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Show by induction that if $f:\mathbb{R}\to\mathbb{R}$ is convex, then for any $x_1,\dots ,x_n$ and $\lambda_1,\dots ,\lambda_n$ with $\sum_{i=1}^n\lambda_i = 1$, $$ f\left(\sum_{i=1}^n\lambda_i x_i\right) \leq \sum_{i=1}^n\lambda_i f (x_i) . $$

I think this is similar to the Jensen's inequality, but am not sure how to formally proof this.

Thank you so much!

EditPiAf
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Liz
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  • This is an example of Jensen's inequality, assuming that the $\lambda_i$ are supposed to be non-negative. You just need to choose the probability space and the random variable $X$ appropriately. – carmichael561 Sep 25 '17 at 19:47
  • @carmichael561 I'm still a little confused, could you explain a little more as a formal answer? Thanks! – Liz Sep 25 '17 at 19:53

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I will assume that the $\lambda_i$ are non-negative, as otherwise the inequality may not hold.

Suppose that $x_1,\dots,x_n$ are fixed real numbers, and define a probability space $\Omega=\{1,2,\dots,n\}$ with $\mathbb{P}(\{i\})=\lambda_i$. Let $X$ be the random variable defined by $X(i)=x_i$. Then $$ \mathbb{E}[X]=\sum_{i}\lambda_ix_i$$ and $$ \mathbb{E}[f(X)]=\sum_i\lambda_if(x_i)$$ so the desired result follows from Jensen's inequality. As Harry49 notes, it is also possible to use induction on $n$.

carmichael561
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