All Questions
Tagged with ca.classical-analysis-and-odes inequalities
176 questions
5
votes
1
answer
429
views
A plausible positivity
After getting stuck with the
previous positivity
(it probably sounds too complex),
I would like to give a version of the problem which is of most interest to me.
Consider a sequence of real numbers
$...
- 13.5k
5
votes
1
answer
224
views
Symmetric inequality on generalized means
Do there exist two functions $f$ and $g$ continuous and strictly increasing $[0,1] \to \mathbf{R}$ such that
$$ f^{-1}\left(\frac{1}{3} f(x) + \frac{2}{3} f(y)\right)<g^{-1}\left(\frac{1}{3} g(x) + ...
- 1,511
5
votes
2
answers
202
views
Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
- 128k
5
votes
2
answers
243
views
Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?
This has received no full solution at StackExchange.
As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\...
- 804
5
votes
1
answer
189
views
Conditions under which inequality holds for all triplets of non-negative reals
I have the following inequality:
$$ \left(\frac{1}{3}
\left(
\left(\frac{a+b}{2}\right)^3 +
\left(\frac{a+c}{2}\right)^3 +
\left(\frac{b+c}{2}\right)^3 \right) \right)^\frac{1}{...
5
votes
1
answer
172
views
uniform one-sided van der Corput inequality
Is the following true (and if yes, where the best proof is written?)?
For any $c>0$ for large enough positive integers $N$ we have $\sum_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$?
Hm,...
- 109k
5
votes
1
answer
921
views
About generalized Minkowski inequality
For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality
$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + f^{-1}\left(\sum\...
- 51
5
votes
1
answer
151
views
On existence of a concave function
Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that
$$ f’’(x) \leq 0\quad \text{and} \...
- 4,143
5
votes
1
answer
644
views
A conjecture about the submatrix of orthogonal matrix
Let $U$ be an $n\times n$ orthogonal matrix, i.e. $U\in\mathbb{R}^{n \times n}$. For any non-empty ordered sets $S_1,S_2\subset\{1,2,...,n\}$, define $U_{S_1S_2}$ to be an $|S_1|\times|S_2|$ submatrix ...
- 1,495
5
votes
3
answers
383
views
The exact constant in a bound on ratios of Gamma functions
The answer to another question (Upper bound of the fraction of Gamma functions) gave an asymptotic upper bound for an expression with Gamma functions:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...
- 343
5
votes
1
answer
445
views
A two-point inequality
Let $M(p,q) = (2p-\sqrt{p^{2}+q^{2}})\sqrt{p+\sqrt{p^{2}+q^{2}}}$ and set $B(t) = M(x+t, \sqrt{t^{2}+(y+bt)^{2}})$. Given any real $x,y,b$ is it true that $\varphi(t) = B(t)+B(-t)$ is decreasing in $...
- 3,946
5
votes
0
answers
204
views
A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
- 852
5
votes
0
answers
277
views
Carleman estimates on monotonicity formulas
I am trying to derive a monotonicity formula for a certain Dirichlet critical point (or even maybe a minimizer) of an energy of the type, say for simplicity, an energy of the from
$$\int_{B_r} (Ae(u)...
- 109
5
votes
0
answers
488
views
Any similar inequality in literature?
I got the following inequality:
$B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary.
$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$,
...
- 215
4
votes
4
answers
474
views
A certain inequality involving square roots of polynomials
I want to prove the inequality
$$\begin{aligned}
&\sqrt{(x - 1)^2 + y^2}\Big[y^2(9x - 6) - 9x^2 + 9x^3\Big]+ y^2(16x^2 - 16x + 7)\\
&- \sqrt{x^2 + y^2}\Big[9x + y^2(9x - 3) + \sqrt{(x - 1)^2 + ...
- 1,109
4
votes
1
answer
353
views
Inequalities involving binary representation of integers
Let $N\geq 1$ be a positive integer and assume that $N=2^{n_1}+2^{n_2}+\cdots+2^{n_{p}}$, $n_{1}>n_{2}>\cdots>n_{p}\geq 0$, is the binary representation of $N$. I believe that the following ...
- 41
4
votes
5
answers
1k
views
An inequality on concave functions
Could somebody help me to answer the following question?
Let $f:R_+ \rightarrow R_+$
be a nonindentically zero, nondecreasing, continuous, concave function with $f(0)=0$. Do we have that for any $s,...
- 227
4
votes
2
answers
245
views
On the monotonicity of the ratio of two logarithmic expressions
According to this comment and this comment, a positive answer to this recent question (about Bernoulli numbers) would be sufficient to prove the following:
$r:=f/g$ is increasing on $(0,\pi/2)$ from $...
- 128k
4
votes
1
answer
387
views
$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
The Fejer-Jackson inequality as follows:
$$\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
261
views
A convexity question
Let $Q=[0,1]\times[0,1]$ and let $a$ be a positive smooth function on $Q$. Does there exist a smooth positive function $u$ On $Q$ such that there holds
$$ \frac{\partial^2}{\partial x_1^2}u <0 $$
...
- 4,143
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
$$ \sum _ {n\ge 1}...
- 60.5k
4
votes
1
answer
149
views
An algebraic inequality in three real variables
Is it true that
$$(v-u)^2+(w-u)^2+(w-v)^2 \\
+\left(\sqrt{\frac{1+u^2}{1+v^2}}
+\sqrt{\frac{1+v^2}{1+u^2}}\right) (w-u)(w-v) \\
-\left(\sqrt{\frac{1+u^2}{1+w^2}}+\sqrt{\frac{1+w^2}{1+u^2}}\right) (w-...
- 128k
4
votes
1
answer
277
views
Find function with inequality for derivative
Let $C\geq 2$ and $L>0$. Does there exist $g \in C^1([0,L])$ such that
\begin{equation*}
g(x)>0, \qquad g'(x)>0, \qquad g'(x) > (g(L) - g(x))C
\end{equation*}
holds for any $x \in [0,L]...
- 115
4
votes
1
answer
326
views
When does the sine transform result in a positive function?
For each $x>0$ is, $$\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt > 0\ \ \ ?$$
In general, it is known that if the kernel is decreasing, then the sine transform is positive. Note that this ...
- 402
4
votes
1
answer
525
views
Controlling subsolutions of a second order linear ODE
Let $f:[0,\infty) \to \mathbb{R}$ obey the differential inequality
$$f'' - 2\alpha f' + 2\alpha f \leq 0$$
where $0 < \alpha < 2$ is some constant. If $f(0) = 0$ and $f'(0) = 1$, can I say that $...
- 205
4
votes
0
answers
197
views
Dynamics of an inequality
The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence
$$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3}
...
- 128k
3
votes
2
answers
597
views
lower bound for $\Re\zeta(1+it)$
Hi
is there any lower bound for $\Re\zeta(1+it)$.
I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.
If it is true, is there any reference to prove it.
thanks
- 163
3
votes
3
answers
1k
views
Poincaré metric on hyperbolic plane
As is well known, we can put a metric on the upper half plane $\mathbb{R}^+ \times \mathbb{R}$
by setting
$$
d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2},
$$
where
...
- 979
3
votes
1
answer
480
views
Is there a uniform upper bound for this oscillatory integral?
I am wondering whether the following uniform upper bound holds:
$|\int_a^{2a}\frac1t\sin(N b^2t)\exp(iNbt^2)dt|\le Cab^2,$
where $0<a<b<1$, $N>N_0(a,b)\gg1$, and $C$ is a constant ...
- 187
3
votes
2
answers
731
views
Non-asymptotic upper bound of right tail of Gamma function
I'm wondering if there is any non-asymptotic upper bound for the following Gamma function:
$$f_a(x)=\int_{x}^{\infty}t^a\exp(-t)dt$$
for $x>0,a>0$? Something like $x^a\exp(-x)$?
- 1,495
3
votes
2
answers
579
views
More recently published comprehensive reference on inequalities in the spirit of Hardy-Littlewood-Pólya
Is there a comprehensive reference book on inequalities in the
spirit of the one written by G.H. Hardy, J.E. Littlewood, and G. Pólya(*), but more up-to-date (i.e., published in more recent years and ...
user60665
3
votes
3
answers
480
views
Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?
What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$?
Or any good reference for tools to tackle this question?
...
- 1,256
3
votes
1
answer
303
views
On Wazewski's theorem on system of differential inequalities
According to Springer's Encyclopedia of Math entry on differential inequalities, T. Wazewski proved in 1950 the following theorem:
Consider the system of differential inequalities given by
$$ \...
- 1,590
3
votes
1
answer
309
views
Extremizing sequence consists of two elements
Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,...
3
votes
1
answer
246
views
What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?
This question was posted in MSE but is still open hence posting in MO.
The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a ...
- 5,470
3
votes
1
answer
240
views
A certain generalisation of the golden ratio
Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$
We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...
- 443
3
votes
2
answers
698
views
Estimate of a ratio of two incomplete gamma functions
I would like to bound from above the expression
$$
\frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}
$$
for $x>y>0$. By plotting the above expression I have found that ...
- 263
3
votes
1
answer
99
views
A bound on an oscillatory solution of an ODE
This question was restated as follows:
Let $V\colon[a,b]\to\mathbb{R}$ be smooth, strictly decreasing and
$V(b) = 0$. Suppose that $f\colon[a,b]\to\mathbb{R}$ is smooth and
satisfies $f''(x)+V(x) f(x)...
- 128k
3
votes
2
answers
287
views
An inequality for an integral transform of a function
Let
$$J_{f;y}(u):=3 u^3 \int_u^1\frac{dt}{t^4} \,e^{-i y t}f(t)- e^{-i u y}f(u),$$
where $y\in(0,\infty)$, $u\in(0,1)$, and
$$f(t):=t+\pi (1-t) t \cot (\pi t).$$
Here are the graphs of $f$ (black), ...
- 128k
3
votes
1
answer
223
views
Ratio of Selberg integral
I'm considering a ratio of incomplete Selberg integral:
$$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...
- 1,495
3
votes
1
answer
71
views
Minoration of linear forms
In the book Number Theory IV from Parshin, one can find this statement (precisely at p. 215) with the comment "it is easy to see that...":
Let $a_{ij}$ ($1\le i,j\le m$) be complex numbers ...
- 3,998
3
votes
1
answer
392
views
Gronwall lemma for a $2$-dimensional system of linear differential inequalities
Let $$z(t)=\begin{pmatrix}x(t) \\ y(t)\end{pmatrix}~,~ A=\begin{pmatrix}0 & -\beta \\ \alpha & -(\alpha + \beta)\end{pmatrix}$$ satisfies the following system of linear differential ...
- 730
3
votes
1
answer
321
views
Bounding a series of nested integrals
Consider the following matrix function
$$
f(t) = \cos(\omega_1t) A_1 + \cos(\omega_2t) A_2, \quad t\ge 0,
$$
where $A_1$, $A_2$ are real square matrices and $\omega_1$, $\omega_2$ positive numbers.
...
- 2,712
3
votes
1
answer
334
views
Improved Hardy Inequality when orthogonal to radial functions?
A formulation of Hardy's Inequality on $\mathbb{R}^d$ states that when $0\le s<d$ and $1< p<\tfrac{d}{s}$ that
$$ \|\frac{u}{|x|^s}\|_{L^p(\mathbb{R}^d)}\lesssim_{s,p,d} \||\nabla|^s u\|_{L^...
- 141
2
votes
1
answer
652
views
Are there such numbers?
Maybe better to ask for help on this question here: Are there eight numbers $0< x_{1},\ldots,x_{4},y_{1},\ldots,y_{4}<1$,
such that
$$
\frac{\max\left(\left(x_{i}-x_{j}\right){}^{2}+\left(y_{i}-...
- 29
2
votes
1
answer
260
views
An elementary functional inequality
Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold
$\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...
2
votes
2
answers
411
views
Given a total variation distance from uniform, how well can we bound the probabilities of sub-intervals?
I made the following claim, which I now see that I don't know how to prove. Can anyone prove it?
Claim. Let $f$ be a concave and non-negative function on $[0,1]$ with $$\int_0^1 f = 1,$$
$$\int_0^1 |f-...
user44143
2
votes
3
answers
260
views
Nonzero solutions of an infinite product
Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$.
Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$
$$f(a;b):=\prod\limits_{k=1}^\infty ...
- 293
2
votes
3
answers
593
views
l^p space inequality related to compressed sensing
I'm trying to read Donoho's 2004 paper Compressed Sensing and am having trouble with a supposedly trivial statement (equation 1.2 on page 3).
He makes the sparsity assumption on $\theta \in \mathbb{R}...
- 198
2
votes
2
answers
150
views
Lower bound for integrals like $\int_1^{t+1}e^{-\sqrt{s}}s^{-1}ds$
Let
$$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