Questions tagged [integral-inequality]
For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.
248
questions with no upvoted or accepted answers
9
votes
0
answers
444
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
$$(f(t))^4\le \left(\int_{-1}^{1}\dfrac{[2(1-|x|)f'(x)-f(x)][2(1-|x|)f'(x)+f(x)]}{4(1-|x|^2)}dx\right)\cdot\left(\int_{-1}^{...
8
votes
0
answers
268
views
Is it true that $\phi(\mu)=\mu F(\mu)^2-\int_{\mu}^{\overline{v}}F(v)[1-F(v)]dv\geq 0$?
Consider a random variable $V$ with distribution function $F$ and density function $f$ with support $[\underline{v},\overline{v}]$, where $0\leq\underline{v}<\overline{v}$. The mean is $\mu$. Here $...
7
votes
0
answers
303
views
How to prove or falsify this inequality?
In STEP 2014 Paper II Question 2, an inequality is assumed for candidates to attempting the question about the approximation of $\pi $
$$\int_{0}^{\pi } (f(x))^2 dx \le \int_{0}^{\pi } (f'(x))^2 dx $$...
7
votes
1
answer
1k
views
Are there any interpretations for the Gronwall's inequality in view of comparison theorem?
One form of the Gronwall's inequality is that
If $\alpha(x),u(x)$ are non-negative continuous functions on $[0,1]$, and $$\forall x\in [0,1], u(x)\leq C+\int_{0}^{x}[\alpha(s)u(s)+K]ds\;(C,K\geq0),$$...
6
votes
0
answers
104
views
Existence theorems for self adjoint elliptic systems
Let's consider an elliptic (vectorial, homogeneous, constant coefficient) system of order 2m
$$
\begin{cases}
Lu=f &\text{in }\Omega\\
Bu=0 &\text{on }\partial\Omega
\end{cases}
$$
which is ...
5
votes
3
answers
301
views
Convolution by $\log$ maps $\mathrm{L}^1$ into $\mathrm{BMO}$
It is stated in Stein's Harmonic Analysis:
Real-Variable Methods, Orthogonality,
and Oscillatory Integrals, Ch. IV, 6.3(i) that
$$
I_{n} f:=f\star\log|\cdot|\in\mathrm{BMO}(\mathbb{R}^n)\qquad\text{if ...
5
votes
0
answers
1k
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 $$ \frac{1}{p} + \frac{1}{q} =1. $$
Prove ...
5
votes
0
answers
292
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 ...
5
votes
1
answer
221
views
A mixture with ingredients of two equivalences with Riemann Hypothesis
Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
4
votes
1
answer
224
views
Minkowski's integral inequality for other norms
In my measure theory course we studied norms $L_p$ and no other norms. For proofs we used exclusively the trick known as Hoelder's inequality which works only on $L_p$ norms. I disliked it very much ...
4
votes
0
answers
126
views
For cubic $P(x)$ with a root in $[0,1]$, find the smallest $C$ such that $\int_0^1 |P(x)|dx\leq C\max_{x\in[0,1]}|P(x)|$
Find the smallest constant $C$ such that for every polynomial $P(x)$ of degree $3$ with a root in $[0,1]$,
$$\int_0^1 |P(x)|dx\leq C\max_{x\in[0,1]}|P(x)|.$$
Here's my rough work. Write $P(x)=a_3x^3+...
4
votes
0
answers
346
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-.)
$$...
4
votes
0
answers
285
views
hard integral inequality with $e^{x^2}$
a) prove the convergency of
$$ \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
votes
0
answers
198
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 ...
3
votes
1
answer
64
views
$L^p_x$ norms of $f(x, x+y)$
Let $f \in \mathcal{S}(\mathbb{R}^2)$. Then do there exists inequalities of the form $$\lVert f(x, A(x,y))\rVert_{L^p_x(\mathbb{R})} \lesssim_{A, p,q} \lVert f(x,y) \rVert_{L^q(\mathbb{R}^2)} $$ where ...
3
votes
0
answers
176
views
Integral inequality from AMM 1992
I would like to know the solution of the following 1992 AMM problem:
Let $f$ be a continuous non-negative function defined on the square $[0,1]^2$. Show that
$$
\int_0^1\int_0^1\int_0^1\int_0^1f(x_1,...
3
votes
0
answers
145
views
Prove $\int_{0}^{\pi} \sqrt[n]{\prod_{k=1}^{n}\csc^2(x-\alpha_k)}dx\geq4π$ for reals $\alpha_1,\alpha_2,...\alpha_n$
Prove for all positive integer $n$ and for all $\alpha_1,\alpha_2,...\alpha_n \in \mathbb{R}$ Prove that
$\int_{0}^{\pi} \sqrt[n]{\prod_{k=1}^{n}\csc^2(x-\alpha_k)}dx\geq4π$
I have tried by applying ...
3
votes
0
answers
62
views
Boundedness of integral operator induced by kernel $K(x,y) := \frac{1}{x+y}$.
Let $t_0 > 0$, $p \in (1,+\infty)$ and define $K:(0,t_0)\times (0,t_0) \rightarrow \mathbb{R}$ by $K(x,y) := \frac{1}{x+y}$. Is it true that for all $f \in L^p\bigl((0,t_0);\mathbb{R}\bigr)$ the ...
3
votes
0
answers
98
views
Energy bound for a closed curve
Let $\gamma : S^1 \rightarrow M$ be a smooth map from a circle of length 1 to a closed manifold $M$ with nonpositive curvature. Could we find a constant $C > 0$ depending only on $M$ such that $$\...
3
votes
0
answers
51
views
Inequality with integral and distribution function
I encountered an inequality with the following variables and functions. $X$ is a random variable drawn from $(-\infty, \infty)$ with cdf $F$, pdf $f$, and mean $\mu=\mathbb{E}[X]$. For any $x$ and $\...
3
votes
0
answers
67
views
How to prove this integral inequality by variational method?
Suppose $f(x)$ is fourth differentiable and satisfies $f(0)=f(1)=f'(0)=f'(1)=0$.Prove the inequality$$\frac{1}{a^4}\int^{1}_{0}[f''(x)]^2\text dx\geq \int^{1}_{0}[f(x)]^2\text dx$$Where $a$ is the ...
3
votes
0
answers
212
views
Prove integral is convex
$X$ are an iid draw from $(-\infty, \infty)$ according to $F$ with mean $\mu$. Further let $A = a(x, \theta)/\cos (\alpha)$ and $B = ((1- \cos(\alpha) - \sin (\alpha))\mu + \sin (\alpha)x)/\cos(\...
3
votes
0
answers
71
views
Estimating $\int_0^{2\pi}\left(\sum_{n=1}^{N} \cos (n^2x) \right)^4\ dx$ for large $N$
In a paper I am reading, it is claimed without proof that for every $\varepsilon > 0$ and $N > N_0(\varepsilon)$
$$
\int_0^{2\pi}\left(\sum_{n=1}^{N} \cos (n^2x) \right)^4\ dx < N^{2 + \...
3
votes
0
answers
233
views
Non-existence of supremum norm bound in terms of $L^2$ norm of gradient
Show there does not exist any constant $C>0$ such that, for any $\phi\in C^{\infty}_c(\mathbb{R}^2)$, the inequality
$$\lVert\phi\rVert_\infty\leq C\lVert\nabla\phi\rVert_2$$
holds.
Obviously, we ...
3
votes
0
answers
91
views
Two-index Hajek-Renyi type inequality
Consider a sequence of mean-zero independent random variables $\{X_i\}_{i=1}^\infty$ and the partial sum $S(n)=\sum_{i=1}^n X_i$. The Hajek-Renyi inequality provides a upper bound of the normed ...
3
votes
0
answers
100
views
A singular integral
So, I was doing some PDE related computations and I obtained the following integral
$$
\iint \frac{(y-y')\,f(x',y')}{(|x-x'|^2+|y-y'|^2)^\frac{3}{2}}\, \left|\frac{x'}{x}\right|^2 \,\mathrm{d} x'\,\...
3
votes
0
answers
35
views
Immersion $H^s(\Omega) \hookrightarrow H^{s'}(\Omega)$ but with fractional laplacian defined on $\mathbb{R}^N$?
I know that it holds this embeding
$$H^{s}(\Omega) \hookrightarrow H^{s'}(\Omega)$$
for $s>s'$ and for any $\Omega \subset \mathbb{R}^N$. In this case, anyway, the fractional laplacian is defined ...
3
votes
0
answers
110
views
Minimizing $\int_0^1 [f''(x)]^2 dx$ with constraints
I would like to find the greatest lower bound and hopefully the function that minimizes $\int_0^1 [f’’(x)]^2 dx$ where $f\in C^2([0,1])$ with $f(0) = f(1) = 0$ and $f’(0) = 1$.
The only thing I could ...
3
votes
0
answers
132
views
Burkholder-David-Gundy ineq. with conditional expectation?
Let $\sigma$ be a progressively measurable stochastic process on a probability space with a standard Brownian motion $W$. The Burkholder-David-Gundy inequality tells us that
$$
\mathbb E \Bigg( \sup_{...
3
votes
0
answers
161
views
Examples of Besov functions of power-logarithmic type $|x|^{\alpha} |\log |x||^{\beta}$
I'm stuck on the following exercise 17.9 from Leoni's text A first course in Sobolev Spaces (second edition). This is from the chapter on Besov spaces, but this is really just a integral inequality ...
3
votes
0
answers
136
views
Is it a well-known inequality?
Let $f\in \mathcal{C}^2(\mathbb{R},\mathbb{R})$ and suppose that
$\int_\mathbb{R}f^2<+\infty$ and $\int_\mathbb{R}f''^2<+\infty$. Then
we can deduce that : $\left(\int_{\mathbb{R}}f'^2\right)...
3
votes
0
answers
101
views
A Gronwall type inequality involving iterated integrals
Let $p(t), a(t)$ be non-negative, continuous functions on $[0,T]$. Suppose that we have:
$$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),$$
where $\kappa, C >...
3
votes
0
answers
132
views
How to prove the following ineqiulity : $\exp\left(\int_0^t f(s)ds \right) \le 1+ \int_0^t e^{\max(1,s)f(s)}ds$
Let $f$ be integrable on $[0, t]$ , $t≥0$ then prove that
$$\exp\left(\int_0^t f(s)ds \right) \le 1+ \int_0^t e^{\max(1,t)f(s)}ds$$
This clearly smells like Jensen's inequality where we instead ...
3
votes
1
answer
60
views
Seeking a Proof or a counter example for the following step regarding inequalties
Please consider if the following pair of inequalities hold
$$
g(X) \le \mathbb{P}\left(\xi > X \right) \le h(X)
$$
where the domain of integration is over some variable $x$ and $X$ is some constant ...
3
votes
0
answers
308
views
Inequality with Products of Integrals
Following from a previous post:
Let $h_i, h_j, f_i,f_j$ continuous functions such that $f$ are increasing and $f(0)=0$, Assume that:
$\forall x$
$$h_i (x)>h_j (x)$$
$$f_i (x)>f_j (x)$$
and
$$...
3
votes
0
answers
972
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
$$ \|{\int f(x_1,...
3
votes
1
answer
496
views
Fractional integral inequality (Hardy-Littlewood)
I am investigating the following integral
\begin{equation}
I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy
\end{equation}
where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
3
votes
0
answers
200
views
Is this function increasing?
I'm stuck in showing that the following function is increasing over the domain $\left[0,x_0\right]$:
$$\Pi\left(z\right) = \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}}\left(2y-b\left(x\right)-x\...
3
votes
0
answers
53
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
1
answer
389
views
Using Gronwall's Inequality with Random Variables
Currently, I've been working with an SDE and trying to get a bound on moments. I have it down to something of the following form:
$$X(t)^p \leq a(t) + \int_0^t X(s)^pY(s) ds + \int_0^t X(s)^p dW_s$$
...
3
votes
0
answers
2k
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
0
answers
121
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
0
answers
392
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
0
answers
47
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
2
votes
0
answers
94
views
How to estimate $\int_0^{L/2} f(x)dx$ from $f(x)\ge 2x-\frac{L}{D} \int_0^x f(t)dt$
$D>0, L>0$ are constant. $f:[0, +\infty) \rightarrow \mathbb R$ is non-negative continuous function. $f(0)=0$ and $D=\int_0^{+\infty} f(x) dx$. In fact, when $x>0$ is large enough (than $L$),...
2
votes
0
answers
71
views
Inverse inequality
I would like to prove the following inequality:
Let $f$ be a function $f:\mathbb R\to\mathbb R$ such that $0\leq f\leq 1$ on $\mathbb R$ and $f=1$ on $[1,+\infty)$ and $f=0$ on $(-\infty,-1]$. Prove ...
2
votes
0
answers
166
views
Understanding Proof of Talagrand's Inequality
I am reading Talagrand's seminal paper Concentration of Measure and Isoperimetric Inequalities in Product Spaces. Lemma 2.1.2 on Page 12 obtains the bound
$$
\int_\...
2
votes
0
answers
96
views
What is the correct version of the Gronwall lemma? Can the sign of u(t) be variable?
In https://encyclopediaofmath.org/wiki/Gronwall_lemma the various forms of the Grönwall lemma in integral form are stated for NON NEGATIVE function $\phi$
And this coincides with what is written my ...
2
votes
0
answers
94
views
Rudin RCA Problem 4.12 (Hint Request)
Clarification and Attempted Solution
I've been stuck on this problem for several days now, and I'm entirely frustrated at this point. I do not know how to estimate the integral in the particular way ...
2
votes
0
answers
67
views
For finite-duration continuous $f(t)$ with $\|f'(t)\|_\infty < \infty$: It is true $\|f'(t)\|_\infty \leq \frac{2\pi \|f'(t)\|_2^2}{\|f'(t)\|_1}$?
For finite-duration continuous $f(t)$ with bounded derivative $\|f'(t)\|_\infty < \infty$: It is true that $\|f'(t)\|_\infty \leq \frac{2\pi \|f'(t)\|_2^2}{\|f'(t)\|_1}$?
I am looking for an upper ...