All Questions
Tagged with ca.classical-analysis-and-odes inequalities
176 questions
1
vote
0
answers
100
views
Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
0
votes
1
answer
71
views
Upper bound on higher order derivatives of $\frac{1}{v(t)}$
Suppose that $ v(t) >l>0$ and
$$
\vert v^{(k)}(t) \vert \leq c \frac{k!}{r^k}.
$$
Can we give an upper bound for
$$
(\frac{1}{v(t)})^{(k)}
$$
?
Attempt:
We first compute the first fourth order ...
- 269
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
4
votes
4
answers
474
views
A certain inequality involving square roots of polynomials
I want to prove the inequality
$$\begin{aligned}
&\sqrt{(x - 1)^2 + y^2}\Big[y^2(9x - 6) - 9x^2 + 9x^3\Big]+ y^2(16x^2 - 16x + 7)\\
&- \sqrt{x^2 + y^2}\Big[9x + y^2(9x - 3) + \sqrt{(x - 1)^2 + ...
- 1,109
0
votes
2
answers
364
views
Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
- 178
6
votes
1
answer
568
views
Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
- 178
2
votes
1
answer
208
views
Proving an exponential sum inequality for symmetric Hamming distance sequences in binary vectors
Background: Let $X = \{0,1\}^k$ represent the set of all binary vectors of length $k$. For two binary vectors $x, y \in X$, the Hamming distance $d_H(x, y)$ is defined as the number of positions where ...
- 349
3
votes
1
answer
309
views
Extremizing sequence consists of two elements
Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,...
7
votes
1
answer
179
views
More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let
$$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
- 128k
6
votes
2
answers
492
views
Does this polynomial have a real zero less than or equal to $1/2$?
Is the smallest root $x$ of
$$
10x^{3}-30x^{2}+\left(30-2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}\right)x\\
+2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}-\sum_{1\le i<j<k\le6}\cos\alpha_{ij}\cos\...
user538574
9
votes
3
answers
2k
views
Smallest root of a degree 3 polynomial
Is it true that the smallest root $t$ of the polynomial
$$
20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
- 171
21
votes
2
answers
2k
views
Boundedness of sum of sin(sin(n))
Playing with desmos I have accidentally noticed that the sequence of partial sums
$$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$
is bounded.
However, I did not succeed in proving this ...
1
vote
1
answer
148
views
An inequality about binomial distribution
Statement
Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that
$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$
...
- 35
0
votes
0
answers
43
views
The reciprocal of the normalized tail of the Maclaurin power series expansion of the hyperbolic sinc function is a convex function
The classical Bernoulli numbers $B_j$ are generated by
\begin{equation}\label{Bernoulli-No-Generating}
\frac{x}{\operatorname{e}^x-1}=\sum_{j=0}^\infty B_j\frac{x^j}{j!}=1-\frac{x}2+\sum_{j=1}^\infty ...
- 1,091
8
votes
2
answers
492
views
A trig inequality
Let $m$ be an odd and $n$ an even positive integers. I need to estimate the maximum value of $\left|\sin(m\theta)\sin(n\theta)\right|$:
$$
c_{m,n}:=\max_{\theta\in[0,\pi]} \left| \sin(m\theta) \sin(n\...
- 697
0
votes
1
answer
75
views
On a differential inequality with an additional constraint
I am stuck on this problem from a research question, which seems to require solving a differential equation, but I am not sure how to deal with integrals like $\int_0^t$ or $\int_t^1$. I will be ...
- 33
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) $f$ ...
2
votes
0
answers
159
views
Upper bound of a product of sines
Consider the function
$$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$
I wonder whether it is possible to compute some nontrivial upper ...
- 826
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
0
votes
0
answers
211
views
Gauss transformation in fractional Sobolev space
Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
- 101
1
vote
1
answer
109
views
Bound on $L^1$ norm of solution of two-point boundary value problem
This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
- 3,065
43
votes
3
answers
2k
views
Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？
For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that
$$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$
Let
\begin{...
- 4,280
3
votes
1
answer
246
views
What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?
This question was posted in MSE but is still open hence posting in MO.
The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a ...
- 5,470
6
votes
1
answer
796
views
A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\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{ ?}$$
- 128k
1
vote
0
answers
70
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
0
votes
1
answer
89
views
On the validity of a certain Grönwall-type inequality
Assume that $u~ \colon \mathbb{R}_+ \to [-M,M]$ is a bounded continuously differentiable function such that $u(0) = 0$ and $$u(t) \leq \int_0^t \lambda(s)~u(s)~\mathrm{d}s + C \label{1}\tag{1}$$ where ...
- 730
6
votes
1
answer
519
views
Cauchy-Schwarz-like inequality with a power $p$ term
We set :
$\phi_1, \phi_2 : \mathbb{R} \to \mathbb{R}^M$ with compact support
$\theta \in \mathbb{R}^M$ non-zero, $\theta_{p-1} = (\operatorname{sign}(\theta_i)|\theta_i|^{p-1})_{1 \leq i \leq M} \in \...
- 121
3
votes
1
answer
391
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
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
1
vote
2
answers
167
views
Prove inequality [closed]
I have to prove the following inequality (See Appendix A of Classical Fourier Analysis of Grafakos).
$$
\left(\frac{(1+s/y)^y}{e^s}\right)^{2y} \leq (1+s)^2 / \exp(s), \quad s\geq 0
$$
and
$$
\left(\...
- 59
0
votes
0
answers
103
views
Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity
Problem:
Consider the autonomous ODE system
\begin{align*}
\dot{x} &= (1-x) (z-xy)\\
\dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\
\dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z
\end{...
- 291
1
vote
2
answers
183
views
What is the easiest way to prove the correctness of this inequality
I have the following inequality for some $0<x<0.1$: $$x^{1/10}-(1-(1-x)^{x^{-0.5}}) \geq 0$$
Is there an easy way to prove the correctness of such inequality? Thanks!
- 11
4
votes
1
answer
149
views
An algebraic inequality in three real variables
Is it true that
$$(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
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
$$ f’’(x) \leq 0\quad \text{and} \...
- 4,135
2
votes
2
answers
189
views
Elementary convexity example
I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...
4
votes
1
answer
326
views
When does the sine transform result in a positive function?
For each $x>0$ is, $$\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt > 0\ \ \ ?$$
In general, it is known that if the kernel is decreasing, then the sine transform is positive. Note that this ...
- 402
2
votes
1
answer
74
views
An inequality about the second-order difference
Fix a continuously differentiable but nowhere twice differentiable function $f$ on $\mathbb{R}$ supported on $[0,1]$. Is it true that for all $x\in[0,1]$ and all $\delta$ sufficiently small
\begin{...
- 419
4
votes
1
answer
353
views
Inequalities involving binary representation of integers
Let $N\geq 1$ be a positive integer and assume that $N=2^{n_1}+2^{n_2}+\cdots+2^{n_{p}}$, $n_{1}>n_{2}>\cdots>n_{p}\geq 0$, is the binary representation of $N$. I believe that the following ...
- 41
1
vote
1
answer
158
views
How do I integrate this inequality that appears in a paper of Rabinowitz?
Sorry if this is too easy for MO, but I found it in a research paper, so I thought that it was worth posting here.
I was reading a paper by Rabinowitz(this one to be more precise) and I came across ...
- 113
1
vote
0
answers
51
views
Polya-Szego inequality w.r.t. partial direction
Let $f\in H^1(\mathbb{R}^d)$ and let $f^*$ be its symmetric deceasing rearragement. Then the Polya-Szego inequality tells us that $f^*$ is also in $H^1(\mathbb{R}^d)$ and
$$\|\nabla f^*\|_2\leq \|\...
- 333
0
votes
1
answer
125
views
Bounding integral expression with Sobolev norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
1
vote
0
answers
107
views
$L^p$ inequality for "positively correlated" random variables
Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following "positive correlation" property: for all non-negative integers $\alpha_1,\ldots,\...
- 7,193
2
votes
1
answer
164
views
Coefficients of certain Taylor series
For $t\in(-1,1)$, let
$$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$
and
$$g(t):=\frac1{f(t)}.$$
Note that the functions $f$ and $g$ are even.
Question 1: Is ...
- 128k
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
$$ \frac{\partial^2}{\partial x_1^2}u <0 $$
...
- 4,135
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
1
answer
206
views
A simple 1-dimensional inequality, maybe Poincaré inequality or Hölder inequality?
I'm reading a paper on the classical Gagliardo-Nirenberg interpolation inequality arXiv link and there is a inequality used
$$
|v-\overline{v}|\le \left\Vert v' \right\Vert_{r,I} \ell^{1-\frac{1}{r}}, ...
- 65
43
votes
1
answer
2k
views
Is $\int_0^\infty{dx\over x^{x^{x^x}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^x}}}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^{x^{x^x}}}}}}}<\cdots$ true?
On MSE I asked whether each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ was less than $2$ and received answers on bounding ...
- 1,459
1
vote
1
answer
146
views
Extremizers of the Sobolev inequality
Background:
I am reading the paper: Best constant in Sobolev inequality by Talenti (see here) and I am trying to understand the following step.
On p. 365, the author is arguing that the solutions to ...
- 537
1
vote
0
answers
138
views
An Elementary Inequality [closed]
A friend of mine found in the internet the following exercise:
Let $a, b, c \geq 0$ with $a + b + c = 1$. Show that
$$
\sqrt{a + b^2} + \sqrt{b + c^2} + \sqrt{c + a^2} \geq 2,
$$
where equality is ...
