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Topic: convex functions (Read 2456 times) |
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trusure
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convex functions
« on: Nov 2nd, 2009, 7:39pm » |
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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
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Obob
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Re: convex functions
« Reply #1 on: Nov 2nd, 2009, 9:24pm » |
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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.
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trusure
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Re: convex functions
« Reply #2 on: Nov 3rd, 2009, 7:21am » |
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So, .. any suggestion?? How I can solve this problem ?
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Eigenray
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Re: convex functions
« Reply #3 on: Nov 4th, 2009, 11:11am » |
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You can take the limit of the finite form of Jensen's inequality. It's also a special case of the measure-theoretic form.
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