# Questions tagged [jensen-inequality]

For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.

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### Contraction of Bellman Operator under general $L_p$ norms

We know that the Bellman Operator $$TV(s) = \max_a r(s,a) + \sum_{s' \in S}p(s'|s,a)V(s')$$ is a contraction under $L_\infty$ norm.For reference one can see the following link Proof that Bellman ...
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### Is there a “reverse” Jensen's inequality up to a constant?

Specifically I was wondering if for fixed $0<p<1$ there exists a constant $C$ such that $(E|X|)^p\leq C_pE(|X|^p)$ for any random variable $X$, giving an inverse of Jensen's inequality. From ...
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### Relating Fatou's Lemma and Jensen's Inequality

If there is a sense in which $\liminf$ is a "concave" function, then we would expect Fatou's Lemma as a consequence of Jensen's inequality: Is there any way to make this precise? For a small ...
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### Proof involving Jensen's inequality on random variables [closed]

Is it possible to use the Jensen's inequality to state the following: $$E\left(\frac{1}{aX+b}\right)> \frac{1}{E(aX+b)}$$ where $X$ is a random variable and $a$ and $b$ are constants? If not, is ...
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### Multivariate Jensen's inequality in a probabilistic setting: When does equality hold?

I am having trouble understanding the concept of Jensen's inequality in a probabilistic setting with multiple variables. Specifically, I am interested in cases where equality holds. I understand that ...
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### Jensen Inequality and Expected value

I am trying to proof that: Let f $\in$ $C^2(R,R)$,$f''>0$, and $E[f(X)]=f(E[X])$. I am trying to proof that X should be a constant. I know that $f(E[X])\leq E[f(X)]$, this holds because of the ...
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