Tagged Questions

For questions about proving and manipulating functional inequalities.

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Inequality: product of integrals

Context: Proving integral inequalities about posterior distributions following different sequences of binary signals. The proofs come down to the following inequalities. Let $\psi(x)$ be a concave ...
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How to show this equality for operator norm?

Let $(X,\Sigma_\mu,d\mu)$ and $(Y,\Sigma_\nu,d\nu)$ be two positive $\sigma$-finite measure space and let $M(d\mu)$ and $M(d\nu)$ be spaces of complex-valued $d\mu$-measurable and $d\nu$-measurable ...
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Prove that $\log x<\sqrt{x}$ for $x\geq 1$

Prove that $\log x<\sqrt{x}$ for $x\geq 1$ Let $f(x)=\sqrt{x}- \log x$. So, $f(1)=1>0$. $f'(x)=\frac{1}{2\sqrt{x}}-\frac{1}{x}>0$ only when $x>4$. When I draw the graph of $f$ in ...
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How to solve $| x - a | + | x + a | < b$ where $a \neq 0$?

The options are : A ) no solution if $b\leq 2|a|$ B ) has a solution set ${ ( -b/2 , b/2 ) }$ if $b > 2|a|$ C ) has a solution set ${ ( -b/2 , b/2 ) }$ if $b < 2|a|$ D ) All of ...
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For $f(x)$ on $(a,b)$ s.t. $f'(x)+f^2(x) \ge -1$ and $\lim\limits_{x \to a} f(x) = - \lim\limits_{x \to b} f(x) = \infty$, prove that $b-a \ge \pi$

$f(x)$ is continuously differentiable here. Using separation of variables, I think I might have shown that the equality form of the statement is true, but I'm a bit wary of trying separation of ...
Prove $n^2 \leq 1.1 ^{n}$ by induction
Prove that for all $n \geq 100$ you have $n^2 \leq 1.1^n$ Base Case: $n = 100$ $(100)^2 \leq 1.1^{100}$ (True) Inductive Case: Suppose $(k-1)^2 \leq 1.1^{k-1}$ for some $k \geq 101$ Prove \$k^...