(sum_k=1 to inf {c_k z_k}) <= sum_k=1 to inf {c_k f(z_k)} "

where c_k>=0, sum{c_k z_k}< infinity and sum{c_k}=1 ?

I proved it if the summation was over finite, but for the infinite form: since convex functions are continuous, so it really is just taking the inequality for finite sums k=1 to n and then taking the limit as n goes to infinity we get the result.

Is that correct ? !!

thanks

Title: Re: convex functions
Post by Obob on Nov 2^nd, 2009, 9:24pm

You can make something like that work, but you have to be a little careful. If an infinite sum sums to 1, the partial sums don't also sum to 1.

Title: Re: convex functions
Post by trusure on Nov 3^rd, 2009, 7:21am

So, .. any suggestion?? How I can solve this problem ?

Title: Re: convex functions
Post by Eigenray on Nov 4^th, 2009, 11:11am

You can take the limit of the [link=http://en.wikipedia.org/wiki/Jensen%27s_inequality#Finite_form]finite form of Jensen's inequality[/link]. It's also a special case of the measure-theoretic form.