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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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331 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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143 views

How to prove or falsify this inequality?

In STEP 2014 Paper II Question 2, an inequality is assumed for candidates to attempting the question about the approximation of $\pi $ $$\int_{0}^{\pi } (f(x))^2 dx \le \int_{0}^{\pi } (f'(x))^2 dx $$...
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154 views

Functional inequality with a strong RHS

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. Show that $$\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$$ I ...
4
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122 views

Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...
4
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278 views

How did he use Gronwall Lemma??

I´ve got these lines from an article: ( where $b:\mathbb{R}_+\to \mathbb{R}_+$ is non-decreasing and $(X_t)$ is an $\mathbb{R}_+$-valued process - it doesn't matter very much, I guess, anyway-.) $$...
4
votes
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211 views

hard integral inequality with $e^{x^2}$

a) prove the convergency of $$ \int_0^{\infty} \dfrac {\cosh x \cdot \sin x} {x\cdot e^{x^2}}$$ b) prove the inequality $$ \int_0^{\frac {7\pi} {12}} \dfrac {\cosh x \cdot \sin x} {x\cdot e^{x^2}}<...
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1k views

Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and $\frac{1}{p}+\frac{1}{q}=\frac{...
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)...
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107 views

How to prove the following ineqiulity : $\exp\left(\int_0^t f(s)ds \right) \le 1+ \int_0^t e^{\max(1,s)f(s)}ds$

Let $f$ be integrable on $[0, t]$ , $t≥0$ then prove that $$\exp\left(\int_0^t f(s)ds \right) \le 1+ \int_0^t e^{\max(1,t)f(s)}ds$$ This clearly smells like Jensen's inequality where we instead ...
3
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80 views

Showing that a $L^2$-norm is positive

Let $\lambda_n = n + \delta_n $ for all $n \in \mathbb{Z}$ where $\delta_n$ are a sequence of real numbers in $\ell^2(\mathbb{Z})$. I am looking to prove that the sequence $(x_n)_{n \in \mathbb{Z}} = (...
3
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0answers
141 views

Inequality with Products of Integrals

Following from a previous post: Let $h_i, h_j, f_i,f_j$ continuous functions such that $f$ are increasing and $f(0)=0$, Assume that: $\forall x$ $$h_i (x)>h_j (x)$$ $$f_i (x)>f_j (x)$$ and $$...
3
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0answers
287 views

Probs. 10 (a), (b), and (c), Chap. 6, in Baby Rudin: Holder's Inequality for Integrals

Here is Prob. 10, Chap. 6, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $p$ and $q$ be positive real numbers such that $$ \frac{1}{p} + \frac{1}{q} =1. $$ ...
3
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568 views

Minkowski inequality for integrals

Let $(X_1,\mu_1)$ and $(X_2,\mu_2)$ be two $\sigma$-finite measure spaces. Let $f(x_1,x_2)$ be a measurable complex valued function. Stein and Shakarchi's 4th book asks to prove that $$ \|{\int f(x_1,...
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184 views

Is this function increasing?

I'm stuck in showing that the following function is increasing over the domain $\left[0,x_0\right]$: $$\Pi\left(z\right) = \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}}\left(2y-b\left(x\right)-x\...
3
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0answers
48 views

Inequality involving homogenous function of degree -1

Let $p\in[1,\infty]$. For $f\in L^p(0,\infty)$ we define $Tf:x\mapsto \int_0^\infty K(x,y)f(y)\,dy$ where $K$ is homogenous of degree $-1$, i.e. $K(\lambda x,\lambda y) = \lambda^{-1} K(x,y)$ for $\...
3
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0answers
117 views

A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to [0,\...
3
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0answers
261 views

An inequality with a characteristic function

It's my first question here, hi. In fact, it derives from my probability theory homework, which appears to be unusually difficult (or I just don't see something): Suppose $X$ is a real valued random ...
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 ...
2
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0answers
85 views

Prove or disprove that integral term is log-concave

Consider the function $h(x_i, x_j) := \int_{x_i}^{\overline{z}}f(z)F(z-x_i+x_j)dz$, where $f(z)$ is a twice continuously differentiable and strictly positive probability density function defined on $...
2
votes
0answers
54 views

On solution of a nonlinear differential inequality

I have the following differential inequality: $$f'(x)\geq cf(x)^a,\quad \forall x\in[0,1]$$ where $0<a<1,\, f(x)\geq 0$. I'm taking the following approach to solve the problem: $$f'(x)\geq cf(x)...
2
votes
0answers
80 views

A Gronwall type inequality involving iterated integrals

Let $p(t), a(t)$ be non-negative, continuous functions on $[0,T]$. Suppose that we have: $$p(t) \leq a(t) + C \int_0^t du e^{-\kappa(t-u)}p(u) \int_0^u ds e^{-\kappa(u-s)} p(s),$$ where $\kappa, C >...
2
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0answers
82 views

$\left|\int_{0}^{2\pi}f(t)\sin nt\text{d}t\right|\leq\frac{4}{n^2}$

Given $f$ such that $f''$ is continious and differentiable on $[0,2\pi]$, and that $f(0)=f(2\pi)$ and $ |f''(x)|\leq1$ for all $x\in[0,2\pi]$ I need to show $$\left|\int_{0}^{2\pi}f(t)\sin nt\text{d}...
2
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0answers
61 views

Proof of Poincaré type inequality

A function is in a space $W^{1,2}_{-\tau}$ if $\int_{\mathbb{R}^n} f^2|x|^{2\tau-n}<\infty$ and $\int_{\mathbb{R}^n} |\partial_kf|^2|x|^{2\tau+2-n}<\infty$ for all $k=1,2,\dots n$. Given a ...
2
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69 views

Inequality with concave functions in expected values

I'm working on an engineering problem and I manage to reduce it to the following claim, but I'm not sure if it is true. It will be great if someone can give me some ideas! Let $u(x)$ is an increasing ...
2
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0answers
123 views

Prob. 15, Chap. 6, in Baby Rudin: If $f$ is a real, continuously differentiable function on $[a, b]$, . . .

Here is Prob. 15, Chap. 6, in the book Principles of Mathematical Analysis, by Walter Rudin, 3rd edition: Suppose $f$ is a real, continuously differentiable function on $[a, b]$, $f(a) = f(b) = 0$,...
2
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0answers
635 views

Reverse of Holder's Inequality

Assume $X$ is Sigma finite. Assume $f$ is an $M$- Measurable function , $1\leq p \leq \infty$ and $g \in L^{p} \implies fg\in L^1$ Prove that $f\in L^P $ I was trying to solve this problem on ...
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0answers
59 views

Show that $\int_0^A (A-x) \cos(x) \ f(x) dx \ge 0$ for $f(x)$ monotone decreasing, non-negative and bounded.

Is it true that \begin{align} \int_0^A (A-x) \cos(x) \ f(x) dx \ge 0 \end{align} for all $A\ge 0$, if for $f(x)$ monotone decreasing, non-negative and bounded. This question came up in the ...
2
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0answers
103 views

Kind of Gronwall Inequality

Does somebody knows if it is possible to obtain an inequality (like for Gronwall inequality) on $f$ if $f$ verify $$ f(t) \leq A+\int_0^{2t} g(s)f(s) ds $$. Where $f$ and $g$ are as smooth as ...
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0answers
68 views

A Simple Stochastic Integral Asymptotics

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth condition $...
2
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0answers
131 views

Poincare type inequality on unit square: can you improve on my constants?

I am trying to bound $\int_{[0,1]^2} u^2$ in terms of its gradient and boundary integrals, $\int_{[0,1]^2} |\nabla u|^2$, $\int_{\partial[0,1]^2} u^2$, with the best possible constants. So far I ...
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0answers
36 views

How does this inequality imply this one?

I am having a little trouble understanding this part of a proof. There is an integral $\text{J}_{n} = \int_0^{\frac{\pi}{2}} x^2\cos^{2n}x dx$ Now, $\text{J}_0 = \frac{\pi ^3}{24} $ The part of ...
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681 views

Integral inequality - lower bound on $L^1$ norm.

I was wondering if one can make an estimate of form: Assume $f\in C^\infty(\overline{\Omega})$ where $\Omega$ is a bounded domain in $\mathbb{}R^d$. Is there a constant $C>0$ independent of $f$ ...
2
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0answers
63 views

Question about the assumption of a version of Grönwall's inequality.

According to Wikipedia, A version of Grönwall's inequality for the integral of continuous functions is the following: Let $I$ denote an interval of the real line of the form $[a,\infty)$ or $[a,b]$ ...
2
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0answers
104 views

integral inequality involving $\sup|f'|$

Let $f:[0,1]\rightarrow \mathbb R$ be continuous function differentiable on $(0,1)$ with property that there exists $a \in (0,1]$ such that $$\int_{0}^a f(x)dx=0$$ Prove that $$\left|\int_{0}^1 ...
2
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0answers
165 views

Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: $\left(\int\left|\left|\psi(g)\right|+\delta\sum_{j=1}^{k}\left(\left|g_{j}...
2
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0answers
117 views

Inequality involving double integral

There's a function $g(x,y):\mathbb{R^+\times \mathbb{R}^+\rightarrow \mathbb{R}^+}$ with $g_1(x,y)>0$, $g_2(x,y)>0$, and $g_{12}(x,y)>0$. I conjecture that $$\int g(x,x)f(x)dx>\int\left(\...
2
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0answers
141 views

L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where $\...
2
votes
0answers
90 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} \...
2
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0answers
389 views

Weak/Variational Gronwall type inequality

I came across the following weak differential inequality while looking through F.Otto's paper on $L^{1}$ contraction and uniqueness of quasilinear elliptic-parabolic equation: \begin{align*} - \int_{...
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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
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 ...
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):= \...
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 $$\phi(t) = \alpha + \int_0^t (\beta \phi(s) + \gamma)ds, \hspace{0.5mm} t\in [0,T], $$ ...
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
0answers
56 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 ...
1
vote
0answers
69 views

Reference for Garding's Inequality

Does anyone know which is the original paper in which Garding's inequality first appeared? By this, I mean https://en.wikipedia.org/wiki/G%C3%A5rding%27s_inequality. Many thanks!
1
vote
0answers
28 views

Calculating upper bound for increments of a function

Let $\varphi_{f,\delta}$ be a function defined by $\varphi_{f,\delta}(u)=\sup\left\{\left|f(y)-f(x)\right|:x,y\in\left[u-\delta,u+\delta\right]\right\}$ for a bounded and symmetric function $f:\left[-...
1
vote
0answers
20 views

Equality regarding power series and an integral

Given the power series $f(z)=\sum^\infty_{n=0}c_n(z-z_0)^n$ I have to prove that $$ \int_0^{2\pi}|f(z_0+re^{it})|^2\;dt =\sum^\infty_{n=0}|c_n|^2r^{2n}$$ Here $ r>0$ is such that $\overline{B_r(z_0)...
1
vote
0answers
34 views

Maximum value of the product of definite integrals

Let $f(x)$ is a continuous function in $[1;3]$ such that $\max\limits_{x\in[1;3]} f(x)=2$ and $\min\limits_{x\in[1;3]} f(x)=\dfrac{1}{2}$. Find the maximum value of $\displaystyle P=\int\limits_1^3 f(...
1
vote
0answers
72 views

Having trouble showing this inequality

Given the initial boundary value problem \begin{align*} &u_t = Du_{xx} + f(u), \quad 0<x<1, t>0 \\ &u(0,t) = u(1,t) = 0, \quad t>0 \\ &u(x,0) = u_{0}(x), \quad 0<x<1 \...