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Questions tagged [integral-inequality]

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

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55 views

Prove that $0.4 ≤ \int_0^1 f(x) dx ≤ 0.5$ for $f(x) = x^{\cos x + \sin x} $. [duplicate]

Consider the function $$f(x) = x^{\cos x + \sin x} $$ defined for $x \ge 0.$ Prove that $$0.4 ≤ \int_0^1 f(x) dx ≤ 0.5.$$ I tried using max-min inequality but it didn't work.
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0answers
22 views

Exponential decay and integration

I am confronted with the following problem: Let $\mu$ be a probability measure on $\mathbb{R}$. We wish to show that for any $p \in \mathbb{N}$ and $r \in \mathbb{R}$, the integral $$ F(r):= \...
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0answers
13 views

If a rotational family is close to a single rotation must its derivative be small?

Let $\mathbb D^n$ be the closed $n$-dimensional unit disk. Let $f:\mathbb D^n \to \text{SO}(n)$ be a smooth map. Let $R \in \text{SO}(n)$ be a fixed rotation. I am trying to prove a quantitative ...
3
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2answers
44 views

If $f(a)=f(b)=0$ and $|f''(x)|\le M$ prove $|\int_a^bf(x)\mathrm{d}x| \le \frac{M}{12}(b-a)^3$

If $f(a)=f(b)=0$ and $|f''(x)|\le M$. Prove $$|\int_a^bf(x)\mathrm{d}x| \le \frac{M}{12}(b-a)^3$$ I have thought about that since $f(a) = f(b) = 0 $ there is $\xi$ such that $f'(\xi) = 0$. Then when ...
4
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2answers
45 views

If $f^2(t) \le 1+2\int_0^tf(s)\mathrm{d}s$ prove $f(t)\le 1+t$

If $f(x)$ is positive and continuous on $[0,1]$ and $f^2(t) \le 1+2\int_0^tf(s)\mathrm{d}s$, prove that $f(t)\le 1+t$. Here's my thinking. $$f^2(t) \le 1+2\int_0^tf(s)\mathrm{d}s \Rightarrow f(t)\le ...
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0answers
15 views

Can we include inequalities while solving under determined simultaneous linear equation in reduced Echelon form?

Is there a way to include the inequalities of variables in calculating family of equations while solving under determined simultaneous linear equations? For Eg: Lets say x + y = 4, y + z = 4. But ...
1
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1answer
23 views

Given a sequence of Lp functions, does the integral commute with the lp norm?

I have been struggling to prove the following: Let $ \{ f_n \}$ be a sequence in $ L^p(E) $ for some $ p \geq 1 $. Then, $$ \left( \sum_{n=1}^\infty | \int_E f_n \mathrm{d}\mu |^p \right)^{ \frac{1}...
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1answer
51 views

Prove that:$\int_{a}^{b}(2x^{3}-3(a+b)x^{2}+6abx)f'(x)dx\geq (a-b)^{3}f(a)$.

Let $f:[a,b]\rightarrow[0,\infty)$ a differentiable function with its derivative continuous and $f(a)=f(b)$. Prove that:$\int_{a}^{b}(2x^{3}-3(a+b)x^{2}+6abx)f'(x)dx\geq (a-b)^{3}f(a)$. I tried to ...
1
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1answer
46 views

$\int_0^1f(x)\cdot x^{n+1}\text{d}x > \int_0^1f(x)\cdot x^n\text{d}x \cdot \int_0^1f(x)\cdot x\text{d}x$

I have convinced myself that $$\int_0^1f(x)\cdot x^{n+1}\text{d}x > \int_0^1f(x)\cdot x^n\text{d}x \cdot \int_0^1f(x)\cdot x\text{d}x$$ is true whenever $f$ is non-negative, $\int_0^1f(x)\text{d}x=...
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2answers
22 views

prove inequality with an integral over region with unit length

I am trying to show that $\log{m}\le{\int_{m}^{m+1}{\log{t}}}dt$, with $m\ge{1}$. I tried simplifying the problem to $0\le{\int_{m}^{m+1}{\log{\big(\frac{t}{m}\big)}}}dt$, but can't seem to get any ...
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0answers
66 views

How is the inequality of these integrals true?

I'm doing an exercise from baby Rudin (chapter 8 exercise 11) and found a suggestion that it might use. $$\left|\int_0^\infty e^{-x}f\left(\frac{x}{t}\right)dx\ -1\right| \leq \int_0^\infty e^{-x}\...
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0answers
81 views

Is it a well-known inequality?

Let $f\in \mathcal{C}^2(\mathbb{R},\mathbb{R})$ and suppose that $\int_\mathbb{R}f^2<+\infty$ and $\int_\mathbb{R}f''^2<+\infty$. Then we can deduce that : $\left(\int_{\mathbb{R}}f'^2\right)...
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0answers
14 views

Linear version of Gronwall's inequality, proof

I am reading a proof of the following theorem: Assume $\phi$ is a continuous function in $[0,T]$ that satisfies $$\phi(t) = \alpha + \int_0^t (\beta \phi(s) + \gamma)ds, \hspace{0.5mm} t\in [0,T], $$ ...
3
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1answer
29 views

Help bounding a “norm”

In Weak Convergence and Stochastic Processes, the authors introduce the following notation: $$\|\xi\|_{2,1} = \int_0^\infty \sqrt{P(\xi > x)}\,\mathrm dx$$ They then admit that this is technically ...
1
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1answer
132 views

Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.

Verify that $$\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$ and $r > 0$. I'm stuck. here is my attempt: $|...
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0answers
16 views

Finding discrete solutions to inequality involving Exponential Integral

I want to identify the least natural number $n$ (of course, it suffices to solve this problem for the reals, and then take the floor) such that $$-c \text{Ei}\left(-e^{\frac{a-d}{c}} (n+1)\right)+a-...
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0answers
23 views

Integral inequality with partial derivative

Fix $\eta \in \Bbb R^n$, then for any $\phi \in C_c^\infty(\Bbb R^n)$, i.e. compactly supported smooth function, $$ \int_{\Bbb R^n} \frac{\partial}{\partial x_j} (\phi(x) (x \eta)) \, dx \le \int_{\...
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0answers
35 views

How to find the sharp constant between norms?

How to prove that $\|u\|_\infty\leq C_p\|u'\|_{L^p},\ \forall\ u\in W^{1,p},\ u(0)=u(T)$ with $\int_0^T u=0$ and $C_p=\frac{1}{2}\left[\frac{T(p-1)}{2p-1}\right]^{\frac{p-1}{p}}$? It is easy to show ...
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1answer
106 views

Application of Gronwall Inequality to existence of solutions

Consider the $N$-dimensional autonomous system of ODEs $$\dot{x}= f(x),$$ where $f(x)$ is defined for any $x \in \mathbb{R}^N$, and satisfies $||f(x)|| \leq \alpha||x||$, where $\alpha$ is a ...
1
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1answer
34 views

Integration inequality, why can we pull e^t out of the integral and leave its e?

In the following, at line 3 $e^t \sin(t)$ is pulled out of the $|\cdot|$ and left as a constant. How does one justify this step?
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1answer
23 views

Inequality used in proof of existence of SDE solutions?

In the proof of existence/uniqueness of SDE the following inequality is used: $$E\left[ \left( \int_0^t a(s,\omega) ds \right)^2 \right] \leq t E\left[ \int_0^t a(s,\omega)^2 ds \right]$$ and I ...
3
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3answers
86 views

Compute the limit $\lim_{n\to\infty} I_n(a)$ where $ I_n(a) :=\int_0^a \frac{x^n}{x^n+1}\,\mathrm{d}x, n\in N$.

For $a>0$ we define $$\space I_n(a)=\int_0^a\frac{x^n}{x^n+1}\,\mathrm{d}x , n\in N.$$ Prove that $0\le I_n(1) \le \frac{1}{n+1}$ Compute $\lim_{n\to\infty} I_n(a)$ My attempt: I regard $I_n(1)=...
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0answers
21 views

What are all functions $f(x)$ that ensure $\int_{a}^{\infty} \frac{f(x)}{\sqrt{x^2-a^2}} \, \mathrm{d} x \le 0$ for all $a$ where $0 \le a \le \infty$

I'm looking to find a set of functions $f(x)$ such that members of the set satisfy the condition $$\int_{a}^{\infty} \frac{f(x)}{\sqrt{x^2-a^2}} \, \mathrm{d} x \le 0 \qquad \textrm{for all }0 \le a \...
7
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3answers
153 views

Show that $\int_{0}^{\pi/6} {\cos (x^2)}\mathrm{d}x\ge\frac12$.

Prove that $\displaystyle\int_{0}^{\frac\pi 6} {\cos ({x^2)}\mathrm{d}x\ge\dfrac12}$. I know this is a Fresnel integral but without going into advanced calculus is there a way to show that this is ...
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0answers
35 views

Prove that if $f \in L^p$ and $g \in L^q$ where $p$ and $q$ are conjugate exponents , then $\lim_{\vert x \vert \to \infty}(f*g)(x)=0$

The convolution of $f$ and $g$ on $R^d$ equipped with the lebsgue measure is defined by $$(f*g)(x)=\int_{R_d} f(x-y)g(y) \, dy$$ Prove that if $f \in L^p$ and $g \in L^q$ where $p$ and $q$ are ...
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3answers
103 views

How to prove that $\int\limits_0^{\pi} e^{\sin^2(x)}dx > {3\over2}\pi$? [closed]

How to prove that $\int\limits_0^{\pi} e^{\sin^2(x)}\ dx > {3 \over 2}\pi$?
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1answer
50 views

An integral inequality with cosine

I tried to prove that $$\int_{a}^{b}\frac{|\cos (x)|}{x}dx\leq \frac{2}{\pi}\log\left(\frac{b}{a}\right)+O(1).$$ Of course $O(1)$ as a function of $b$, i.e. a bounded function of $b$. $a$ is ...
0
votes
1answer
28 views

Integral inequality for sin function

Let $0<r<1$ and $t\geq 0$ real numbers. Is it true that $$\int_t^{t+r} \sin(x)\, dx \leq \int_{\frac{\pi}{2}-\frac{r}{2}}^{\frac{\pi}{2}+\frac{r}{2}}\sin(x)\, dx \,? $$ I suspect that yes, ...
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0answers
27 views

Integral inequality of two variables function

Suppose $u(x,y)$ is continuous in $D=\{(x,y)| 0\le x \le 1, 0\le y \le 1\}$, $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial^2 u}{\partial x \partial y}$ is absolutely ...
1
vote
1answer
67 views

Complex Integral Inequality $\int_{-1}^{1}|f(x)|^{2}dx\leq\pi\int_{0}^{2\pi}|f(e^{i\theta})|^{2}\frac{d\theta}{2\pi}$

Question Let $$f(z) = \sum_{k=0}^{n}c_{k}z^{k}$$ be a polynomial, where $c_{k}\in\mathbb{C}$. Prove that $$\int_{-1}^{1}|f(x)|^{2}dx\leq\pi\int_{0}^{2\pi}|f(e^{i\theta})|^{2}\frac{d\theta}{2\pi} =...
1
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1answer
46 views

A definite integral inequality

Suppose $f(x)$ has continuous derivative on $[-\pi, \pi]$, $\,f(-\pi)=f(\pi)\,$ and $\,\int_{-\pi}^{\pi}\, f(x)\, dx=0$. Then prove that: $$ \int_{-\pi}^{\pi} [\,f'(x)]^2\, dx \ge \int_{-\pi}^{\pi} f^...
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1answer
24 views

$\mathbb E[S_n'^4]\le C\cdot n^2$

Let $X_1,...,X_n$ be iid random variables on $(\Omega,\mathcal A,\mathbb P),\ \ \mathbb E[X_1]=\mu\in\mathbb R,\ \ \mathbb E[X_1]^4<\infty,\ \ X_i':=X_i-\mu,\ \ S_n'=X_1'+...+X_n'.$ Prove that $\...
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0answers
47 views

Why is the laplacian a closed operator in $W^{2,p}(\mathbb{R}^n)$?

I have read that the laplacian is a closed operator in $W^{2,p}(\Omega)$,(that is, $\Delta : W^{2,p} \to L^p$) where $\Omega$ satisfies some conditions (I need the case $\Omega = \mathbb{R}^n$ so ...
0
votes
1answer
25 views

Inequality between Expectation and Quantil

For a sample of independent observations $X_1,X_2,...,X_n$ on a continuous distribution $F$, let the ordered sample values be $X_{(1)},X_{(2)},...,X_{(n)}$. From the theory of order statistics, the ...
1
vote
1answer
32 views

Integral Inequality with L-2 Norm

On page 135 of The Mathematical Theory of Finite Element Methods (Brenner and Scott), I encountered the following inequality: $\left | \int_{\Gamma} \overline{v} - v \, ds \right | \leq |\Gamma |^{...
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0answers
19 views

Integration Inequalities [duplicate]

The question we need to prove is $$1.222\le 1+3^{-2}+5^{-2}+\cdots\le 1.252.$$ I know how to prove the first part $$1+3^{-2}+...+43^{-2}\ge 1.222.$$ But I do not know how to prove the second part. If ...
5
votes
1answer
115 views

Lower bound for an integral of a polynomial over an interval of finite length

Assume that $Q$ is a degree $k$ polynomial and $I$ an interval of finite length $c$. Can you please give me some hints on how to show that $$ \left( \sum_{m=0}^k \frac{c^{m}}{m!} |Q^{(m)} (t_0) | \...
0
votes
2answers
22 views

Integral with complex perimeter – convergence.

Kinda tough one – been thinking about this integral for a while. $$\int_0^{\infty} \left| x^{-2s} \right| \text{d}x$$ for any complex $s$ and $x \in (0;\infty)$ obv. Intuition 'tells' me, this ...
2
votes
0answers
34 views

$\frac{1}{n^s}-\int_n^{n+1} \frac{\mathrm{d}t}{t^s} \leq \frac{s}{n^{s+1}}$ looking for different proof

Let $n \in \mathbb{N}_{>0}$ and $s \in \mathbb{R}_{>0}$ then I am interested in the following inequality : $$\frac{1}{n^s}-\int_n^{n+1} \frac{\mathrm{d}t}{t^s} \leq \frac{s}{n^{s+1}}$$ Here ...
4
votes
2answers
87 views

How to prove $\frac{1}{2n+2}<\int_0^{\frac{\pi}{4}}\tan^nx\,dx<\frac{1}{2n}$

How to prove $$\frac{1}{2n+2}<\int_0^{\frac{\pi}{4}}\tan^nx\,dx< \frac{1}{2n}$$ Set $A_n=\int_0^{\frac{\pi}{4}}\tan^nx\,dx$, then we have $A_n+A_{n+2}=\frac{1}{n+1}$ and we have $A_{n+2} < ...
4
votes
1answer
59 views

If $f$ is continuous and $3\geq f(x)\geq 1$ for all $x\in[0,1]$, show the integral inequality.

If $f$ is continuous and $3\geq f(x)\geq 1$ for all $ x\in [0,1]$, show that $$ 1 \leq \int_0^1f(x)dx\int_0^1\Bigg(\frac{1}{f(x)}\Bigg)dx \leq \frac{4}{3}. $$ Thanks!
2
votes
1answer
49 views

$\int_\Omega\min(f,g)\text{d}\mu\ge\frac{1}{2}\big(\int_\Omega\sqrt{f\cdot g}\ \text{d}\mu\big)^2$

Let $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space, $\mathbb P,\mathbb Q$ be probability measures on $(\Omega,\mathcal A)$ with density functions $f,g: \Omega \to (0,\infty)$ concerning $...
0
votes
1answer
20 views

Prove an inequality based on a scalar product

Let $V=\{f:[-\pi,\pi]\to\mathbb{R}| f\text{ continuous}\}$ with: $$f\times g=\int_{-\pi}^{\pi}f(x)g(x)dx$$ Show that $\forall f \in V$: $$\left|\int_{-\pi}^{\pi}f(x)\sin(x)dx\right|\leq\...
0
votes
0answers
20 views

Jensen type inequality for a non-convex function.

I suppose that a function $f$ is $\geq 0$ on $[-1,1]$, decreasing and $f(t)(1+t)$ is concave. Moreover for every $a,b \in [-1,1]$, $a<b$, we have a (simple positive) measure $\mu_{a,b}$ such that $...
0
votes
0answers
40 views

Other half of Gronwall

I am looking to show that if y(t) is continuous on (a, b) and satisfies $$y(t)\leq H+K\int_{t}^{t_{0}} y(s)ds,\quad\forall\ a<t\leq t_0$$ where $t_0 \in (a,b)$, then $$y(t)\leq He^{K(t_0-t)},\...
0
votes
1answer
40 views

Show that $\left|\oint_{\gamma_R}f(z) \ dz\right|\leq\max_{z\in\gamma_R}\left|f(z)\right||\gamma_r|.$

I have that $f(z)=\frac{e^{3iz}}{z^2+12}$ and what I actually need to show is that the integral below goes to $0$ as $R\rightarrow \infty$. The $\gamma_R$ curve is the semi circle, counterclockwise ...
2
votes
1answer
58 views

Applying Gronwall lemma in Majda-Bertozzi book.

I am studying Majda-Bertozzi book about incompressible flows. I have applied Gronwall lemma several times, but I do not know how to do in the following case: We have $$ |\nabla v(\cdot,t)|_{L^{\infty}}...
1
vote
1answer
37 views

Is this integral inequality correct?

There is a Young inequality that says: given $a,b \in \mathbb{R}$ and $p,q> 1$ such that $\frac{1}{p} + \frac{1}{q}=1$ we have: $$|ab| \leq \frac{1}{p}|a|^p + \frac{1}{q} |b|^q$$ The question is: ...
2
votes
3answers
70 views

Prove that $1/6 < \int_0^1 \frac{1-x^2}{3+\cos(x)}dx < 2/9$

Prove that $$\frac{1}{6}<\int_0^1 \frac{1-x^2}{3+\cos(x)}dx < \frac{2}{9}. $$ I tried using known integral inequalities (Cauchy-Schwarz, Chebyshev) but I did not arrive at anything. Then I also ...
1
vote
2answers
86 views

Inequality involving supremum norm on integral

Given is that $T:C([0,a])\rightarrow C([0,a]),\space (Ty)(x)=\frac{x^2}{2}+\int_{0}^{x}ty(t)dt, \space||y||_1= sup_{x \in[0,a]} |y(x)|$. I want to prove that $||Ty-Tz||_1\leq \frac{a^2}{2}||y-z||_1$. ...