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Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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Upper bound of a product of sines

Consider the function $$f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$ I wonder whether it is possible to compute some nontrivial upper ...
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On polynomial equation of fourth order depending on two parameters and bound on a maximal root

I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$: \begin{eqnarray} F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p^...
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Demonstrating bound on integral

I found this question on another forum and it got me interested because of how tight the bound is: prove that $$\int_0^\infty \frac{\arctan(x)}{x^2 + 4}\,dx > \frac{\pi}{4}.$$ The difference ...
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Gauss transformation in fractional Sobolev space

Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that  \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
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