# 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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### Integrability condition of composite function in Jensen's inequality [duplicate]

In Jensen's inequality $\phi(\mathbb{E}[X])\leq \mathbb{E}[\phi(X)]$, it is required that $\phi(X)$ is integrable. But why? If this integral is infinity, the inequality is still correct.
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### Approximate Saturation of Jensen's Inequality

Let $f$ be a convex function and $X$ a random variable. Jensen's inequality states: $$\mathbb{E}[f(X)]\geq f(\mathbb{E}[X]).$$ When $f$ is an affine function, this inequality is saturated, i.e. it ...
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### Prove the following from Jensen's inequality

$n \cdot \left( \frac{n+1}{2} \right)^{\left( \frac{n+1}{2} \right)} \leqslant \sum_{k=1}^{n} k^k \text{ for } n \in \mathbb{N}$ I've tried to transform the left part of inequality, but nothing worked ...
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### Elementary proof that $\mathbb{E}[1/X] \geq 1/\mathbb{E}[X]$ for finitely supported positive random variables.

Let $X$ be a random variable with finite support contained in $(0,\infty)$. By Jensen's inequality we have $\mathbb{E}[1/X] \geq 1/\mathbb{E}[X]$. I am curious if there exists an elementary proof of ...
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### Showing $\frac1{ab+4}+\frac1{ac+4}+\frac1{ad+4}+\frac1{bc+4}+\frac1{bd+4}+\frac1{cd+4}\geq\frac65$, for positive $a,b,c,d$ with $ab+bc+cd+da=4$

Let us take $a\geq b\geq c\geq d>0$ such that $ab+bc+cd+da=4$. Show that $$\frac{1}{ab+4}+\frac{1}{ac+4}+\frac{1}{ad+4}+\frac{1}{bc+4}+\frac{1}{bd+4}+\frac{1}{cd+4}\geq\frac{6}{5}$$ $a+b+c+d \ge 4$...
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### Case of equality in Jensen's Inequality

I haven't really understood parts of the proof given for $\textrm{Problem 6.2}$ of Steele's Cauchy-Schwarz Master Class. Suppose that $f : [a, b] → \mathbb{R}$ is strictly convex and show that if \...
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### Show $\left(x_1x_2\ldots x_n\right)^{\frac{x_1+\cdots +x_n}{n}}\le x_1^{x_1}x_2^{x_2}\ldots x_n^{x_n}$

For $x_1\ldots x_n \in \mathbb{R_{+}}$ show: $$\left(x_1x_2\ldots x_n\right)^{\frac{x_1+\cdots +x_n}{n}}\le x_1^{x_1}x_2^{x_2}\ldots x_n^{x_n}$$ Convexity and logarithmic function are the first ideas ...
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### How to prove $x^{x/(1+x)}/(1+x)\geq1/2$

I need to show that $\frac{x^{\frac{x}{1+x}}}{1+x} \geq \frac{1}{2}$ for all $x\geq0$. I am fairly certain this statement is true, but have no idea how to go about proving it. I know that it attains a ...
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### Finding $\small{\min\limits_{a+b+c=2}\sqrt{a-2bc+3}+\sqrt{b-2ca+3}+\sqrt{c-2ab+3}.}$

Given non-negative real numbers $a,b,c$ satisfying $a+b+c=2.$ Find the minimal value of expression $$P=\sqrt{a-2bc+3}+\sqrt{b-2ca+3}+\sqrt{c-2ab+3}.$$ Source: AOPS-Jokehim. See also here. Here is ...
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### Prove $\frac{1}{\sqrt{2a+b+c}}+\frac{1}{\sqrt{2b+a+c}}+\frac{1}{\sqrt{2c+b+a}}\le \frac{9}{2}\cdot\frac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}}.$

Let $a,b,c>0.$ Prove that $$\frac{1}{\sqrt{2a+b+c}}+\frac{1}{\sqrt{2b+a+c}}+\frac{1}{\sqrt{2c+b+a}}\le \frac{9}{2}\cdot\frac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}}.$$ WLOG, assuming that $a+b+c=1$ and we'...
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### Does $E^2(X)/E(Y) \le E(X^2/Y)$ hold

Let $f(x),g(x)$ be $\mu$ measureable functions with $g(x) > 0$ almost surely. Does $$\frac{\left(\int f(x)\mu(x)dx\right)^2}{\int g(x)\mu(x)dx} \leq \int \frac{(f(x))^2}{g(x)}\mu(x)dx$$ hold? I ...
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### Jensen inequality with fraction expectation, determinant, and log function

I have a question when I try to have some analysis on the following expression. \begin{align} \mathbb{E}\left[\log_2\det\left(\mathbf{I}_L+\frac{1}{N_0+\sigma^2}\mathbf{X}\right)\right], \end{align} ...
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### Conditional Jensen's inequality proof correctness. Queries regarding convex functions.

Let $(Ω, \mathcal{F}, P)$ be a probability space and let $\mathcal{G} ⊂ \mathcal{F}$ be a sub-$σ$-algebra. Conditional Jensen's inequality: Let $φ : R → R$ be a convex function, $X$ and $φ(X)$ be ...
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### Single inflection point theorem

Let $f$ be a twice differentiable function on $\mathbb{R}$ with a single inflection point, let $S$ be a fixed real number and let $$g(x) = f(x) + (n-1)f(\frac{S-x}{n-1}).$$ If $x_1, x_2,...,x_n$ are ...
The original problem posed was the following: On a sphere of radius $1$ are given four points $A,B,C,D$ such that $$AB\cdot AC\cdot AD \cdot BC \cdot BD \cdot CD = \frac{2^9}{3^3}$$ I am aware of a ...