# Tagged Questions

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### Integral Hölder bound

I was wondering if it is possible to find the following bound or if not, find a counterexample of it. Let $f\in C_0^1$ (compactly supported continously differentiable, in particular $\alpha$-Hölder ...
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### Easy question on integrals

I have some problems understanding this inequality: $$\int_{x-\varepsilon x}^x \vartheta\left(t\right)dt \leq \vartheta\left(x\right)x\varepsilon$$ where $\vartheta\left(x\right)$ is the Čebyšëv (or ...
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### Inequality $\int^{1}_{0}(u(x))^2\,\mathrm{d}x \leq \frac{1}{6}\int^{1}_{0} (u'(x))^2\,\mathrm{d}x+\left(\int_{0}^{1} u(x)\,\mathrm{d}x \right)^2$

I've been scratching my brain on this one for about a week now, and still don't really have a clue how to approach it. Show that for $u \in C^1[0, 1]$ the following inequality is valid: ...
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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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### Geometric interpretation of an integral inequality

Let $f: [a, b] \to \mathbb [0, \infty)$ be an integrable function. By Cauchy-Schwartz: $$\left(\int_a^b f(x) dx\right)^2 \leq (b-a) \int_a^b f(x)^2 dx$$ with equality iff $f$ is constant. If we ...
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### Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
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### Inconventional Integral inequality

$$\int_a^bw(x)|f(x)||g(x)|\;dx \le \left(\int_a^bw(x)\;dx\right) \max_{a\le x\le b}|f(x)|\cdot \max_{a\le x\le b}|g(x)|$$ I don't really understand this integral inequality. How do I go about ...
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### Show that $\int_{\pi/4}^{\pi/2} \frac{\sin x}{x}\,dx\leq \frac{\sqrt{2}}{2}$

Show that $$\int_{\pi/4}^{\pi/2} \dfrac{\sin x}{x}\,dx\leq \dfrac{\sqrt{2}}{2}$$ Any Ideas, how to start ?!
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### Positive integral everywhere implies positive function a.e

I would like to get feedback on my demonstration of this simple statement : Let $f$ be an integrable function on the measure space $(X,S,\mu)$. \begin{align} \text{If }\int_E f \, d\mu \geq 0\text{ ...
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### How to prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$

Prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$ by using Riemann integral?
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### Asymptotic behaviour of the integral of the quadratic mean of the coordinates on the hypercube

I have to compute the limit $\lim_{n\to +\infty}I_n$, where: $$\qquad I_n=\int_{[0,1]^n}\sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2}\,d\mu.$$ I believe that its value is just $\frac{1}{\sqrt{3}}$, since the ...
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### Why is Volume^2 at most product of the 3 projections?

Is there a simple proof for $$\text{Vol}^2(P)\le \prod_{i=x,y,z} \text{Area}(\text{Proj}_i(P)),$$ where $P\subset \mathbb R^3$ and $\text{Proj}_z(P)$ denotes the projection of $P$ to the $z=0$ ...
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### Triangle inequality for integrals with complex valued integrand

This is a step in a lecture note I'm reading. It should be simple because the author considers it obvious but I can't see it. What am I missing? Suppose $U$ and $V$ are integrable over measure space ...
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### Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
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### How can we prove $\int_1^\pi x \cos(\frac1{x}) dx<4$ by hand?

Is there any way we can prove this definite integral inequality by hand: $$\int_{1}^{\pi}x\cos\left(1 \over x\right)\,{\rm d}x < 4$$ I don't where to start even, please help. That ...
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### Differential equation with sec

With $(a)$ I got that $-y^2 dx = \sec^2x\ dy$, but it makes no sense. Hence, no Idea how to handle $(b)$.
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### $\int_0^1\frac{f^2(t)}{t(1-t)}dt \leq \frac{1}{2}\int_0^1 f'(t)^2 dt$

Let $f\in C^1([0,1],\mathbb R)$ such that $f(0)=f(1)=0$ Prove that $\displaystyle \int_0^1\frac{f^2(t)}{t(1-t)}dt \leq \frac{1}{2}\int_0^1 f'(t)^2 dt$ First thing that bothers me is that the ...
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### How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$ [duplicate]

Let $f$ be $C^1$ in $[-\pi, \pi]$ and satisfies $\int_{-\pi}^\pi f(x)dx=0$, periodic boundary condition. Then, prove that $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$. I try to prove ...
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### Show $\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$

Assume $f$ and $g$ are monotonically increasing on $[0,a]$, Show that $$\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$$ If I differentiate both sides w.r. to $a$ then; ...
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### Integral inequality with first two moments equal to $1$.

Let $f\in \mathcal{C}^0([0,1],\mathbb{R})$ such that $$\int_0^1 f(x)\text{d}x = \int_0^1 xf(x)\text{d}x=1.$$ Show that $\int_0^1 f(x)^2 \ge 4$. I tried to use Cauchy-Schwartz inequality such that ...
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### Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}}$$ Clearly I have to use some kind of inequality, but ...
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### Prove a inequality about integral and summation

If $f(x)$ is monotonic increasing on the interval $a\leq x < \infty$, could we prove following inequality formally? f(a+k) \leq \int_{a+k}^{a+k+1} f(t) dt \leq f(a+k+1) ...
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### An integral inequality question.

If we have two functions $f,g:[a,b]\to\mathbb{R}$ and we know they are bounded, so: $\sup_{x\in[a,b]}|f(x)|=K$, and $\sup_{x\in[a,b]}|g(x)|=M$. Where $K,M$, are positive finite constants, which of ...
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Let $f:[0, 1]\rightarrow \mathbb{R}$ be a function that is continous on $[0,1]$ and derivable on $(0, 1)$. If $\int_0^1 f(x)^2\le 1$ and $\int_0^1 f'(x)^2\le 1$, show that $\left|f(x)\right|\le ... 1answer 116 views ### Integral inequality with a function twice differentiable Let$f:[0,1]\longrightarrow\mathbb{R}$be a function twice differentiable with continous second derivative and$f(1)=f(0)$. The inequality: $$\int_{0}^{1}(f''(x))^2dx\geq ... 1answer 37 views ### Integral inequality in \Bbb R^n I came across this problem : Let f\colon [a,b]\rightarrow \mathbb{R}^n a continuous vector valued function. Then it is true that:$$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int ... 2answers 102 views ### Trigonometric Inequality$\cos 1 +\cos2+\ldots +\cos n < 0.55$can be solved with the help of Integrals? How can I prove for every$n \in \mathbb{N}$$$\cos 1 +\cos2+\ldots +\cos n < 0.55$$ Any idea, any solution? Thanks! EDIT Can be solved this inequality with the help of integrals, because I met ... 2answers 94 views ### Show that if$\displaystyle\int_0^1f(x)dx=a$, then$\displaystyle\int_0^1\sqrt{f(x)}dx\ge a^{2/3}f$is continuous on$[0,1]$and there is$a>0$such that,$0\le f(x)\le a^{2/3}$for$x\in[0,1]$. Show that if$\displaystyle\int_0^1f(x)dx=a$, then$\displaystyle\int_0^1\sqrt{f(x)}dx\ge ...
I have a short question. how can i proof $\int_{a}^b \vert \alpha x + \beta \vert^2 \, dx \leq (b-a) \left( \vert \alpha a + \beta \vert^2 + \vert \alpha b + \beta \vert^2 \right)$.