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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.

7
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333 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}^{...
6
votes
0answers
146 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 $ $$\int_{0}^{\pi } (f(x))^2 dx \le \int_{0}^{\pi } (f'(x))^2 dx $$...
5
votes
0answers
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
votes
0answers
128 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
votes
0answers
279 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
0answers
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}}<...
4
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0answers
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
85 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)...
3
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0answers
109 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
votes
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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
votes
0answers
145 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
votes
0answers
313 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
votes
0answers
592 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,...
3
votes
0answers
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
votes
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
votes
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
votes
0answers
283 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
27 views

Minimizing the value of integral under certain conditions

I have been working on the following minimization problem: Minimize $I(f) := \int_{-\infty}^{\infty}{|f(x)|^2+|f'(x)|^2dx},$ where $f\in\mathcal{A} = \{f:\mathbb{R}\to\mathbb{R}$ is differentiable ...
2
votes
0answers
80 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 ...
2
votes
0answers
35 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
votes
0answers
86 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
81 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
83 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
votes
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
votes
0answers
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
votes
0answers
136 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
650 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 ...
2
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0answers
60 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
votes
0answers
106 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 ...
2
votes
0answers
70 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
votes
0answers
135 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 ...
2
votes
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 ...
2
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0answers
686 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
votes
0answers
64 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
votes
0answers
105 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
votes
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
votes
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
votes
0answers
144 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
91 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
votes
0answers
401 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_{...
1
vote
0answers
29 views

Conditions to ensure an integral to be positive

Suppose that $g:[0,1]\to\mathbb{R}$ is a differentiable function with $g(0)=0$. I want to find conditions on $g$ such that $$ \int_0^1 f(x)g(x)\int_x^1f(t)\,dt\,dx=\int_0^1f(t)\int_0^xf(x)g(x)\,dx\,...
1
vote
0answers
43 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
24 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
18 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
17 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
69 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
71 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)...