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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31 views

Looking for name/references for basic integral inequality

I am looking for a reference (or theorem name) for the fact that an integral is larger if its non-negative integrand and region of integration are both strictly larger. I am looking at this ...
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23 views

Inequality on expectation

I will show the exact inequality I'm dealing with. So, say that I have a collection of random variables on a probability space $\{X_i\}_i^N: X_i=V_{n,i}/\sqrt{d_n}\in L^2(P)$. I think that this ...
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18 views

Visual interpretation of integral inequality

Is there any nice visual proof of the following proposition? Let $$F(x) = (p(x), q(x))$$ then $$||\int_0^t F \|| \leq \int_0^t ||F||$$ were $$\int_0^t F \ $$ is a vector that results from ...
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13 views

Deriving a bound on a function by taking an integral of a bound on it's derivative

Let $f$ be real-valued continuous function on $[a,b]$, differentiable on $(a,b)$. Let $g$ be Riemann integrable on [a,b]. Assume that $f'(x) \geq g(x)$ when $x \in (a,b)$ Does this imply that $f(x) ...
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2answers
4k views

Proof of Wirtinger inequality

Quoting from Ana Cannas da Silva's book on Symplectic Geometry: "As an exercise in Fourier series, show the Wirtinger inequality: for $f\in C^1([a,b])$, with $f(a)=f(b)=0$ we have $$ \int_a^b\Big|\...
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1answer
33 views

Conjectured decreasing of sequence of integrals of products of Fourier transformations

Let $f$ be a real function with the following properties: $$f(x) \geq 0$$ $$f(x) = 0 \textrm{ for } x < 0 $$ $$ \int_0^\infty dx f(x) = 1$$ and define $$\hat{f}(k) = \int_{-\infty}^{\infty}dx \, ...
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2answers
21 views

Lower bound $\int_0^\infty e^{-t-\frac{t^2}{2\sigma^2}}dt$ by $1-\frac{1}{\sigma^2}$

I am trying to show a lower bound $\int_0^\infty e^{-t-\frac{t^2}{2\sigma^2}}dt \geq 1-\frac{1}{\sigma^2}$. It seems like one could try integration by parts and get $$ \int_0^\infty e^{-t-\frac{t^2}{2\...
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1answer
7 views

Is my estimation of the equality given below true?

When I deal with an estimation in analysis,I hope the equality given below is true.The geometry series on the numerator and the estimates $\sum_{n=1}^N {1\over n}\geqslant clogN$ we all know indulge ...
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2answers
175 views

(Putnam) Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and

The following is a Putnam math competition problem: Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and $ \int_{1}^{3}f(x)dx = 0 $. What is the max value of $\int_{1}...
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19 views

Proving increasing function for some conditions

Let $f\in C([0,\infty)$ be a real valued function and $u\in H^{1}(\mathbb{R}^{N})$ for $N\geq 3$. Assume that $\frac{f(s)}{|s|}$ is an increasing function for any $s\in\mathbb{R}\setminus\{0\}$. I ...
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27 views

$f''(x)\ge m>0$, show that $ |\int_0^{+\infty}\exp(if(t))\mathrm{d}t | \le 8/\sqrt{m}$

How can this inequality be proven? I thought of writing $e^{if(t)}$ as $ \frac{1}{f'(t)} \cdot f'(t) e^{if(t)}$ to integrate by parts but I can't finish.
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2answers
86 views

How prove this inequaliy $\sup_{x \in [a, b]} |f(x)| \leq \frac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt $

let $f \in C^1([a, b])$ with $a, b \in \mathbb{R}, a < b$ show that $$\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt$$ I've tried to use the ...
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62 views

I wonder if such integral inequality exists

Given $ \Omega$ a compact subset of $\mathbb{R}^n$ and $f\in H^1(\Omega,\mathbb{C})$ with zero average, I wonder if there exists an inequality of the form $$   \int_\Omega \phi(|f|^2)\varphi(|f|^2)\ ...
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1answer
72 views

Optimal coefficient in Cauchy-Schwartz inequality?

This may be trivially wrong, but I can't see a counterargument. Let $f,g:\mathbb{R}\to\mathbb{R}$ be measurable complex-valued functions such that $f/g$ is defined. Does there exist a coefficient $\...
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32 views

Question about an Integral inequality (with norm, Holder or Minkovski)

I'm reading an article that associated with Integral type inequality, suppose $$B\left(x\right)\in L^{1}\left(0,T;W_{\mathrm{loc}}^{1,\alpha}\left(\mathbb{R}^{N};\mathbb{R}^{N}\right)\right),$$ in ...
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1answer
39 views

Inequality involving integrals of trigonometric functions

Prove the inequality $$\left|\int_0 ^{\pi/4} \frac{\tan x~dx}{3-\sin(x^2)}\right|≤ \frac{1}{4}\log_e 2.$$ I have tried many different ways to get this inequality but failed.
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2answers
61 views

upper bounds for $\int_a^{b} \frac{\exp(x)}{x}\ dx$

Let $a<b$ be a positive real numbers. Are there tight upper bounds for $\int_a^{b} \frac{\exp(x)}{x}\ dx$, specially asymptotic bounds when $a, b,\frac{b}{a}\to\infty$?
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1answer
63 views

Prove the following integral inequality

I came across the following inequality in "Classical and New Inequalities in analysis" by Mitrinivic and Pecaric. The inequality is stated as follows. Suppose that $F$ is an increasing function on $[...
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1answer
22 views

Strict subadditivity of upper Riemann/Darboux integral

Calling $\int_{a}^{b*} f = \inf\limits_{P \in P([a,b])} U_{f,P}$ Where $U_{f,P} = \sum\limits_{k=1}^{n} \sup\limits_{x \in [x_{k-1},x_{k}]}f(x) \cdot ( x_{k}-x_{k-1})$ is the upper sum on a given ...
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1answer
52 views

An integral inequality related to the normal distribution

Through numerical experiments, I conjecture $$I=\int_0^\infty \big(\big|1-e^{\sigma(-x_1+x)}\big|-(1-e^{-\sigma x_1})\big)e^{-\frac{(x_1+x)^2}2}dx>0,$$ $\forall x_1>0,\, \sigma>0$. Is this ...
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1answer
74 views

Prove that $12(a\sin a+\cos a-1)^2\le 2a^4+a^3\sin(2a)$,$\forall a\in (0,\infty)$

Prove that $$12(a\sin a+\cos a-1)^2\le 2a^4+a^3\sin2a,\forall a\in (0,\infty).$$ The solution in the book where I found this goes like this : from CS for integrals we have that $$\left(\int_0^a x\cos ...
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1answer
73 views

Prove intergral inequality

If $f$ is a Riemann-integrable function on $[a,b]$ for which $\int\limits_a^b f(x) dx = 0$, and $m \leq f(x) \leq M$ for all $a \leq x \leq b$, then prove that $$\int\limits_a^b f(x)^2 dx \leq - m M ...
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1answer
24 views

Finding an upper bound for a probability of minimum by moment inequality

Suppose we want to show that $$P[|S_j - S_i| \wedge |S_k - S_j| \ge \lambda] \le \frac{1}{\lambda}^{4\beta}C_{i,k}$$ for some $\lambda>0$, $\beta \ge 0$ and some $C_{i,k}$. Then how is this ...
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2answers
53 views

Comparing $L^p$-norms of a function for two different values of $p$.

Let $(S,\Sigma,\mu)$ be a measure space with $\mu(S)<\infty$. Let $p\in(1,\infty]$ and suppose $f\in L^p(\mu)$. A simple application of the Holder's inequality shows that for any $0<q<p$, $f$ ...
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3answers
170 views

Under what conditions do we have $\int_{0}^{\infty} |f(x)|^2 dx \leq C \int_{0}^{\infty} x^2 |f^{\prime}(x)|^2 dx$?

I have been trying to prove the inequality $$\int_{0}^{\infty} |f(x)|^2 dx \leq C \int_{0}^{\infty} x^2 |f^{\prime}(x)|^2 dx$$ for some constant $C$, under the most general set of assumptions ...
32
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2answers
801 views

Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: $$\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)...
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1answer
68 views

Show that $\int_0^af(x)\,dx + \int_0^bf^{-1}(x)\,dx \geq ab\; \forall a,b \in \mathbb{R}^+.$

Let $f: [0,\infty) \rightarrow [0,\infty)$ be an onto, strictly increasing function. This problem looks like $\textbf{Young's Inequality}$ but it does not say that $f(0)=0$ or does the fact that I am ...
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412 views

How to prove Integral inequality with Hardy's inequality

Let $f\in C_{0}^{\infty}((-1,1))$. Prove that for any $t\in (-1,1)$ we have $$(f(t))^4\le \left(\int_{-1}^{1}\dfrac{[2(1-|x|)f'(x)-f(x)][2(1-|x|)f'(x)+f(x)]}{4(1-|x|^2)}dx\right)\cdot\left(\int_{-1}^{...
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0answers
46 views

Inequality involving integration on the unit square

I came across an inequality the other day, and I assume that it is either a special case of something well known or else something from a math contest. In an effort to write out this post, I came up ...
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1answer
41 views

Find $\vert \vert T_{k} \vert \vert$ where $T_{k}f(s)=\int_{0}^{1}k(s,t)f(t)dt$

Let $T_{k}:L^{p}[0,1] \to L^{p}[0,1]$, where $k \in C([0,1]^{2})$ and $T_{k}f(s)=\int_{0}^{1}k(s,t)f(t)dt$ Show $\vert \vert T_{k} \vert \vert \leq \sup\limits_{s}(\int^{1}_{0} \vert k(s,t) \vert ...
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1answer
76 views

How to prove the following inequalities.

The first inequality is $\int_{0}^{\infty} x^af(x)dx \leq a(\int_{0}^{\infty} xf(x)dx) $ for $0<a<1$ and $\int_{0}^{\infty} x^af(x)dx \geq a(\int_{0}^{\infty} xf(x)dx)$ for $a>1$ The ...
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1answer
58 views

Let $g$ be a differentiable, continuous function $[0,1]$ and $a≤g'(x)≤b$ for all $x\in [0,1]$

Let $g$ be a differentiable, continuous function $[0,1]$ and $a≤g'(x)≤b$ for all $x\in [0,1]$ Then prove that : $$\frac{b^2}{12}≥\int_0^{1}g^{2}(x)dx-\left(\int_0^{1}g(x)dx\right)^{2}≥\frac{a^2}{...
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19 views

Weak Lp spaces.

I'm trying to show that $|x|^{-\lambda}$ belongs to the weak $L_p(\mathbb{R^{n}})$ spaces, for $p=n/\lambda$, any help would be greatly appreciated.
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17 views

Use Grönwalls Inequality to construct majorant, that satisfies linear IVP

Recently we introduced Grönwalls Inequality: Let $\omega(t), a(t)$ and $b(t)$ be nonnegative, integrable functions, such that $a(t)\omega(t)$ is integrable. Furthermore, let $b(t)$ be monotonically ...
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1answer
66 views

Prove: Error of Riemann Sum is Decreasing for a Particular Function

Let $n$ be a fixed integer and define $f(x)=\frac{1}{n}\sum_{k=1}^{n} \left( \frac{(k/n)^x-1}{x} \right)$ and $g(x)=\int_{0}^{1} \frac{t^x-1}{x} dt$ for $x>0$. Prove that $f(x)-g(x)$ is decreasing ...
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1answer
364 views

Is this function decreasing in $x$?

Consider $$x \mapsto \frac{\int_{x-b}^{x+b} e^{-\frac{z^2}{2} }\text d z}{\int_{x-c}^{x+c} e^{-\frac{z^2}{2} }\text d z}$$ decreasing in $x\in [0,\infty)$, if $c > b > 0$ ? Edit: What I tried: ...
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48 views

If $u\left(x\right)\leq c+\int_{0}^{x}u\left(t\right)v\left(t\right)dt$ then $u\left(x\right)\leq c\exp\left(\int_{0}^{x}v\left(t\right)dt\right).$

I'm trying to solve the next problem: Let $c\in\mathbb{R}_{+}$and $u,v$ be continuous and postive functions from $\mathbb{R}_{+}$ to $\mathbb{R}$ such that for all $x\in\mathbb{R}_{+}$, $u\left(x\...
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1answer
24 views

p-norm inequality for two random variables

I read the following result in a book, however I believe that there is a mistake in the proof. Do you know of any book that proves this result, or do you have an idea on how to prove it? Let $X$ and $...
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0answers
39 views

Understanding theorem concerning stability of ordinary differential equations

We recently introduced the following theorem in my current lecture: Theorem (Stability): Let $f(t,u)$ and $g(t,u)$ be two continuous functions on a cylinder $D = I \times \Omega$ where the interval ...
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1answer
51 views

Inequality of integral : $\int_0^{1}((\sqrt{3}f(x))^{2}-2(f(x)^{3})dx≤1$

If $f(x)>0$ be continue function then this inequality true ? a) $\int_0^{1}((\sqrt{3}f(x))^{2}-2(f(x)^{3})dx≤1$ b) $(\int_0^{1}\sqrt{3}f(x))^{2}-\int_0^{1}2f(x)^{3}dx≤1$ I was used Cauchy ...
3
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1answer
44 views

An integral inequality for a real-valued differentiable monotone function on $[0,1]$

I have the following question from a past analysis qualifying exam: Let $f$ be a real-valued differentiable monotone function on $[0,1]$. Define $$g(x)=\frac12[f(x)+f(1-x)]$$ for $0\le x\le1$. ...
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1answer
31 views

Proof of Hardy Integral Inequality in N Dimensions

This comes from a recent lecture I've had. I have questions about on specific step in the short proof. This inequality is noted as the "Subcritical Hardy Inequality on the Whole Space". Statement: ...
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41 views

Integral inequality with some parameters

I'm studying a paper and I'm not able to prove the following inequality: $$\int_{U(r)} \frac{1}{(2\pi)^{k+1}}\prod_{j=1}^{k+1}\bigg(2\frac{\sin(b(x_j-y_j))}{x_j-y_j}\bigg)^2\,d\sigma(y) \leq c(k)b^{2(...
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1answer
46 views

under which situation inequality between two integral yeilds pointwise inequality?

suppose I have $$\int{f(x)} < \int{g(x)}$$ when I can conclude this : $f(x_{0})< g(x_{0})$ for some $x_{0}$ or is there such a $x_{0}$? how I can find it? if there is not answer, how ...
0
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0answers
35 views

About an integral inquality

Set $\phi (x) = u(x)+iv(x)$, $x=(x_1,...,x_N)$, a $T$-periodic function in $H^1_{loc}(\mathbb{R}^N)$, that is $\phi (x) = \phi (x_1 + T ,..., X_N + T)$ for all $x$ and where $u = Re\ \phi$ and $v = ...
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3answers
161 views

Proving a Definite Integral Inequality without Geometrical Intuition

I solved an integral inequality problem using geometrical methods. However, I just cannot satisfy with them and want a without-geometrical-intuition proof, and I couldn't find one. Proof the ...
2
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1answer
41 views

Proving that $\int_0^1 \left(\frac{\partial T}{\partial z}(t,z)\right)^2\mathrm{d}z \geq 2 \int_0^1 T^2(t,z)\mathrm{d}z$

Exercise : Assume that $T$ satisfies the equation $T_t(t,z) = aT_{zz}(t,z)$ for $t>0, z \in (0,1)$ and $a > 0$ a constant. Moreover, suppose that $T(0,z) = T_0(z)$ for $z \in [0,1]$, where $...
4
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2answers
90 views

Let $f:[0,1]\to[1,3]$ be continuous. Prove $1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}$

Let $f:[0,1]\to[1,3]$ be continuous. Prove $$1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}.$$ The left is just Cauchy's inequality with integral form, but ...
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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 ...
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0answers
44 views

differential inequality involving p.s.d matrix

Problem description Let $h(t) > 0$ be a continuous real function and $x(t) \in \mathbb{R}^{3}$ be also a continuous function. Let $$T(t) = h + \dot{x}^T\dot{x}$$ It is known that $$ \dot{T} = -\...