All Questions
16 questions
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
$$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
- 1,053
2
votes
1
answer
321
views
A strange functional inequality
Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\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
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
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 $$\forall u\in [0,U],\alpha\in\left]0,1\right],\int_0^u\int_{[-1,1]^2}\int_{[-1,...
- 573
4
votes
1
answer
166
views
Estimate an improper integral
Suppose that $f$ satisfies $a$-Hölder condition on $[0,1]$ ($0<a<1$). Fix $0<b<a$. For any $x\in [0,1]$ and sufficiently small $\delta>0$, is the following inequality established?
$$ \...
- 419
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 $n\rightarrow\infty$?
$$\int_0^{1} I_{2 t - t^2}\left(\frac{n - 1}{2}, \frac{1}{2}\right) dt=\Omega\left(\...
- 1,677
1
vote
1
answer
181
views
Optimization problem with definite integral inequality constraints
Question: How can we prove that there exists a real constant $c\ge 1$ such that the following inequality holds for all integers $d>1$ and all real numbers $r\in\left[1,\sqrt{d}\right]$?
$$\int_{-1}^...
- 1,677
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
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
$$\operatorname{gd}(x)\...
1
vote
1
answer
299
views
Examples of Steffensen's inequality at undergraduated level studies
I've known few days ago the known as Steffensen's inequality, see the article Steffensen's inequality from Wolfram MathWorld and the cited bibliography. It seems that there are applications (I don't ...
0
votes
1
answer
350
views
Uniformly Bounded (updating)
Suppose that $a_1<1$, $a_1+a_2+a_3>1.$ For $x,y,z>0,$
(1) define a fucntion
$$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
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 <...
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
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
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
$$
I_2(f) := \iint_{\...
- 183
1
vote
1
answer
100
views
Can this equality hold for a nonzero $b$?
Please may you kindly assist me on this integration exercise: For real $a, b$ with $a \neq 0$, consider the equality
$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\...
- 19