The inequalities tag has no usage guidance.

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votes

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199 views

### An inequality for $L^p$-functions [on hold]

I am interested in the following inequality:
\begin{equation}
\int\limits_\Omega \left\vert f_1 - f_2 \right\vert^p ~ d\mu \le C_p \left[ \int\limits_\Omega \left\vert f_1 \right\vert^p ~ d\mu + ...

**2**

votes

**3**answers

215 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 ...

**2**

votes

**0**answers

54 views

### Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question:
Bounding the absolute sum of entries of the ...

**2**

votes

**1**answer

71 views

### A continuity/bootstrap argument

I am trying to understand how one can prove the following assertion using a continuity argument:
Let $0<\epsilon<\epsilon_0$. Let $I=[t_0,R]$ be a compact interval. Suppose that $S:I\to ...

**5**

votes

**4**answers

775 views

### Is the componentwise square-root of a positive-definite matrix also pos.-def.?

Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a_{ij} = a_{ji} \geq 0$ and
$B=(b_{ij})$ with $b_{ij} = \sqrt{a_{ij}}$.
Is $B$ positive-definite whenever $A$ is?
In other words:
...

**15**

votes

**0**answers

600 views

### A curious eigenvalue inequality

Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I ...

**4**

votes

**1**answer

421 views

### inequality with exponents

We are given a graph $G$, each vertex $v$ has an assigned value $\gamma_v\in [0,1]$, and it happens that for every $v$ we have $\gamma_v+\sum_{u\in \delta(v)} \gamma_u = 1$. Assume that $\sum_v ...

**3**

votes

**1**answer

174 views

### Inverse Ackermann Function

The inverse Ackermann function is defined over the natural numbers as follows: ($[x]$ means that we round up x to the nearest integer, while $\log^*$ is the iterated log function discussed here: ...

**1**

vote

**1**answer

52 views

### Inequality implies locally uniform convergence of a series

We have the inequality
$$\alpha_n(t) \le 4\pi(3\pi)^{1/3} \exp\left\{\int_0^t(1+3F_P(\sigma)) \, d\sigma \right\} \cdot \int_0^t P(\sigma)^2(3CD_1^2)^{1/3}\alpha_{n-1}(\sigma) \, d\sigma$$
for ...

**5**

votes

**1**answer

207 views

### Estimate of incomplete binomial integral

Let $0\le k \le n$. Prove that
$$
n\binom{n}{k}\int_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2.
$$
As far as I know
1) it is proved for $\frac{k}{n+1}\le 1/2$ and
2) not proved for $1/2 ...

**4**

votes

**1**answer

255 views

### Time-efficient way of calculating the least number of 1s in a representation of $n$ using only the operations $+,!$

This was inspired by the following paper:
J. Arias de Reyna, J. van de Lune, "How many $1$s are needed?" revisited, arXiv link.
It might help explain my question better, because my question is ...

**8**

votes

**1**answer

252 views

### An inequality on the simplex involving $x^x$

Is anything known about the behavior of the function $$f(x)=\prod_{i=1}^n x_i^{x_i}$$ on the standard simplex, i.e. the set $\{x\in\mathbb{R}^n:\sum_{i=1}^n x_i=1, x_i\geq0\}$? I ask because I have ...

**11**

votes

**2**answers

375 views

### regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that
$$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$
is constant on $L.$ Could somebody ...

**6**

votes

**0**answers

215 views

### An inequality which involves a sum of integrals

Please help me to prove
$$
\sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad ...

**3**

votes

**1**answer

106 views

### Sharpened Pinsker inequality for special case

Let $B(p)$ denote the Bernoulli distribution over $\{0,1\}$ and $B(p)^n$ the corresponding product distribution over $\{0,1\}^n$. For $n>1$ and $0<x<1$, define
$$P_n(x):=B(\frac12+\frac ...

**2**

votes

**1**answer

84 views

### Inverse Hadamard determinant inequality

As far as I remembered there is an inverse Hadamard inequality for the determinant of the form
$$
|D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)}
$$
providing all values in $(\cdot)>0$.
...

**4**

votes

**2**answers

2k views

### Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...

**1**

vote

**1**answer

88 views

### Tail inequality for orthomartingales/martingale difference random fields

It is known that if $(S_i= \sum_{j \leqslant i }X_i, \mathcal F_i)$ is a martingale,
then for each
$
\beta>1$, $\delta\in (0,\beta-1)$ and $\lambda>0$, and each integer $N \geqslant 1$, the ...

**1**

vote

**1**answer

99 views

### Nonlinear elliptic problem involving the p-laplacian, Hölder inequality

I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan.
I have a problem understanding one step ...

**-1**

votes

**1**answer

124 views

### Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]

Inequality
What values of $n$ satisfy the following inequality?
$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$
$p$ are prime numbers and the notation $p_i$ indicates the $i$th ...

**1**

vote

**1**answer

70 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
$$ ...

**4**

votes

**1**answer

235 views

### The values of $n$ which satisfy an inequality about prime numbers

For which values of $n$ does the following inequality hold for?
$$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$
$p$ are prime numbers and the notation $p_i$ indicates the ...

**3**

votes

**1**answer

144 views

### How to use Gronwall's inequality? [closed]

I am just trying to understand the role of Grownwall's Lemma to show global wellposedness results, in the paper I have been reading. And So I hope this is OK for MO.
Let $u\in C(\mathbb R, ...

**0**

votes

**1**answer

381 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 ...

**2**

votes

**1**answer

93 views

### Sufficient conditions for weak majorization

Suppose $x_1\ge x_2\ge \cdots \ge x_n\ge 0$ and $y_1\ge y_2\ge\cdots\ge y_n\ge0$ be reals such that for any positive integer $p$,
$$
\sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p.
$$
Question: Is ...

**11**

votes

**1**answer

600 views

### Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...

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votes

**2**answers

446 views

### A log inequality for positive definite trace-one matrices

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following ...

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vote

**0**answers

54 views

### Finding the infimum of the range of a certain non-negative function associated to a $ C^{*} $-algebra

Let $ A $ be a non-trivial $ C^{*} $-algebra and $ n \in \mathbb{N} $. Setting $ \mathcal{D} \stackrel{\text{df}}{=} A^{n} \setminus \{ (0_{A},\ldots,0_{A}) \} $, we can define a function $ f: ...

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votes

**0**answers

55 views

### Extremum of the cyclic sum of polynomial ratios (same degree)

I've noticed a few times (probably nothing new) that cyclic sums (assuming $x, y, z > 0$) like:
$\frac{x^2+y^2}{yz} + \frac{y^2+z^2}{zx} + \frac{z^2+x^2}{xy}$,
where in each of the 3 ratios, all ...

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votes

**0**answers

256 views

### Probabilistic inequality problem: how do I find its least upper bound function?

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let's restrict our attention to the vectors $\overrightarrow{x} = ...

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votes

**2**answers

282 views

### A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...

**1**

vote

**1**answer

65 views

### On generalization of rearrangement inequality

In fact, I'm intrested in such a question:
If $\sigma$ and $\pi$ are two permutations of $\{1,2,\ldots,n\}$
what is necessary and sufficient condition to be the following statement true. For all ...

**5**

votes

**1**answer

304 views

### An elementary inequality for a recursive double sequence

Here is what looks like (but is not) an Olympiad problem. Is it really that tough, or am I overlooking a simple solution?
I have a system of sequences $\sigma_0(m),\sigma_1(m),\ldots\,$ defined for ...

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votes

**6**answers

2k views

### Understanding Gibbs's inequality

Short version
Gibbs's inequality is a simple inequality for real numbers, usually
understood information-theoretically. In the jargon, it states that
for two probability measures on a finite set, ...

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votes

**0**answers

67 views

### Refined versions of Azuma's inequality

Is there any version of Azuma's inequality where the bound $c_k$ as mentioned in https://en.wikipedia.org/wiki/Azuma's_inequality comes in the numerator of the fraction in the negative expoential.

**0**

votes

**2**answers

69 views

### Level sets and integral of functions of two variables

Let $f_1,f_2$ be two positive functions on $\Omega_1, \Omega_2 \subset R^2$ with $f_1|_{\partial \Omega_1}=f_2|_{\partial \Omega_2}=0$. For every $\lambda>0$, denote the the area of the domain ...

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votes

**0**answers

50 views

### Maximal inequality for Markov process

For a Markov process $\{X_n\}$ is there any inequality available for
$$ E[\sup_{0 \leq n \leq k} X_{n}]$$
in terms of moments of $E[X_n], 0 \leq n \leq k$

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votes

**0**answers

53 views

### Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that ...

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vote

**2**answers

259 views

### Looking for (information about) long diamonds

I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be ...

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votes

**0**answers

72 views

### Weyl-type inequality for non-Hermitian matrices?

What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?

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votes

**2**answers

157 views

### Inequality from a point in plane to a triangle OR Inequality on a quadrilateral

If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that :
$\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge ...

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vote

**0**answers

15 views

### Bounding Hidden Markov model Bayesian filter error with inexact models

In context of a hidden Markov model, I am interested in bounding the error of a Bayesian filter when using inexact state transition and observation models.
Consider a hidden Markov model (HMM) with ...

**11**

votes

**1**answer

431 views

### An inequality for the spectral radius of matrices used by J. Bochi

I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...

**30**

votes

**2**answers

2k views

### An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...

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votes

**2**answers

185 views

### concentration inequality for entropy from sample

Consider a measure $\mu$ on a finite set, and let $x_1, \ldots, x_n$ be i.i.d samples from $\mu$. Then the expression $S_n = -\frac{1}{n} \sum_{i=1}^n \log \mu(x_i)$ converges by a.s. to the entropy ...

**0**

votes

**1**answer

75 views

### Estimate $\left|\sum_{n,m}a_n \bar b_m\right|\leq C \left(\sum_n|a_n|^2\right)^{1/2} \left(\sum_n|b_n|^2\right)^{1/2}$ [closed]

It is well-known the Hilbert's inequality for double sum:
$$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$
Give $a_n, b_n$ two sequences of complex numbers. I am ...

**5**

votes

**1**answer

140 views

### Rate of Convergence of Borwein Algorithm for computing Pi

In a book "Pi and the AGM" in 1987, authors, Jonathan Borwein and Peter Borwein, introduced a magical algorithm to compute $\pi$. However there is a problem that I couldn't understand and couldn't ...

**2**

votes

**0**answers

105 views

### Logarithmic bound for Diophantine equation

Let $a_1 \geq a_2 \geq a_3$ be given positive integers and let $N(a_1,a_2,a_3)$ be the number of solutions $(x_1,x_2,x_3)$ of the equation $$\dfrac{a_1}{x_1}+\dfrac{a_2}{x_2}+\dfrac{a_3}{x_3} = ...

**0**

votes

**0**answers

69 views

### Conditional version of martingale difference concentration inequality

Let $M_n$ be a $\mathscr{F}_n=\sigma(\eta_m,\theta_m, m\leq n)$ measurable martingale difference sequence. Then is it possible to find a exponential tail bound for the following
$$P(|M_{n+1}| > ...

**89**

votes

**17**answers

8k views

### Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...