The inequalities tag has no usage guidance.

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votes

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689 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...

**14**

votes

**3**answers

636 views

### An inequality concerning Lagrange's identity

we know Lagrange's identity
$$(a^2_{1}+a^2_{2}+a^2_{3})(b^2_{1}+b^2_{2}+b^2_{3})=(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3})^2+\sum_{i=1}^{2}\sum_{j=i+1}^{3}(a_{i}b_{j}-a_{j}b_{i})^2$$
then we have Cauchy-...

**4**

votes

**1**answer

510 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 \...

**7**

votes

**2**answers

224 views

### Why $M_1 \subset M_2 \not \Rightarrow N_{M_1} (\lambda) \leq N_{M_2} (\lambda)$ for eigenvalue problem? (EDIT)

We know that for a direct problem with Dirichlet Boundary Condition (with Laplacian operator) that if two domains $M_1$ and $M_2$ are such that $M_1 \subset M_2$, then $\lambda(M_2) \leq \lambda(M_1)$,...

**3**

votes

**1**answer

110 views

### Analytic Combinatorics: upper bound for sum of absolute values of two complex functions: $|z f'(z)| + |2 f(z) - zf'(z)| \leq 2f(|z|)$

Let $f \colon \mathbb C \to \mathbb C$ be a complex-valued analytic function with non-negative coefficients of Taylor series at 0 (suppose that radius of convergence is $+\infty$ for simplicity):
$$
...

**2**

votes

**1**answer

241 views

### upper bound on derivatives of a function defined on an arc

This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here.
You can skip examples below and read from "General setting" at ...

**4**

votes

**2**answers

169 views

### A Stochastic Taylor Expansion/Asymptotics

Question:
Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and
$$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$
$(\mu,\sigma)$ obeys the linear growth ...

**4**

votes

**1**answer

88 views

### Does the Nash inequality hold on manifolds with Lipschitz boundary?

Let $N$ be a smooth manifold without boundary of dimension $n$. $M$ is a manifold with Lipschitz boundary if $M \subset N$, $M$ and $N$ are of the same dimension, and in the charts of $N$, the ...

**12**

votes

**0**answers

290 views

### Inequalities for marginals of distribution on hyperplane

Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...

**2**

votes

**0**answers

130 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
$$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...

**-1**

votes

**1**answer

132 views

### Prove this conjecture inequality 2 [closed]

Let $n$ be postive integer,I conjecture
$$(1+2n)^n\ge 1^n+2^n+4^n+6^n+\cdots+(2n)^n \tag{1}$$
This problem when I solve this equation
$$(1+2n)^n=1^n+2^n+4^n+6^n+\cdots+(2n)^n\tag{2}$$
if this $(1)$...

**13**

votes

**0**answers

392 views

### Determinant inequality involving Hermitian, positive definite matrices

Question:
Let $A,B,C\in M_{n}(C)$ be Hermitian and positive definite matrices such that:$$A+B+C=I_{n}$$
Show that:
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question ...

**1**

vote

**0**answers

31 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 ...

**14**

votes

**5**answers

2k views

### How to prove a known inequality from a book

The following inequality is from page 125 of D.S. Mitrinovic, J. Pecaric, A.M. Fink, Classical and new inequalities in analysis, Kluwer
Academic Publishers, Dordrecht/Boston/London, 1993.
If $a_i>...

**3**

votes

**0**answers

142 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 $$\dfrac{a_1}{x_1}+\dfrac{a_2}{x_2}+\dfrac{a_3}{x_3} = 1$$...

**0**

votes

**1**answer

62 views

### Matrix norm inequality for C*-Algebras [closed]

Let A a $C^*$-Algebra. I have already shown that the maps $Tr, \sigma: M_n(A)\rightarrow A$ given by $Tr((a_{ij})):=\sum_{i}a_{ii}$ and $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, }j}a_{ij}$...

**7**

votes

**1**answer

202 views

### Is this simple-looking moment inequality true?

Let $p \ge 1$ be an integer. Does there exist a constant $C_p$ such that for every random variable $X \ge 0$,
$$
\mathbb{E} \left[ \left(X - \mathbb{E} \left[ X \right] \right)^{2p} \right] \le C_p \...

**4**

votes

**1**answer

210 views

### Prove this conjecture inequality

This following problem is from my Conjecture many years ago,
Question :
Let $a,b>0,n\in N^{+},n\ge 3$,such
$$a^n+b^n+(2n+2)(ab)^n\le 2n$$
Conjecture: then $a+b\le 2$
or
$a+b>2.a>0.b>0,...

**0**

votes

**2**answers

215 views

### Absolute value inequality with complex numbers

Following a problem I found on mathstack, with no solution, and no comment, so I think this inequality is not easy, so I post it here (because I think there are more some good math job, maybe someone ...

**13**

votes

**2**answers

459 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 ...

**2**

votes

**0**answers

77 views

### Octahedron and System of trigonometric equations

Could somebody help me to prove the following?
$$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$
$$\sum_{k=1}^6 \cos (\phi_k)=0$$
$$\sum_{k=...

**1**

vote

**0**answers

63 views

### Lower bound the ratio of the combinatorial quantities

Suppose $p < q$, $s = p^{d}$ for some fixed $d \in (0,1)$, let $p$ goes to infinity, define the following quantity,
\begin{aligned}
\quad f(j) = \sum_{i = 0}^{\min(j,s)}{s \choose i}{p-s \choose i}{...

**19**

votes

**1**answer

1k 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 ...

**2**

votes

**1**answer

78 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 ...

**13**

votes

**2**answers

452 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
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...

**3**

votes

**0**answers

200 views

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

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 + \int\...

**2**

votes

**3**answers

240 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

**1**answer

85 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 [0,\...

**5**

votes

**4**answers

783 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:
$\...

**3**

votes

**1**answer

180 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: http:/...

**1**

vote

**1**answer

63 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 $n=2,3,\...

**5**

votes

**1**answer

214 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 <\frac{...

**4**

votes

**1**answer

261 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

267 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 ...

**6**

votes

**0**answers

218 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

111 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 x2)^n$...

**2**

votes

**1**answer

91 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

92 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

106 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

129 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

78 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

241 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 $i$-...

**3**

votes

**1**answer

151 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 $u\in C(\mathbb R, L^{2})$...

**0**

votes

**1**answer

385 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

98 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

615 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, ...

**8**

votes

**2**answers

469 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 ...

**1**

vote

**0**answers

59 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: \...