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Topic: convex functions (Read 2457 times) 

trusure
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convex functions
« on: Nov 2^{nd}, 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 2^{nd}, 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 3^{rd}, 2009, 7:21am » 
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So, .. any suggestion?? How I can solve this problem ?


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Eigenray
wu::riddles Moderator Uberpuzzler
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Re: convex functions
« Reply #3 on: Nov 4^{th}, 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 measuretheoretic form.


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