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Suppose $X$ is a finite set and $f:X\rightarrow \mathbb{R}$ satisfies $\sum_{x\in X}f(x)=0$. Let $p\in\Delta(X)$ be a probability measure on $X$. Does the following statement hold? $$ \sum_{x\in X} f(x)(p(x))^2 \ge 0 \Rightarrow \sum_{x\in X} f(x)p(x) \ge 0. $$

I find it hard to find any counterexamples. I know that when $|X|=2$ the statement holds. Also, Cauchy-Schwarts does not seem to lead me anywhere. I feel like some generalized version of Jensen's inequality could work?

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Counter-example: $X=\{-1,0,1\}, f(-1)=-1, f(0)=\frac 5 8, f(1)=\frac 3 8, p(-1)=\frac 1 3 , p(1)=0, p(1)=\frac 2 3$.

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  • $\begingroup$ Thanks! I am quite stupid it seems. $\endgroup$
    – Lemma1
    Feb 28 at 9:31

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