# Tagged Questions

For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.

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### Prove or disprove this inequality

Let $p, q, a$, and $b$ be natural numbers such that $p<q$, $1<b<a$ and $b\nmid a$. Is is true that $(bp+aq)^3> (a^3+b^3)q^3$? This is what I tried: expanding the left-hand side, we ...
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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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### Determining Bounds to calculate mass

Let $E$ be the solid region defined by the inequalities $x \ge 0$, $0\le z \le \sqrt(x^2 + y^2)$, $x^2 + y^2 + z^2 \le 4$ Suppose that $E$ has mass density $\mu(x,y,z) = xz$. Calculate the ...
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Let $\psi \in C_0^{\infty}(\mathbb{R}^3)$. How to prove (or where I can find this proof) that $$\int_{\mathbb{R}^3}\frac{1}{4r^2}|\psi(x)|^2d^3x\le \int_{\mathbb{R}^3}|\nabla\psi(x)|^2d^3x$$ ? ...
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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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### Rewrite order of $\int_0^1\int_0^{\sqrt{y}} \int_y^1 \, dz \, dx \, dy$ to $dx\,dy\,dz$ and $dy\,dz\,dx$

I need to change the order of $$\int_0^1\int_0^{\sqrt{y}}\int_y^1\,dz\,dx\,dy$$ to $dx\,dy\,dz$ and $dy\,dz\,dx.$ I can extract the inequalities to get $1≤z≤y$, $0≤x≤\sqrt y$, $0≤y≤1$, but I get ...
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### Inequality releating squared absolute value of an integral to the integral of the squared absolute values of the integrand

Is this inequality $\left| \int_{0}^{x} f(t) \ dt \right|^2 \leq \int_{0}^{x} |f(t)|^2 \ dx$ true for $x\in [0,1]$. In case it is how to prove it? If there is no square in both sides it is easy since ...
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### Bounds on $f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \, dx}{ \int_0^\infty \cos(b x) e^{-x^k}\, dx}$

Suppose we define a function \begin{align} f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \,dx}{ \int_0^\infty \cos(b x) e^{-x^k} \,dx} \end{align} can we show that \begin{align} |f(k ;a,b)| \...
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### A proof of $|J_{\nu}(x)|\leq x/(2\nu-1)$
I am looking for a proof of the following inequality for Bessel functions : $$|J_{\nu}(x)|\leq \frac{x}{2\nu-1} \quad \left(\text{for}~\nu>1,~0\leq x \leq \frac{\pi}{2}\right).$$ Many thanks !