All Questions
14 questions
14
votes
3
answers
664
views
(Sharp) inequality for Beta function
I am trying to prove the following inequality concerning the Beta Function:
$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$
where as usual $B(a,b) = \...
- 315
11
votes
1
answer
725
views
Chebyshev's other inequality
It is a simple fact, the granddaddy of correlation inequalities that if $f,g$ are monotone functions on $[0,1]$ then $$\int_0^1 f(x)g(x) dx \ge \int_0^1 f(x) dx \int_0^1 g(x) dx.$$ In their famous ...
- 827
10
votes
1
answer
1k
views
What would the best treatment of Gehring's lemma look like?
In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...
- 2,041
7
votes
2
answers
697
views
Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)
According to the entry "Differential inequality" of the Encyclopedia of Mathematics
http://www.encyclopediaofmath.org/index.php/Differential_inequality
the following result is due to Chaplygin (1919)...
- 1,371
7
votes
1
answer
352
views
Tight upper bounds on trigonometric polynomials
According to D. Hajela's chapter in Open Problems in Communications and Computation the following question was open as of the late 1980s. I have been unable to find any references so any results or ...
- 10.4k
6
votes
1
answer
796
views
A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx
\le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
- 128k
6
votes
2
answers
513
views
Need a reference for a trigonometric inequality
In my old high school notebook (20 years ago), the following inequality appears with its proof:
$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$
for any real $x$ and positive ...
- 335
4
votes
1
answer
387
views
$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
The Fejer-Jackson inequality as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
I conjecture that the inequality as follows holds:
$$\sum_{...
- 1,776
4
votes
2
answers
283
views
Bounding the series of the geometric means of the terms of a given positive series
Let $ \{ a _ k \} _{k\in\mathbb{N} _ +} $ be a sequence of non-negative numbers, and let $MG(a_1,\dots,a_n)$ denote the geometric mean of the first $n$ terms. Then, the inequality
$$ \sum _ {n\ge 1}...
- 60.5k
3
votes
2
answers
579
views
More recently published comprehensive reference on inequalities in the spirit of Hardy-Littlewood-Pólya
Is there a comprehensive reference book on inequalities in the
spirit of the one written by G.H. Hardy, J.E. Littlewood, and G. Pólya(*), but more up-to-date (i.e., published in more recent years and ...
user60665
1
vote
1
answer
109
views
Bound on $L^1$ norm of solution of two-point boundary value problem
This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
- 3,065
1
vote
1
answer
3k
views
Is there a standard proof that the L^1 norm > constant * sup norm for functions with derivative bounded above by K on the unit disk in R^n?
Suppose that you have a bounded function $f(x)$ on a compact domain in $\mathbb{R}^n$. It's easy to see from Holder's inequality that
$$
||f||_1 \leq \operatorname{Volume}(D) ||f||_\infty.
$$
There ...
- 747
1
vote
0
answers
138
views
An Elementary Inequality [closed]
A friend of mine found in the internet the following exercise:
Let $a, b, c \geq 0$ with $a + b + c = 1$. Show that
$$
\sqrt{a + b^2} + \sqrt{b + c^2} + \sqrt{c + a^2} \geq 2,
$$
where equality is ...
0
votes
0
answers
71
views
Reference request for equivalent Lipschitz smoothness conditions
For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
