# Tagged Questions

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### $L_2$ error between a non-negative monotone function and its mean?

I have been recently trying to prove a lemma which seems true in every single example I have tried, yet that I didn't manage to prove so far unless making extra (not desirable) assumptions. A ...
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### Integral Inequality involving the Euclidian Norm

I have spent several hours trying to establish the inequality shown in the attached photo. Here we assume that $\vec{r}(t)$ is a vector function in $R^n$, and is integrable on $[a,b]$. I am in need of ...
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### How to prove Godunova's inequality?

Let $\phi$ be a positive and convex function on $(0,\infty)$. Then $$\int_0^\infty \phi\left(\frac{1}{x}\int_0^x g(t)\,dt\right)\frac{dx}{x} \leq \int_0^\infty \phi(g(x))\frac{dx}{x}$$ The ...
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### $|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

Let $f\in C^1([0,\pi],\mathbb R)$ such that $\displaystyle\int_0^\pi f(t) dt=0$ Prove that $\forall x\in [0,\pi],\displaystyle|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2(t)dt}$ Failed ...
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### estimating $L^p$ norm of $\frac{x}{|x|^3} \ast (-\text{div}_x \ldots)$

I've been having problems with following an argument in a paper I'm reading, I hope someone can help me understand the point. Suppose $f(t,\cdot,\cdot)$ is a $C_0^1 (\mathbb{R}^6)$ function for all ...
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### How can we prove this integral inequality ? $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin{(2n+1)t}}{\sin{t}}\right|dt<\pi\left(1+\frac{\ln{n}}{2}\right)$

Use this $$\dfrac{1}{2}+\sum_{k=1}^{n}\cos{(kx)}=\dfrac{\sin{\left(n+\dfrac{1}{2}\right)x}}{2\sin{\dfrac{x}{2}}},x\neq 2m\pi,m\in\mathbb{Z}$$ to show that ...
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### Prove $|\int_a^b$$f(x)dx| \leq \int_a^b$$|f(x)|dx$

Prove $$\left|\int_a^b f(x)dx\right| \leq \int_a^b |f(x)|dx.$$ My thoughts: first I think we must show that if $f \geq 0$ is Riemann integrable on $[a,b]$, then $\int_a^b f(x)dx \geq 0$. Then we ...