# Tag Info

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### Integral inequality: Prove $\int_0^1 f\int_0^1 1/f \leq 1$ for a certain function $f$

Matematika, shmatematika: any minimally decent CAS should immediately detect and tell the human operator that, for $r=1$, the inequality reads $$(\int_X F-\int_Y G)(\int_X 1/F-\int_Y 1/G)\le 1$$ ...
• 54.4k

### Integral inequality: Prove $\int_0^1 f\int_0^1 1/f \leq 1$ for a certain function $f$

The inequality in question is equivalent to the following: $$L(X):=Ef(X)\,E\frac1{f(X)}\le1,$$ where $$f(x):=x\sqrt{1-r/x^2}$$ and $X$ is a random variable (r.v.) such that $X^2>r$ and $EX=0$. By ...
• 85.3k

• 2,968

### Proof of a matrix implication

By considering cases of real/complex eigenvalues, the problem is reduced to a system of polynomial inequalities in each of the four cases. For example, the case of real eigenvalues of $A^2B$ and non-...
• 27.5k
1 vote

### Is there an inequality relation between KL-divergence and $L_2$ norm?

This is probably obvious, but just wanted to explicitly mention the following. It is perhaps more natural to look at the modified $L_2$ norm $L_2'(p,q) = L_2(\sqrt{p}, \sqrt{q})$. Note that ...
• 1,033
1 vote

### Elementary convexity example

Here is a much simpler proof, actually of the more general fact that $$f(x):=x^p(1+\ln^+ x)^s$$ is convex in $x\ge0$ for any real $p\ge1$ and $s\ge0$, where $\ln^+ x:=\ln\max(1,x)$. For the left and ...
• 85.3k