# Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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### A question about mutual information

Let $A$ and $B$ be two, possibly dependent, random variables, and let $X$ be a random variable independent of $(A,B)$. For simplicity, let's concern ourselves with discrete random variables. Is the ...
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### Is $\int_0^\infty{dx\over x^{x^{x^x}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^x}}}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^{x^{x^x}}}}}}}<\cdots$ true?

On MSE I asked whether each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ was less than $2$ and received answers on bounding ...
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### Obtaining the "best possible" inequality by tuning hyper-parameters

I encountered the following problem in one of my research projects which can be encapsulated as follows. Let's say we have a set $\mathcal{C}$ of functions $f$ defined from $\mathbb R_+$ to $\mathbb R$...
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### Extremizers of the Sobolev inequality

Background: I am reading the paper: Best constant in Sobolev inequality by Talenti (see here) and I am trying to understand the following step. On p. 365, the author is arguing that the solutions to ...
There is the following exercise in Vershynin's book on High-Dimensional Probability. Exercise 3.1.6: Let $X = (X_1, \dots, X_n) \in \mathbb{R}^n$ be a random vector with independent coordinates $X_i$ ...