All Questions
Tagged with integration inequalities
16 questions with no upvoted or accepted answers
7
votes
0
answers
317
views
An inequality which involves a sum of integrals
Please help me to prove
$$
\sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \...
- 71
6
votes
0
answers
129
views
A reference for an integrability property?
In a recent paper of mine (Compensated integrability), I established a functional inequality which has nice consequences. For instance, it contains the isoperimetric inequality, and it gives a new ...
- 52.3k
4
votes
0
answers
207
views
Integral inequality of Polya
In the book Math Problems AMM (1957), Problem 230, there is the next inequality of D. Polya:
let $a,b>0$, $0\leq x \leq a $,
$f(x)$ --- being not a linear function, and $f(0)=0$, $f(a)=b$, $f(x)\...
- 1,550
4
votes
0
answers
349
views
Fractional integral inequality (Hardy-Littlewood-Sobolev)
I am investigating the following integral
\begin{equation}
I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy
\end{equation}
where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
3
votes
0
answers
209
views
An "elementary" inequality
The following is left unproven in a monograph. The author refers to it as "elementary exercise" but I am unable to prove it. Any insight is appreciated.
$$
\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
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
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
0
answers
82
views
Reverse Inequality
I was doing some numerical integration when I figured the function I was dealing with (i.e., the function I was integrating) evaluated to big numbers on a tiny portion of the interval (over which I ...
- 369