Questions tagged [inequalities]

for questions involving inequalities.

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5
votes
1answer
156 views

a square root inequality for symmetric matrices?

In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
-2
votes
0answers
36 views

Well-posedness in modified H2 space

Can I modify the $H^2$ space such that: $$\tilde{H}^2 := \left[ u(\Omega): ||u||^2_{L_2(\Omega)} + ||\nabla u||^2_{L_2(\Omega)} + ||\Delta u||_{L_2(\Omega)}^2 < \infty \right]$$ and then use the ...
0
votes
0answers
48 views

Trace inequality normal derivative

For $v(\Omega) \in W^1_2$ and $\Omega \in C^1$ we have a trace inequality: $$\Vert v \Vert _{L_2(\partial \Omega)} \leq C_\Omega \Vert v \Vert _{W_2^1},$$ which can be found in many places in the ...
6
votes
1answer
199 views

Matrix inequality : trace of exponential of Hermitian matrix

I want to know whether the following inequality holds or not. \begin{align} (\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1} \end{align} where $A, B$ are Hermitian ...
-4
votes
0answers
53 views

inequality by changing summation and product [closed]

Let $0 \le n_1 ,n_2 , ... , n_k < a_i \in \mathbb{R}$ for $i=1,..,m$ and $k \ge 1$ and $m \ge 1$. Prove that the following inequality holds: $$ \prod_{i=1}^m\left(a_i - \frac{1}{k}\sum_{j=1}^{k}...
4
votes
1answer
156 views

Functional inequalities involving the condition $\left(\int_0^t f(x)dx\right)^2 \ge \int_0^t f(x)^3dx$

I was reading the solution to a functional inequality in an article when the author made the following remark without giving any proof: let $f(x): [0, \infty]\to[0, \infty]$ be locally integrable and ...
2
votes
0answers
66 views

An elementary proof of Davies' inequality

In the paper Lipschitz continuity of functions of operators in the Schatten classes, Davies proved the following matrix inequality. Let $a_i,b_i>0$ for $1\leq i\leq n$ and $A$ be an $n\times n$ ...
0
votes
1answer
80 views

Finding a connection between two types of convergence

Please, help me find connections between two types of convergence: Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences: 1) $X_n \...
0
votes
1answer
122 views

Is this probability inequality true?

This question may be simple, though I'm not managing to find an answer. Let $X$ and $Y$ be two dependent random vectors in in $\mathbb{R}^d$, with joint probability density $\mu(x,y)$ (with respect to ...
4
votes
1answer
113 views

Minimization of a discrete valued function

$$ \min_{f} \sum_{i=1}^n \max \left( 0, 2f(i) - f(i-1) -f(i+1)\right), $$ where the minimum is taken over all the functions $f$ from $\{0,1,2,\ldots,n+1\}$ to $[0,x]$, $x <1$, such that $f$ is non-...
1
vote
2answers
101 views

Expected value of a truncated binomial

Let $X\sim B(n,p)$ be a binomial random variable and fix $0<k<n$. Are there any well-known bounds for $\mathbb{E} (X-k)^+$, where $(X-k)^+ =\max\{0,X-k\}$? I am particularly interested in ...
4
votes
1answer
112 views

Is there a non-convex function with non-decreasing average rate of change?

$\newcommand{\R}{\mathbb R}$ Let $f$ be a function from $\R$ to $\R$. It is said that $f$ is midpoint-convex if for any real $x$ and $y$ we have $f(x-y)+f(x+y)\ge2f(x)$ or, equivalently, $f(x+y)-f(x)\...
4
votes
0answers
215 views

Inequalities for trace/eigenvalues of product of multiple 2x2 matrices

Consider the matrix product $\prod_i^n A_i$, where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
1
vote
1answer
51 views

A uniform mixture of order statistics

Let $0<k<n$ be integers, and let $X$ be a random variable obtained as follows: sample $n$ points independently and uniformly at random in the unit interval, and select (uniformly) one of the $k$...
1
vote
2answers
40 views

Cyclic inequality for 2 dimensional simplex elements

Let $p=(p_{1},p_{2},p_{3})\in\Delta$, with $\Delta:=\lbrace p\in(0,1)^{3}\ |\ p_{1}+p_{2}+p_{3}=1 \rbrace$. I aim to prove (not knowing whether it is true though) that \begin{equation} p_{1}^{p_{3}-p_{...
0
votes
0answers
130 views
+50

On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means

In this post (the content of this post is now cross-posted from Mathematics Stack Exchange see below) we denote the radical of an integer $n>1$ as the product of disctinct primes dividing it $$\...
-1
votes
0answers
43 views

Berry-Esseen unit ball

How can I show that sampling a random vector from a uniform distribution over a $d$-dimensional unit ball is similar to sampling a random vector from a uniform distribution over $d$-dimensional ...
1
vote
0answers
53 views

Maximum value of $\int (aF^2(x)g(x)+G^2(x)f(x))dx$ over all $f,g$ densities satisfying $\int F(x)g(x)dx=1/2$

I want to maximise $$I(f,g):=\int_{-\infty}^\infty (aF^2(x)g(x)+G^2(x)f(x))dx$$ where $a>0$ is a given constant, over all possible probability densities $f,g$ satisfying $$\int_{-\infty}^\infty F(x)...
-2
votes
1answer
174 views

Bounding $L^p$ norms in terms of lower-order $L^q$ norms

Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
0
votes
1answer
288 views

How I can prove or disprove that $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=1$ in rational? [duplicate]

The motiviation of this question is to look if there is such solution in rational number to the identity which montioned here, I have done many attempts using wolfram alpha to find such pairs of ...
1
vote
0answers
150 views

What is decoupling theory means on Tao Blog ? And what is its purpose in mathematics? [closed]

I accrossed on Tao Blog a new theory for me which it is called "Decoupling theory", But I didn't find in the web its definition and its purpose , I find only this article in wiki but this very far ...
3
votes
1answer
98 views

Euler–Maclaurin formula in $\mathbb{Z}^d$

I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as $$ \sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x) $$ where $d\ge 2$ is an integer, $a,b \...
3
votes
1answer
137 views

Inequality on the Hellinger distance between Poisson and mixture of Poisson

Let $H$ denote the Hellinger distance; i.e., for two discrete distributions $p,q$ (identified with their pmf) over $\mathbb{N}$, $$ H(p,q)^2 = \frac{1}{2}\sum_{n=0}^\infty \left(\sqrt{p(n)}-\sqrt{q(n)}...
2
votes
1answer
80 views

When is a set defined by multivariate polynomial inequalities convex?

Consider the set of real numbers given by $$S = \{(a,b,c,d,e,f,g,h) \in [0,1]^8 : 0 \le \frac{e(g-h)}{b(g-f)} \le 1 \text{ and } 0 \le \frac{e(h-f)}{(1-b)(g-f)} \le 1\}$$ Note that this set can also ...
0
votes
1answer
136 views

Probability of a random variable greater than its expected value

We have a lot of probabilities lower bounding as (e.g. chernoff bound, reverse markov inequality, Paley–Zygmund inequality) $$ P( X-E(X) > a) \geq c, a > 0 \quad and \quad P(X > (1-\theta)E[...
1
vote
1answer
141 views

Does kernel regression preserve monotonicity?

Consider the Kernel regression estimator: $$\hat{y}(x)=\frac{\sum_{i=1}^n{K(x-x_i)y_i}}{\sum_{i=1}^n{K(x-x_i)}},$$ where $x,x_1,\dots,x_n\in\mathbb{R}^d$, $y_1,\dots,y_n\in\mathbb{R}$, where $K:\...
9
votes
3answers
367 views

Polynomial inequality of sixth degree

There is the following problem. Let $a$, $b$ and $c$ be real numbers such that $\prod\limits_{cyc}(a+b)\neq0$ and $k\geq2$ such that $\sum\limits_{cyc}(a^2+kab)\geq0.$ Prove that: $$\sum_{cyc}\...
4
votes
0answers
361 views

A symmetrization-majorization inequality for i.i.d. zero mean random variables

Let $k\geq 2$ and $$f(x_1,\ldots,x_k)=\Bigl(\prod_{i\leq k}(1+x_i)+\prod_{i\leq k}(1-x_i)\Bigr)\log\Bigl(\prod_{i\leq k}(1+x_i)+\prod_{i\leq k}(1-x_i)\Bigr)-k(1-x_1 x_2)\log(1-x_1 x_2).$$ If random ...
5
votes
3answers
331 views

Polynomial inequality $n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3$

Let $n\ge 3$ be an integer. I would like to know if the following property $(P_n)$ holds: for all real numbers $a_i$ such that $\sum\limits_{i=1}^na_i\geq0 $ and $\sum\limits_{1\leq i<j<k\leq n}...
0
votes
2answers
82 views

Bounding $E[\|\Sigma^{-1/2}(X-\mu)\|_2^3]$ for 2-dimensional Bernoulli

Let $X\in\{0,1\}^2$ have mean $\mu=\left[\begin{smallmatrix}p_1\\p_2\end{smallmatrix}\right]$ and $\Pr[X_1 = X_2 = 1] = p\le \min\{p_1,p_2\}$. (Note we must have $1-p_1-p_2+p\ge 0$ for the ...
1
vote
0answers
80 views

Upper and lower bounds on the entries of a matrix power

Say I have a non-negative square $n\times n$ irreducible stochastic matrix $A$ (i.e., each column sums to 1), for which the following holds: $$A_{ij} > 0 \iff A_{ji} > 0.$$ I know that no more ...
2
votes
1answer
49 views

Bounds on the second derivative of a natural cubic spline in terms of the data

Suppose we have real numbers $x_1 < \cdots < x_n$ and $v_1, \ldots, v_n$. Let $f$ be the natural cubic spline such that $f(x_i) = v_i$. Is there a simple explicit bound on $\|f''\|_\infty$ in ...
9
votes
2answers
963 views

An inequality involving square roots and sums

I've been trying to prove (maybe even disprove) the following inequality: $$ \sum_{n=1}^{N} \frac{a_n}{\sqrt{\sum_{i=1}^{n}a_i}} \leq C \sqrt{\sum_{n=1}^{N}a_n} $$ Where $ a_1,...,a_N\geq 0 $ are ...
-2
votes
1answer
54 views

Triangle inequality in sum of summations [closed]

I have $$a_i^2-a_i^3=\sum_{b\in B}b+\sum_{c\in C}c,\\B\ne C,\\B, C \subset R$$ Can I say that $$|a_i^2-a_i^3|\le\sum_{b\in B}|b|+\sum_{c\in C}|c|$$?
2
votes
1answer
116 views

Every element of $A$ and $B$ differ in at least $k$ positions

Let $m,n$ be positive integers, $m,n>1$ and $X = \{(x_1,x_2, ..., x_m) \in \mathbb{Z}^m :1 \le x_i \le n, \forall 1 \le i \le m\}$. $A$ and $B$ are two disjoint subsets of $X$, such that if $a ...
3
votes
1answer
271 views

Inequality for $AB + BA$ when $A,B\geq0$, reference request

Let $A,B\geq0$ be positive semidefinite matrices of arbitrary size $n\times n$. Denote by $\alpha$ and $\beta$ their largest eigenvalues. It is well-known that the eigenvalues of the expression $AB +...
8
votes
2answers
501 views

On the conjecture : $|m^2-n^3|>\frac{1}{5}\sqrt[6]{m^2+n^3}$

Let $m,n$ be positive integers, such that $m^2\neq n^3$. I conjecture the inequality $$|m^2-n^3|>\dfrac{1}{5}\sqrt[6]{m^2+n^3}.$$ I've tried a lot of numbers, and they all seem to work, but how ...
3
votes
1answer
132 views

Isoperimetric inequality for analytic functions on an annulus

Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that $$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...
2
votes
1answer
112 views

Proof of a discrete isoperimetric inequality

The following inequality appears in the proof of certain isoperimetric-type inequalities for analytic functions in two dimensions: $$\sum_{m=0}^{\infty}\frac{|c_m|^2}{m+1} \leq \pi \left(\sum_{m=0}^{...
4
votes
3answers
97 views

Find distribution that minimises a function of its moments

Imagine a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-centred moment $x_{n}$. The mean $x_{1}$ is fixed (and positive). How can I find $f(x)$ that ...
2
votes
2answers
215 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,\...
3
votes
0answers
104 views

On Pitt's inequality (weighted Fourier inequality)

One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$, $$ \sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(...
2
votes
0answers
47 views

Deciding whether a system of linear integer inequalities has infinitely many solutions

I have a quick question that I am struggling to find a solution to: Given a system of linear integer inequalities $A\textbf{x} \leq \textbf{b}$, where $A\in \mathbb{Z}^{m\times n}$ and $\textbf{b}\...
1
vote
0answers
57 views

Maximizing quadratic forms

Consider the maximization problem $$\text{maximize} \quad Q(x)= \sum_{i<j} \Big(\sum_{k} a_{ik}a_{jk}\Big) x_i x_j \quad \text{subject to} \quad \sum_{i}x_i^2=1,$$ and let $M$ be maximum value ...
1
vote
1answer
161 views

Resolution of an inequality on integers

I’m trying to resolve respect to $k$ the following inequality, $$ k\left(\log k +\log \log k-\alpha+O\left(\frac{\log \log k}{\log k}\right)\right)\geq x, $$ in order to obtain, under the condition $...
7
votes
3answers
420 views

A generalization of discrete Hibert's transform (Montgomery's inequality)

In the paper "Hilbert's inequality", Montgomery and Vaughan proved that a generalization of the discrete Hilbert transform is bounded in $\ell^2$. The inequality reads as follows $$ \Big| \sum_{k\neq ...
1
vote
1answer
168 views

Inequality in Iwaniec-Kowalski

I am reading about Dirichlet polynomials in the book Analytic Number Theory by the said authors. Can anyone justify the following inequality? Assume that $a(n),b(m)$ are sequences of non-negative ...
4
votes
0answers
202 views

An inequality in harmonic analysis with the BMO flavour

I am asking myself this question (which seems to be a natural generalization of Remark 4.4 of these lecture notes). Question. Let $I_s, s \in \mathcal{S}$ be a collection of intervals included in ...
1
vote
1answer
98 views

Understanding a family of Sobolev-type inequalities

I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following: Denote the following inequality as $S_{r,s}^{\theta}$: $\...
2
votes
2answers
240 views

An “obvious” probability lemma about random words

Fix some positive integers $p,n,k$. Let $w$ be chosen uniformly at random from $[k]^n$ (the set of $n$ length words/sequences where each entry is in $\{1,\ldots,k\}$). Let $A_i$ be the event that $...

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