All Questions
Tagged with integration inequalities
53 questions
43
votes
1
answer
2k
views
Is $\int_0^\infty{dx\over x^{x^{x^x}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^x}}}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^{x^{x^x}}}}}}}<\cdots$ true?
On MSE I asked whether each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ was less than $2$ and received answers on bounding ...
- 1,459
38
votes
4
answers
3k
views
Binomial again, and again
Let $\lceil a\rceil=$ the smallest integer $\geq a$, otherwise known as the ceiling function. When the arguments are real, interpret $\binom{a}b$ using the Euler's gamma function, $\Gamma$.
Recently, ...
- 43.2k
24
votes
2
answers
1k
views
Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$?
$\newcommand\abs[1]{\lvert#1\rvert}\newcommand\Abs[1]{\left\lvert#1\right\rvert}$Question: Let $f(x)$ be a continuous real-valued function defined on the interval $[0, 1]$. Does the following ...
- 4,280
9
votes
2
answers
622
views
Integral inequality: Prove $\int_0^1 f\int_0^1 1/f \leq 1$ for a certain function $f$
Let $g$ be a piecewise smooth, zero average, function over $[0,1]$ such that $\min g^2>0$. I would like to show that
$$
\int_0^1 g\sqrt{1-r/g^2}\int_0^1 \frac{1}{g\sqrt{1-r/g^2}} \leq 1
$$
for all $...
- 264
8
votes
3
answers
628
views
Expected distance between two uniform points in distinct rectangles
Are there any good approximations (especially upper bounds) for the quantity $E(\|X_1-X_2\|$), where each $X_i$ is uniformly distributed in a rectangle $[a_i,b_i]\times[c_i,d_i]$? It does not appear ...
- 4,049
7
votes
3
answers
515
views
Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$
The question is to prove:
$$
\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1.
$$
Numerically it seems to hold true. So I have made some attempts to ...
- 419
7
votes
0
answers
317
views
An inequality which involves a sum of integrals
Please help me to prove
$$
\sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \...
- 71
6
votes
0
answers
129
views
A reference for an integrability property?
In a recent paper of mine (Compensated integrability), I established a functional inequality which has nice consequences. For instance, it contains the isoperimetric inequality, and it gives a new ...
- 52.3k
5
votes
2
answers
301
views
Euler–Maclaurin formula in $\mathbb{Z}^d$
I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as
$$
\sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \...
- 446
4
votes
1
answer
166
views
Estimate an improper integral
Suppose that $f$ satisfies $a$-Hölder condition on $[0,1]$ ($0<a<1$). Fix $0<b<a$. For any $x\in [0,1]$ and sufficiently small $\delta>0$, is the following inequality established?
$$ \...
- 419
4
votes
0
answers
207
views
Integral inequality of Polya
In the book Math Problems AMM (1957), Problem 230, there is the next inequality of D. Polya:
let $a,b>0$, $0\leq x \leq a $,
$f(x)$ --- being not a linear function, and $f(0)=0$, $f(a)=b$, $f(x)\...
- 1,550
4
votes
0
answers
349
views
Fractional integral inequality (Hardy-Littlewood-Sobolev)
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
2
answers
645
views
Upper bound for complex integral
I am interested in obtaining a good upper bound for the absolute value of the following integral
$$
\left| \int_{0}^{\pi/3} e^{-itn} \left( 1-e^{it} \right)^{k} dt \right|,
$$
when $n>k>0$ are ...
- 43
3
votes
3
answers
480
views
Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?
What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$?
Or any good reference for tools to tackle this question?
...
- 1,256
3
votes
1
answer
205
views
Bound on an integral representing a difference of two relative entropies
Let $ f : [0,1] \to \mathbb{R} $ be a function satisfying: 1.) $ |f(x)| \leqslant a $ for some $ a < 1 $, and 2.) $ \int_0^1 f(x) {\mathrm d}x = 0 $. I would like to know whether the following ...
- 503
3
votes
1
answer
223
views
Ratio of Selberg integral
I'm considering a ratio of incomplete Selberg integral:
$$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...
- 1,495
3
votes
1
answer
321
views
Bounding a series of nested integrals
Consider the following matrix function
$$
f(t) = \cos(\omega_1t) A_1 + \cos(\omega_2t) A_2, \quad t\ge 0,
$$
where $A_1$, $A_2$ are real square matrices and $\omega_1$, $\omega_2$ positive numbers.
...
- 2,712
3
votes
0
answers
209
views
An "elementary" inequality
The following is left unproven in a monograph. The author refers to it as "elementary exercise" but I am unable to prove it. Any insight is appreciated.
$$
\int f \log f d\mu \le 2 \left[\...
- 519
3
votes
0
answers
103
views
Extension of the Gagliardo Inequality
The Gagliardo Inequality generalizes Fubini's Theorem: let $f_j$ be $d-1$ non-negative measurable functions over ${\mathbb R}^{d-1}$. Let us form the function
$$f(x)=\prod_{j=1}^df_j(\widehat{x_j}),$$
...
- 52.3k
2
votes
1
answer
137
views
Approximating a limit of an integral
How can we prove the following asymptotic lower bound for the regularized Beta function when $n\rightarrow\infty$?
$$\int_0^{1} I_{2 t - t^2}\left(\frac{n - 1}{2}, \frac{1}{2}\right) dt=\Omega\left(\...
- 1,677
2
votes
2
answers
150
views
Lower bound for integrals like $\int_1^{t+1}e^{-\sqrt{s}}s^{-1}ds$
Let
$$I(t) = \int_{1}^{t+1}\exp\left\{-c\frac{s^{1-\beta}}{1-\beta}\right\}s^{-2\beta}ds,$$
where $c$ is some positive constant and $\beta\in(0, 1)$.
Since the integral $I(t)$ given above could not be ...
- 63
2
votes
2
answers
169
views
A general question on comparison of integrals and a specific problem
When working on an applied math topic, I have come across the following general problem.
Let $f(x_1, x_2, ..., x_n)$ be a real function of $n$ real variables $x_1, x_2, ..., x_n$ which is ...
- 21
2
votes
1
answer
563
views
Prove or disprove $ \int_0^\infty \int_{-x}^0 f(x)f(y)\,dy\,dx > \int_0^\infty \int_{-\infty}^{-x} f(x)f(y)\,dy\,dx. $
Consider a symmetric, unimodal distribution $f(x)$ such that $\int_0^\infty f(x)\,dx > 1/2$. I want to prove or disprove the following:
$$
\int_0^\infty \int_{-x}^0 f(x)f(y)\,dy\,dx > \int_0^\...
- 175
2
votes
1
answer
321
views
A strange functional inequality
Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
- 4,074
2
votes
1
answer
157
views
$\int_0^u\int_{[-1,1]^2}\int_{[-1,1]^2}\frac{1}{r}e^{-\alpha^2|x-y|^2/r} \, dx\,dy\,dr\leq Cu^{\epsilon}\alpha^{-2\beta}$
I am looking for a proof for the following fact: for $U>0,\beta>0,$ there exists $C>0,\epsilon>0$ such that $$\forall u\in [0,U],\alpha\in\left]0,1\right],\int_0^u\int_{[-1,1]^2}\int_{[-1,...
- 573
2
votes
1
answer
125
views
For which value of $C(f)$ would the following inequality hold?
I am wondering what would be the value of $C(f)$ for the following inequality to hold? E.g., $C(f)$ could be some quantity related to the Lipschitz constant or the size of the domain.
$$\left(\int f(x,...
- 23
2
votes
1
answer
116
views
Bounding a function with second moments
Let $f(x,y)$ be a non-negative function with $x,y \in \mathbb R^3$ that satisfies
$$
I_1(f) := \iint_{\mathbb R^3 \times \mathbb R^3 } f(x,y) \, dx \ dy < \infty
$$
and
$$
I_2(f) := \iint_{\...
- 183
2
votes
0
answers
101
views
An inequality related to Problem 10210 AMM 1992 No. 3
Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that
$$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
- 1,053
2
votes
0
answers
124
views
Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?
Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
- 5,470
2
votes
0
answers
162
views
Taylor coefficients of the integral of the ordered exponential
Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of
$$
X_A'(t) = A(t) X_A(t), \qquad X(0) = I.
$$
In other words $X_A$ is the ordered exponential of $...
- 577
2
votes
0
answers
85
views
inequality for two integral expressions
Given bounded positive functions $f, g, h \in L^1$, with $g, h$ finitely supported, I would like to compare the following two expressions:
$$\begin{aligned}
a_1 &= \int_{0}^{\infty} \! dx \, f(x) \...
user114668
2
votes
0
answers
180
views
Removing integral from norm by inequality
My first question on Math Overflow.
For my Mathematics Bachelor thesis I am looking at a paper called "Deep Limits of Residual Neural networks" by Matthew Thorpe and Yves van Gennip. (arxiv....
2
votes
0
answers
296
views
An integral inequality
Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be bounded with derivative $g'$. I have shown that the following inequality holds for all $w\in\mathbb{R}$,
\begin{equation}\bigg|w\int_0^1\int_{-\infty}^{\...
- 21
1
vote
2
answers
120
views
Integral inequality implies $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
Let $f:[0, \infty)\to [0, \infty)$ be non-increasing and satisfy for all $t>t_{0}$, $$f(t)+C\int_{t_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t_{0}}\int_{t_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ ...
- 459
1
vote
1
answer
299
views
Examples of Steffensen's inequality at undergraduated level studies
I've known few days ago the known as Steffensen's inequality, see the article Steffensen's inequality from Wolfram MathWorld and the cited bibliography. It seems that there are applications (I don't ...
1
vote
2
answers
135
views
Lemma about the weighted interpolation inequality
In this article Interpolation inequalities with weights
Chang Shou Lin the following lemma is stated and proved.
Lemma: Suppose $\dfrac{1}{p}+\dfrac{\alpha}{n} > 0$, then there exists a constant $C$...
- 677
1
vote
1
answer
257
views
If $h$ is a decreasing function then $\psi$ is an increasing function
Let $h : [0,1] \to [0,1]$ be a $\mathcal{C}^1$ function such that $h'(x)<0$ for all $x \in (0,1)$. Consider the function
$$
\psi(x) = \frac{\int_0^1yh(|x-y|) dy}{\int_0^1 h(|x-y|)dy}
$$
I am trying ...
- 31
1
vote
1
answer
112
views
Inequalities for the Gudermannian function of the type $\operatorname{gd}(x)\operatorname{gd}\left(\frac{1}{x}\right)<\text{upper bound}$, where $x>0$
After I've read the solution of Problem 4327 (see [1]) I wondered if an inequality for a similar upper bound in the RHS is feasible for the known as Gudermannian function
$$\operatorname{gd}(x)\...
1
vote
1
answer
100
views
Can this equality hold for a nonzero $b$?
Please may you kindly assist me on this integration exercise: For real $a, b$ with $a \neq 0$, consider the equality
$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\...
- 19
1
vote
1
answer
120
views
estimate involving Gaussian data
Let $x\in \mathbb{R}^{d}$, $d\geq 1$ and $p>\frac{2}{d}$. For any $R>0$ and any $0<s<\frac{dp}{2}-1$
\begin{align}
&\int_{\mathbb{R}}\int_{|x|>R}\left(\frac{1}{1+t^2} \right)^{\frac{...
- 131
1
vote
1
answer
181
views
Optimization problem with definite integral inequality constraints
Question: How can we prove that there exists a real constant $c\ge 1$ such that the following inequality holds for all integers $d>1$ and all real numbers $r\in\left[1,\sqrt{d}\right]$?
$$\int_{-1}^...
- 1,677
1
vote
2
answers
218
views
Infimum upper bound
Let $h:[0,1]\to[0,1]$ be a $\mathcal{C^1}$ function such that $h'(x)<0$ for all $x\in(0,1)$. I am trying to show that (not sure if it is true):
$$
\inf_{f\in \mathcal{H}}\left(\int_0^1\int_0^1 h(|...
- 31
1
vote
0
answers
37
views
Inequality for function on Spinor bundle
I have a function $H(x,\psi)$ defined on the spinor bundle $\mathbb{S}$ with $H_\psi$ being the continuous derivative in fiber direction having the following properties:
(H-1) There exists $0<\...
- 11
1
vote
0
answers
75
views
Integral inequality related to the (mixed?) moments of two functions
For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set
$$
S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}.
$$
Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce
$$
...
- 577
1
vote
0
answers
422
views
Integral of matrix determinant with respect to Lebesgue measure
$\newcommand\norm[1]{\lVert#1\rVert}
\newcommand\opnorm[1]{\norm{#1}_{\text{op}}}
\newcommand\Frnorm[1]{\norm{#1}_{\text F}}$Define
\begin{align*}
S_t=\{
(A,B)\in\mathbb{R}^{n\times n}\times\...
- 1,495
0
votes
1
answer
103
views
Demonstrating bound on integral
I found this question on another forum and it got me interested because of how tight the bound is: prove that
$$\int_0^\infty \frac{\arctan(x)}{x^2 + 4}\,dx > \frac{\pi}{4}.$$
The difference ...
- 689
0
votes
2
answers
154
views
Perhaps an application of Hardy's inequality
Let $f \in H_{0}^{1}(0,1)$ and $\lambda >0$ big enough. Consider $0 <\alpha < 1$ and some $k > 0$. I would like to show the following inequality
$$
\int_{\lambda^{-k}}^{1}|f(x)|^{2}dx \leq ...
- 241
0
votes
1
answer
127
views
upper bound for infimum of integral
I was reading this post , where the following question is discussed:
Let $h:[0,1] \rightarrow[0,1]$ be a $C^{1}$ function such that $h^{\prime}(x)<0$ for all $x \in(0,1)$. Then,
$$
\inf _{f \in \...
0
votes
1
answer
210
views
Integral estimate (inequality) with a Schwartz function
$\DeclareMathOperator\supp{supp}\newcommand\abs[1]{\lvert#1\rvert}\newcommand\Bigabs[1]{\Bigl\lvert#1\Bigr\rvert}$Given a Schwartz function $f \in \mathcal{S}(\mathbb{R})$ with $\supp(f) \subseteq [-A,...
user168186
0
votes
1
answer
350
views
Uniformly Bounded (updating)
Suppose that $a_1<1$, $a_1+a_2+a_3>1.$ For $x,y,z>0,$
(1) define a fucntion
$$H(x,y,z)=\frac{x^{\frac{1}{2}}\int_0^{\infty}\frac{1}{t^{a_1}~ (1+t)^{a_2+1}~
(1+t+z)^{a_3}}\exp\big\{-\frac{...
- 119