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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.

4
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1answer
99 views

Proving $\frac12 < \int_0^1 \frac1{\sqrt{4-x+x^3}} dx < \frac{ \pi }6$

I have a question where I have to show $$\frac12 < \int_0^1 \frac1{\sqrt{4-x+x^3}} dx < \frac{ \pi }6 \approx 0.52359$$ using the result $$\frac12 < \int_0^{1/2} \frac{1}{\sqrt{1-x^{2n}}} ...
0
votes
0answers
14 views

An integral inequality on a disc about twice derivatives.

Denote $f:\mathbb{R^2}\rightarrow\mathbb{R}$ has a continuous twice derivative on the disc $D=\{(x,y):x^2+y^2\leq1\}$, which satisfy $f(\partial D)=\{0\} $, where $\partial D=\{(x,y):x^2+y^2=1\}$, ...
0
votes
0answers
10 views

About the sufficient conditions for an upperbound for an integral ratio.

Let f and g be functions, I am interested in finding (if there is any) the sufficient conditions on f and g such that we can satisfy an inequality of this type: $$\frac{\int_{0}^{\infty}f(x)dx}{\int_{...
0
votes
1answer
37 views

Maximum of $\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$

Let $p>q>0$ and $C=\{f:[0,1] \to \mathbb{R} \mid f \text{ is continuous} \}$. Determine $$\max_{f \in C}\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$$ and the functions for which this maximum occurs. ...
2
votes
1answer
75 views

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $f:[0, \infty) \to [0,\infty)$ be a differentiable function with $f'$ continuous. If $f(f(x))=x^2$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $f.$ Since we ...
1
vote
0answers
65 views

Proof-Verification: $ \int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x)\,{\rm d}x \leq \frac{1}{2019}\int_a^b f(x)\,{\rm d}x$

Problem Let $f(x)$ be continuous and increasing over $[a,b]$. Prove $$\displaystyle \int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x){\rm d}x \leq \frac{1}{2019}\int_a^b f(x){\rm d}x.$$ Proof By ...
1
vote
1answer
46 views

$f,g \in [0,1] \times [0,1]$, $\int f - g \mathrm{d}x = 0$ and are monotonically increasing, then $\int |f-g| \mathrm{d}x \le \frac{1}{2}$

$f,g$ are monotonically increasing in $[0,1]$ and $0\le f , g \le 1$. $\int_0^1 f - g \mathrm{d}x = 0$. Prove that $$\int_0^1 |f - g|\mathrm{d}x \le \frac{1}{2}$$ In my previous question, $g(x) = x$....
1
vote
2answers
37 views

$\int_{-\epsilon}^{\epsilon}e^{-f(x)}dx\leq C\epsilon e^{-f(0)}$, f convex [closed]

Let $f:(-1,1)\rightarrow \mathbb{R}$ be a $C^{2}$ convex function. Is it true that $$ \int_{-\epsilon}^{\epsilon}e^{-f(x)}dx\leq C\,\epsilon e^{-f(0)}, $$ for any $\epsilon>0$ small enough. Here $C&...
0
votes
1answer
44 views

Integral inequality $\int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$

Let $f \in C^1([0;1],\mathbb{R})$ such that $f(0)=0$. $$\text{Prove that} \qquad \int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$$ My attempt: Let $$g(x)=\begin{cases} ...
1
vote
1answer
36 views

Integral inequality issue

Given that the bilinear form on the set $$V:= \{v\in C^2[0,1] v(0)=0=v(1)\}$$ is defined as $$[u,v]=\int_0^1 [pu'v'+quv]dx,$$ where $p\in C^1[0,1], p(x)\ge p_0>0, q(x)\ge 0, q\in C[0,1]$, I want ...
0
votes
2answers
65 views

Integral inequality $\int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}x|f'(x)|dx$

Let $f \in C^1([0;1],\mathbb{R})$ such that $f(1)=0$. $$\text{Prove that} \qquad \int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}x|f'(x)|dx$$ My attempt: \begin{align} \int_{0}^{1}x|f'(x)|dx = \int_{0}^{1}...
1
vote
0answers
35 views

Gagliardo-Nirenberg inequality for fractional Sobolev spaces

Wikipedia states two versions of the Gagliardo-Nirenberg inequality for nonfractional Sobolev spaces. I'm interested in generalizations to fractional (Slobodeckij) Sobolev spaces. Such a ...
0
votes
1answer
40 views

Bounding integral of square root by square root of integral

Let $f(x)\geq 0$ be a function over $[0,\infty)$. How can I lower bound $\int_{x=0}^{u}\sqrt{f(x)}dx$ by $c \sqrt{\int_{x=0}^{u}f(x)dx}$ where $\sqrt{\int_{x=0}^{u}f(x)dx}<\infty$ and $c>0$ is ...
0
votes
2answers
53 views

Inequality for bounded locally integrable functions

$\textbf{The Problem:}$ Let $f\geq 0$ be a bounded function and $E\subset\mathbb R^d$ have finite measure. Prove that there exists $R>0$ such that for all $0<r<R$ we have $$\int_{E}f(x)dx\...
10
votes
2answers
181 views

$\int_a^bf^2(x)\,dx\le \frac{2}{3}\int_a^bf(x)\,dx$ for a convex differentiable function

If $f:[a,b] \to \mathbb{R}, f(a)=0,f(b)=1$ is a convex increasing differentiable function on the interval $[a,b]$ . Prove that $$\int_a^bf^2(x)\,dx\le \frac{2}{3}\int_a^bf(x)\,dx$$ Since f is ...
5
votes
2answers
71 views

Proof of maximal principle on Laplace Equation involving Poisson integral formula

This question appeared on a past PDE exam I found while studying for my finals: Let $u(r,\theta)$ be solution to the Laplace equation in polar coordinates: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}...
1
vote
1answer
34 views

Littlewood's inequality for $L^p$ spaces

I have tried to prove the following inequality, but I couldn't do yet. Prove the following interpolation estimate: $$\| u\|_q \leq \| u\|_p^{\theta} \| u\|_r^{1- \theta}$$ where $p≤q≤r$, $θ∈[0,1]$ ...
1
vote
1answer
66 views

Heisenberg’s inequality

I have tried to write a proof for the following inequality, but I couldn't constitute a rigorous one yet. Suppose $f$ is absolutely continuous on $[−a,a]$ for all $a \in \mathbb R$ so that $f′$ ...
1
vote
0answers
23 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):= \...
0
votes
0answers
14 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
votes
2answers
47 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
votes
2answers
51 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 ...
1
vote
1answer
33 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}...
0
votes
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
vote
1answer
52 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
vote
1answer
47 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=...
0
votes
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 ...
0
votes
0answers
67 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}\...
3
votes
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)...
2
votes
1answer
2k views

Problem 9 - Chapter 5 - Evans' PDE (First Edition) - Clarification

I've been trying to work out the solution to Question 9 in Chapter 5 of Evans, and I'm having some difficulties. I've been looking at the solution posted here: Question $5.9$ - Evans PDE $2$nd ...
9
votes
1answer
4k views

Problem 9 - Chapter 5 - Evans' PDE (First Edition)

In the $1$st edition, this was question $5.9$. The question is: Integrate by parts to prove: $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\...
1
vote
0answers
15 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
votes
1answer
30 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 ...
0
votes
1answer
2k views

Lyapunov's inequality in Probability

I have a question about the proof of the inequality. The well known result stats Let $Z$ be a RV and let $0<s<t$. Then $$E(|Z|^s)^{1/s} \leq E(|Z|^t)^{1/t}$$ The proof follows almost ...
1
vote
1answer
133 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: $|...
1
vote
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-...
1
vote
1answer
138 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 ...
0
votes
0answers
25 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_{\...
0
votes
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 ...
1
vote
1answer
35 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?
2
votes
3answers
87 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)=...
2
votes
3answers
98 views

Do a specific inequality hold under integration?

As we know that if we have the inequality $f\leq g$, it does not imply $f'\leq g'$. Now Let $f(x)\leq g(x)$ for each $x\in [a,b]$, where $0<a,b<\infty$. Is it possible to prove $$\int_{a}...
1
vote
1answer
24 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 ...
0
votes
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
votes
3answers
154 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 ...
0
votes
0answers
38 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 ...
-2
votes
3answers
104 views
0
votes
1answer
51 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
29 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, ...