# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### Dominance between two functions

Let two functions $f(z)$ and $g(z)$ with $z\in[0,c]$ with $c$ a constant such that $c<b$. I'd like to check whether $f(z)-g(z)>0$. I've tried to set $f(z)$ to its minimal value and $g(z)$ to its ...
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### Positivity of a quartic form

If I have a quartic form that I can write as $$P(x,y)=(x^2/2,y^2/2,xy)M(x^2/2,y^2/2,xy)$$ where $M$ a a $n \times n$ symmetric matrix, what is the simplest way to derive whether the form is positive ...
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### Solution for an inequality

I want to solve this inequality for $z$ $$(z+1) \left(1-e^x\right)-e^y\geq 0$$ where $-\infty <x\leq \log \left(\frac{1}{z+1}\right)$ and $-\infty <y\leq 0$. I am struggling because $z$ ...
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### On some iterated inequalities and $x \geq 5$
Let $x \in \mathbb{N}$. Suppose that I have a function $f:\mathbb{N}\rightarrow\mathbb{Q}$, with initial bounds $$2 - \frac{2}{x_0} < f(x_0) = \frac{2{x_0}}{x_0 + 1} \leq 2 - \frac{5}{3x_0}.$$ ...
Does there exist $f=f(x)$ satisfying $f(x)\ge0$ for $x\in\mathbb{R}$, $f(x)=f(-x)$ for $x\in\mathbb{R}$ (i.e. $f$ is even), $\int_{\mathbb{R}} f(x)\,dx<\infty$, and \$\int_{\mathbb{R}} x^2\,f(x)\,dx&...