# Questions tagged [jensen-inequality]

For questions about proving and using Jensen's inequality for convex functions. To be used necessarily with the [inequality] tag.

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### Help me with this probability theory inequality---I have been stuck for weeks

I'm a bit stuck on a proof that seems to hold when I test in in graphing calculators, would love to hear some tips on strategies I might use. Suppose we have two finite sets $\Omega$ and $\Phi$, with ...
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### JBMO-$2014$ Inequality question [duplicate]

Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $${\left(a+\frac{1}{b}\right)^2}+{\left(b+\frac{1}{c}\right)^2} +{\left(c+\frac{1}{a}\right)^2}≥3(a+b+c+1)$$ My solution: By Jensen'...
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### A 'continuous version' of Jensen's inequality

Let $X$ denote a random variable that is smoothly distributed on $[0, 1]$ with PDF $f$. Consider $$g(c) = \mathbb{P}(X< c) \phi(\mathbb{E}[X|X < c]) + \int_c^1 \phi(x)f(x) dx$$ where $\phi$ is ...
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### Prove that $\frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z} \geq \frac{1}{2}$

Assume that positive numbers a, b, c, x, y, z satisfy $cy+bz =a; az + cx = b$$bx + ay = c$. Prove that $\frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z} \geq \frac{1}{2}$ I've tried appling ...
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### Counterexample for Jensen's inequality for rank one convex function

We all know the Jensen's inequality is for all probability measures $\mu$ and all convex $h:\mathbb{R}^N\to\mathbb{R}$ it holds that \begin{equation} h\left(\int Xd\mu(X)\right)\leq \int h(X)d\mu(X). \...
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### Jensen's inequality but with geometric mean

If the inequality $f({\sqrt x}{\sqrt y})⩽{\sqrt f(x)}{\sqrt f(y)}$ is satisfied for all non negative $x,y$ in the domain, what can we say about the convexity of $f$ ? Or are there any other properties ...
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### Is $C^2$ necessary for convexity $\implies$ $f''(x) \geq 0$?

Let $f$ be a twice differential function defined on open interval $(a,b) \subseteq R$. Let $[c,d] \subseteq (a,b)$. I am able to prove that if $f''(x) \geq 0$ on $[c, d]$ then $f$ is necessarily ...
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Let $d_p(u(x), v(x)) = \left( \int_0^m \left| u(x)-v(x) \right|^p dx \right)^{1/p}$ be the distance according to the $\ell_p$ norm of two functions $u,v:[0,m]\to [0,M]$. Is the following true for the ...
The only proof of Jensen inequality (and most general version) that I know is a direct consequence of the Fenchel-Moreau Theorem : If $X$ is a locally convex Hausdorff topological space, let $\mu$ be ...