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4
votes
0answers
158 views
+50

Any similar Lagrange's identity inequality

we known 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 ...
3
votes
0answers
136 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
0answers
56 views

Generalized Poincaré Inequality on H1 proof

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
-5
votes
0answers
53 views

how to prove the inequality? [on hold]

Question:How to prove the following inequality? Is there anyone who can tell me ?thank you $$(x-y)^{\gamma}\leq x^{\gamma}- y^{\gamma}, \forall x\geq y\geq 0$$ where $\gamma>3/2$
0
votes
1answer
47 views

Matrix norm inequality for C*-Algebras [on hold]

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{, ...
4
votes
1answer
454 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
0answers
63 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 ...
2
votes
1answer
224 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 ...
7
votes
1answer
193 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
1answer
197 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 ...
0
votes
2answers
209 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
3answers
455 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
1answer
76 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
-3
votes
0answers
71 views

Bernoulli's Inequality when $-2≤x<-1$ [migrated]

Why is it that Bernoulli inequality $(1+x)^r>1+rx$ is said to be true for every integer $r≥0$ and every real $x≥-1$; why the range $-2≤x<-1$ is not included? It seems that, by induction (or ...
1
vote
0answers
61 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 ...
19
votes
1answer
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
1answer
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
2answers
449 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
0answers
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 + ...
2
votes
3answers
235 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
1answer
78 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
4answers
779 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
1answer
178 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
1answer
59 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
1answer
212 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
1answer
258 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
1answer
263 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
0answers
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
1answer
107 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
1answer
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
2answers
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
1answer
89 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
1answer
105 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
1answer
128 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
1answer
74 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
1answer
239 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
1answer
149 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
1answer
383 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
1answer
96 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
1answer
605 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
2answers
457 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
0answers
56 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: ...
0
votes
0answers
58 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 ...
0
votes
0answers
349 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} = ...
1
vote
1answer
69 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
1answer
337 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 ...
12
votes
6answers
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, ...
0
votes
0answers
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
2answers
70 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 ...
0
votes
0answers
53 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 $E[X_n], 0 \leq n \leq k$