# Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

1,673
questions

3
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### An inequality about factorial function

Let $d,s,k$ be integers such that $d<s+2$, $s=o(k)$. For sufficiently large integer $k$, is the following inequality right?
$$\frac{(k-2d+1)^{k+s-d}}{(k-d)!\cdot (k-2)_s} \ge 1$$
We write $(k)_s = ...

0
votes

0
answers

44
views

+100

### Maximum of $\sum_{n=1}^N z^T X(P_n X + I)^{-1}z$ over unit trace, positive semidefinite matrices?

Let $z$ denote a unit vector.
Fix a finite collection of positive semidefinite matrices $\mathcal{P}$.
Define the function and set
$$
f_{\mathcal{P}}(X) = \sum_{P \in \mathcal{P}} z^T X(PX + I)^{-1} z,...

0
votes

0
answers

135
views

### Inequalities on the distribution of the maximum of the normalized sum $\max_{k = 1,\dots,n} \frac{|S_k|}{\sqrt{k}}$

Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. random variables with $\mathbb{E}(X) = 0$,$\mathbb{E}(X^2) = \sigma^2$ and finite moments. Let $S_k = \sum_{i = 1}^{k} X_i$ and consider the normalized ...

0
votes

0
answers

87
views

### Sufficient conditions for the rank of the Hadamard product to be as large as possible

Let $A$ and $B$ be hermitian $n \times n$ positive semidefinite matrices. Assume that the rank of $A$ is 2. We know that
$$ \operatorname{rk}(A \circ B) \leq \operatorname{rk}(A) \operatorname{rk}(B) =...

0
votes

0
answers

77
views

### A conjectured generalization of Oppenheim's inequality, inspired by Horn-Yang's theorem

In this post, $A$ and $B$ are hermitian $n \times n$ positive semidefinite matrices.
It is well known that if $A$ has rank $n$ and if $B$ has only positive entries on its diagonal, then the rank of ...

2
votes

0
answers

83
views

### Elliptic regularity theory in $\mathbb{R}^2$

I recently encountered two papers discussing elliptic PDEs and variational methods. The first paper claims that according to regularity theory, the solution to $-\Delta u = ug(u^2)$ in $\mathbb{R}^2$ ...

2
votes

2
answers

159
views

### Gronwall's inequality in discretized time

$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...

2
votes

1
answer

275
views

+50

### A maximal inequality

Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. symmetric random variables, with $-1\leq X_i\leq 1$, $\mathbb{E}(X_i) =0$, $\mathbb{E}(X_i^2) = 1$. We have that:
$$
P\left(\bigcap_{k = 1}^{n}\frac{|\sum_{i = ...

3
votes

0
answers

125
views

### Stirling number, Delannoy number, and binomial coefficients in a sum

I want to compute/prove that the following sum is positive:
$$ \sum_{i = 0}^n \left[\frac{D(n - i, i)}{d} \sum_{j = m}^d s(d, j) \binom{j}{m} (d - i)^{j - m}\right] > 0
$$
where $s(d, j)$ is the ...

-1
votes

0
answers

87
views

### Sufficient conditions for $1/f\in L^p$

This is a very simple question that I have not found a satisfactory answer for. When is the reciprocal of a function $f:[-1,1]\to\mathbb{R}$ in $L^p([-1,1])$? In other words, when is $\int1/f^p\,dx<...

1
vote

0
answers

105
views

### Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?

Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...

1
vote

0
answers

56
views

### Inequality involving minors of an orthogonal matrix

Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following ...

2
votes

0
answers

52
views

### stochastic process and integral

Let $(X_n(t))_{t\in [1,+\infty], n\geqslant 1}$ be a sequence of nonnegative random variables and $(\mathcal{F}_s)$ a filtration ($\mathcal{F}_s \subset \mathcal{F}_r$ for $s\leqslant r$). We assume ...

1
vote

1
answer

146
views

### Characteristics of numbers that satisfy inequality related to diophantine approximation with perfect squares

This question has come up in my algorithms and physics research. I apologize if this is very basic, but I am new to number theory and it seems this is a number-theoretic question. What can we say ...

2
votes

1
answer

98
views

### Simultaneous Concentration of $\sum_{i = 1}^{n} X_i^2$ and $\sum_{i = 1}^{n} X_i$ with $X_i$ iid. Poisson

Consider $n$ independent Poisson(1)-distributed random variables $(X_i)_{1 \leq i \leq n}$.
This is a (hopefully more interesting) follow-up question to Super-exponential concentration for $\frac{\...

2
votes

1
answer

104
views

### How to lower bound the absolute value of the difference of two Kullback-Leibler divergences given the constrains below?

Given that $\min (D[p_1||p_3],D[p_2||p_4])=a$, how to find a lower bound of the difference $\left \vert D[p_1\parallel p_2]-D[p_3\parallel p_4] \right\vert$ with respect to $a$? If the condition is ...

1
vote

1
answer

91
views

### Small total variation distance between sums of random variables in finite Abelian group implies close to uniform?

Let $\mathbb{G} = \mathbb{Z}/p\mathbb{Z}$ (where $p$ is a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$.
For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\...

1
vote

0
answers

40
views

### Inequality with characteristic and moment generating functions

Suppose $\mu$ is a symmetric probability measure on $\mathbb{R}$. Define its characteristic and moment generating functions:
\begin{align*}
\phi_{\mu}(t) = \int_{\mathbb{R}}{\cos(xt) d\mu(x)}
\\
M_{\...

2
votes

1
answer

89
views

### Maximal sum of a function defined on 2-dimensional grid if the sum is bounded on product sets

Disclosure: I have asked this question on MSE (https://math.stackexchange.com/questions/4895621/maximal-sum-of-a-function-such-that-the-sum-is-bounded-on-product-sets) but received no comments in a ...

9
votes

1
answer

546
views

### Entropy arguments used by Jean Bourgain

My question comes from understanding a probabilistic inequality in Bourgain's paper on Erdős simiarilty problem: Construction of sets of positive measure not containing an affine image of a given ...

1
vote

0
answers

73
views

### Multilinear non-commutative Khintchine inequality

Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...

0
votes

0
answers

137
views

### Notation $\le_{a,b,n,\ldots}$ in Analysis

In modern Analysis, especially Functional Analysis, one proves, or one uses inequalities of the form
$$F(X)\le_{a,\ldots,n}G(X).$$
The meaning of the subscripts in the inequality sign means that there ...

0
votes

1
answer

60
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 ...

2
votes

1
answer

282
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(...

0
votes

0
answers

84
views

### Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables

Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...

0
votes

1
answer

44
views

### Norm of a $2$-tuple of operators

Let $E$ be a complex Hilbert space and $K_1,K_2$ are bounded linear operators on $E$.
Let $\omega(K_1)$ and $\omega(K_2)$ be the numerical radius of $K_1$ and $K_2$ respectively. That is
\begin{align*}...

3
votes

1
answer

418
views

### Does $f(t) \le \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$ imply $f=0$?

Let $\beta \in (0, 1)$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that
$$
f(t) \le \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s,
\quad \...

2
votes

1
answer

125
views

### Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$

Let $\alpha \in (0, \infty)$ and $\beta \in (0, 1]$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that
$$
f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(...

3
votes

0
answers

369
views

### Some specific case of the abc-conjecture related to the Mason-Stothers proof of Serge Lang for polynomials?

I stumbled upon a version of the Mason-Stothers theorem, which can be modified to make a similar statement for natural numbers, which I think I can prove here. It is not the abc-conjecture, but it has ...

5
votes

1
answer

307
views

### Does the Poincaré inequality hold on annular domains?

Does the following Poincaré inequality hold
$$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$
where $B_r$ denotes a ball of radius ...

5
votes

0
answers

231
views

### Monotonicity of ratio of symmetric polynomials

The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...

0
votes

0
answers

59
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) $f$ ...

-1
votes

2
answers

216
views

### Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$

Suppose $f \colon [0, 1] \to \mathbb{R}$ is continuously differentiable, and satisfies $f(0) = f(1) = 0$ and $\|f'\|_2 \leq 1$.
I am wondering if it there is a constant $C > 0$ such that for all ...

0
votes

0
answers

69
views

### Maximizing the trace of the resolvent of a Wishart matrix over positive unit trace matrices?

Let $G$ be a standard $d \times d$ Wishart random matrix and consider the problem of maximizing the function
$$
f(M) = \mathbb{E}\Big[\mathrm{tr}((G + M^{-1})^{-1})\Big],
$$
over the class of real ...

0
votes

1
answer

59
views

### Does point process ordering ever imply conditional intensity ordering?

Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...

2
votes

0
answers

142
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 ...

0
votes

0
answers

82
views

### Some new questions on Rademacher complexity

For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable.
Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...

0
votes

3
answers

267
views

### A generalisation of Tchebychev inequality

Let $f,g \in C(\mathbb R)$ with $\exists M \in \mathbb R^*, \forall (x,y) \in \mathbb R^2, M\times (f(x)-f(y))(g(x)-g(y)) \geq 0$.
Is it true that exists $ u$ any real function, and $a,b$ monotone ...

2
votes

2
answers

215
views

### $L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?

Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...

3
votes

0
answers

78
views

### Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set

Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...

2
votes

2
answers

180
views

### $L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...

2
votes

0
answers

156
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 $...

0
votes

0
answers

91
views

### On polynomial equation of fourth order depending on two parameters and bound on a maximal root

I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$:
\begin{eqnarray}
F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p^...

1
vote

0
answers

70
views

### Integral inequality related to the (mixed?) moments of two functions

For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set
$$
S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}.
$$
Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce
$$
...

2
votes

1
answer

170
views

### Estimating ${\left(\sum_{i=j}^k {x_i}\right)^2} \times \left\lvert\sum_{i=j}^k {a_i}\right\rvert$

Given two sets; $X = \{x_i : x_i \geq 0; i \in [\sqrt{n}]\}$ and $A = \{a_i : |a_i| \leq 1; i \in [\sqrt{n}]\}$ of size $n^{\frac{1}{2}}$ each, with the following properties
\begin{equation}\label{...

5
votes

1
answer

143
views

### Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$

Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...

0
votes

1
answer

98
views

### Demonstrating bound on integral

I found this question on another forum and it got me interested because of how tight the bound is: prove that
$$\int_0^\infty \frac{\arctan(x)}{x^2 + 4}\,dx > \frac{\pi}{4}.$$
The difference ...

0
votes

0
answers

59
views

### Upper bound for an additional Product formula

We have three sequences of positive integers $l$, $p$ and $q$ such that:
$$
p_1 \geq p_2 \geq \cdots \geq p_k\text{ and } q_1 \geq q_2 \geq \cdots \geq q_k \geq \cdots \geq q_h \text{ where: } k < ...

0
votes

1
answer

138
views

### An inequality of Huygens

I am preparing a paper on Huygens' approximations to $\pi$ and I have to prove the following inequality: if $0 \le x\le \pi/2$ then
$$
\pi\ge \frac{\pi}{x}\sin(x)\bigl(20+51\cos(x)-2\cos^2(x)+6\cos^3(...

0
votes

0
answers

208
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}...