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Suppose $\varphi$ is convex on $(a,b)$. I want to show that for any $n$ points $x_1,\dots,x_n \in (a,b)$ and nonnegative numbers $\theta_1,\dots,\theta_n$ such that $\sum_{k=1}^n \theta_k = 1$ we are able to boost $\varphi$'s convexity property up to look like Jensen's Inequality for simple functions:

$$\varphi\left( \sum_{k=1}^n \theta_k x_k \right)\leq\sum_{k=1}^n \theta_k\varphi\left( x_k \right)$$

So far I observed that if $\sum_{k=1}^n \theta_k = 1$ then $(1-\theta_n) = \sum_{k=1}^n \theta_{n-1}$. So, we use this to manipulate the RHS in order to apply $\varphi$'s convexity:

$$\varphi\left(\theta_nx_n + (1-\theta_n)\sum_{k=1}^{n-1} \frac{\theta_k x_k}{1-\theta_n} \right)\leq \theta_n\varphi(x_n) + (1-\theta_n)\varphi\left(\sum_{k=1}^{n-1} \frac{\theta_k x_k}{1-\theta_n}\right)$$

I think induction is waiting for me, but I can't see the next step. Perhaps I am not thinking clearly enough about what the right hand side means in this last line. Any help would be nice.

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You are basically done. You just need to apply the inductive hypothesis on your last term.

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  • $\begingroup$ Ahh, because $\sum_{k=1}^{n-1} \frac{\theta_k}{1-\theta_n}=1$. Thanks. $\endgroup$ – Prototank May 8 '14 at 4:09

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