All Questions
Tagged with ca.classical-analysis-and-odes inequalities
176 questions
2
votes
2
answers
358
views
An ODE comparison problem
Recently I met an ODE problem but after thinking for quite a while I still could not find an answer. Here is the question, which looks very simple:
Let $y=y(t)$ be a smooth function defined on $[0,\...
- 1,350
2
votes
1
answer
138
views
The blow-up rate of a nonlinear oscillator
(Related to this Math.SE question.)
For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$
...
- 692
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
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 ...
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
$$
|v-\overline{v}|\le \left\Vert v' \right\Vert_{r,I} \ell^{1-\frac{1}{r}}, ...
- 65
2
votes
1
answer
198
views
Bounds for the beta CDF
This question is closely related to a previous question that I asked here:
An inequality involving the beta distribution
Let $a,b$ be strictly positive integers, and let $F_{a,b}(x)$ denote the CDF ...
- 4,049
2
votes
1
answer
286
views
An inequality for abstract Cauchy problem
Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates an analytic $C_0$-semigroup on a Banach space $X$. How we can prove an inequality of ...
- 97
2
votes
1
answer
675
views
Moment generating function of random unit vector
Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of
$$\mathbb{E}[\exp(X^Tv)]$$
for any $v$?
- 1,495
2
votes
1
answer
214
views
Ratio of hypergeometric function
Given $a>b>0$, is there any upper bound of the following ratio of hypergeometric function?
$$\frac{_2F_1(a,1-b;a+1;x)}{_2F_1(a,1-b;a+1;y)}$$
for $1>x>y>0$ ideally in the form like some ...
- 1,495
2
votes
1
answer
164
views
Coefficients of certain Taylor series
For $t\in(-1,1)$, let
$$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)}.$$
Note that the functions $f$ and $g$ are even.
Question 1: Is ...
- 128k
2
votes
1
answer
253
views
Differential inequalities for a strictly diagonal dominant system of linear ODEs
Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column ...
- 135
2
votes
1
answer
267
views
Hardy-type inequality for point boundary
Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...
- 83
2
votes
1
answer
74
views
An inequality about the second-order difference
Fix a continuously differentiable but nowhere twice differentiable function $f$ on $\mathbb{R}$ supported on $[0,1]$. Is it true that for all $x\in[0,1]$ and all $\delta$ sufficiently small
\begin{...
- 419
2
votes
1
answer
248
views
Ratio of expectation involving random unit vectors
Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, ...
- 1,495
2
votes
1
answer
1k
views
Inequality concerning absolute value of a polynomial
Let
$$f(z) = (1-1/t) z^w + z/t - 1$$
with integers $t\geq2$ and $w\geq2$.Let $r=1+1/(tw^3)$. How do I show
$$\left\lvert f(r e^{i\varphi}) \right\rvert \geq \left\lvert f(r) \right\rvert$$
for any $\...
- 577
2
votes
0
answers
159
views
Upper bound of a product of sines
Consider the function
$$ 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.
$$
In other words $X_A$ is the ordered exponential of $...
- 577
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\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 $$
for $\epsilon>0$, $f \in L^\...
user139844
2
votes
0
answers
73
views
Question about Gidas-Ni-Nirenberg result
Background:
So I know that the Euler Lagrange equation associated with the Sobolev inequality takes the following form,
$$-\Delta u = u^p$$
where $p=2^*-1$ and here we assume that $u>0$ on $\mathbb{...
- 537
2
votes
0
answers
85
views
inequality for two integral expressions
Given bounded positive functions $f, g, h \in L^1$, with $g, h$ finitely supported, I would like to compare the following two expressions:
$$\begin{aligned}
a_1 &= \int_{0}^{\infty} \! dx \, f(x) \...
user114668
1
vote
1
answer
547
views
Coefficient bounds of an inequality
Hello,
Given positive integers $k$ and $n$. Are there upper bounds on coefficients $A$ and $B$ such that they depends only on $k$ (eg., $2 k^k$) and for all non-negative integer sequences $(a_i)_{1}^...
- 175
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
1
vote
1
answer
444
views
Upper bound of a ratio of integrals
I'm wondering how to upper bound the following ratio of integrals:
$$\frac{\int_{\Delta_a}(\prod_{i=1}^n\lambda_i)^{p-1}\prod_{i<j}|\lambda_i-\lambda_j|}{\int_{\Delta_b}(\prod_{i=1}^n\lambda_i)^{p-...
- 1,495
1
vote
1
answer
529
views
sobolev embedding theorem in the smooth metric measure space
we know the sobolev embedding theorem of Saloff-Coste
$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu
$
wtih $Ric\ge-(n-1)K$, for ...
- 199
1
vote
3
answers
379
views
monotonicity from 4 term-recursion.
In determining the monotonicity of coefficients in a series expansion (which appeared in one of my study), I come across the following problem.
Let $p\ge 2$ be an integer, and $$6p^3(i+3)d_{i+3}=6p^...
- 1,858
1
vote
1
answer
454
views
A polynomial inequality
Given $x_1,x_2, \ldots, x_n \ge 0, \alpha \ge 1$, show that
$\sum_{i}\alpha x_i(\sum_{j \le i}{x_j})^{\alpha-1} \ge (\sum_{i}x_i)^\alpha$
We're pretty sure the inequality holds for the given ...
- 13
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
1
vote
1
answer
507
views
Upper bound of the fraction of Gamma functions
Is there a simple upper bound of the following fraction of Gamma functions for any $a,b\geq1/2$:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$
An upper bound in the following form is ...
- 1,495
1
vote
1
answer
114
views
additive discrepancy under a multiplicative constraint
Consider four sequences of numbers, $0 \le a_i, b_i, c_i, d_i \le 1$, suppose they satisfy the following constraints:
(1). $\sum_{i=1}^K a_i \ge 1/2 + \epsilon$;
(2). $\sum_{i=1}^K d_i \le 1/2 - \...
- 23
1
vote
1
answer
356
views
Statistical inequality
Let $X$ be a finite discrete variable and $X\ge0$. Is it true that
$$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$
where $\operatorname{...
- 95
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
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
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
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
$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\...
- 19
1
vote
1
answer
135
views
Positive Definiteness of a certain function
Recall that a function $f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}^d$, the matrix $(a_{ij})$ with entries
$$a_{ij} = ...
- 9,853
1
vote
2
answers
345
views
how to solve a singular integral equation involving the kernel $1/x$
Dear all,
Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...
- 1,649
1
vote
2
answers
204
views
One inequality connected with the linear second order ODE
Is the following statement true?
Let $ a>0, b>0, h>0 $, $x(t)$ be the solution of the differential equation
$ \ddot{x}+a \dot{x}+bx=h$
with initial conditions $x(0)=u<0 , \dot{x}(0)...
- 97
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
1
answer
113
views
Prove $\int_\Omega \left(\rho_{1} \ln \frac{\rho_{1}}{\rho_{2}}\right)dx dy \leq C\int_\Omega |\rho_1-\rho_2|dxdy$ for $0 \le \rho_1, \rho_2 \in L^1$
Let $\rho_1, \rho_2 \in L^1(\Omega;\mathbb R_+)$ such that $\int \rho_i|\ln \rho_i| < \infty$. Is it true that there exists a constant $C>0$ such that
\begin{align*}
\int_\Omega \left(\rho_{1} \...
user140746
1
vote
2
answers
405
views
Controlling solutions of a second order linear differential inequality
A slightly less general version of this question was asked, in a subsequent comment, by the OP of the question at
Controlling subsolutions of a second order linear ODE
Let $f:[0,\infty) \to \mathbb{...
- 128k
1
vote
1
answer
402
views
Is there a complex analog of this sharpened Cauchy Inequality?
Let $x$ and $y$ be two points on the unit sphere $S^{n-1}$ in Euclidean space ${\mathbb{R}}^n$. Suppose that the angle $\theta$ between the points $x$ and $y$ is acute, so that the dot product $x\cdot ...
user2529
1
vote
1
answer
1k
views
How to estimate derivatives of multivariate polynomial near a manifold
I have to provide a (Markov or Bernstein-based?) inequality that gives an upper bound for the partial derivatives of a multivariate polynomial calculated near a real smooth surface in terms of value ...
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
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:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
1
vote
2
answers
167
views
Prove inequality [closed]
I have to prove the following inequality (See Appendix A of Classical Fourier Analysis of Grafakos).
$$
\left(\frac{(1+s/y)^y}{e^s}\right)^{2y} \leq (1+s)^2 / \exp(s), \quad s\geq 0
$$
and
$$
\left(\...
- 59
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
$$\|\nabla f^*\|_2\leq \|\...
- 333
1
vote
0
answers
107
views
$L^p$ inequality for "positively correlated" random variables
Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following "positive correlation" property: for all non-negative integers $\alpha_1,\ldots,\...
- 7,193
1
vote
0
answers
138
views
An Elementary Inequality [closed]
A friend of mine found in the internet the following exercise:
Let $a, b, c \geq 0$ with $a + b + c = 1$. Show that
$$
\sqrt{a + b^2} + \sqrt{b + c^2} + \sqrt{c + a^2} \geq 2,
$$
where equality is ...
1
vote
0
answers
84
views
How to show that $\Delta W \leq −2(n − 4)V$?
I am reading a preprint and trying to understand the proof of Lemma 3.5. On Pg. 19 above eqn (3.49) the authors claim that $\Delta W \leq −2(n − 4)V$ where the functions $W$ and $V$ are defined below,
...
- 537
1
vote
0
answers
151
views
Log-concavity inequality
Let $x,y,$ and $t$ be fixed real numbers, $1<x<y$, $0<t<1$. Does the following inequality hold for some $c$
$$\frac{\log{(tx+(1-t)y)}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{(sw+(1-s)z)}}{...
- 3,209