The inequalities tag has no usage guidance.

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### How much can KL divergence decrease by diluting the reference distribution

Let $\Omega$ be a countable set and $\mu,\nu\colon\to[0,1]$ be distributions on $\Omega$, that is we have $\sum_{x\in\Omega}\mu(x)=1$ and likewise for $\nu$. The Kullback-Leibler divergence of $\mu$ ...

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### How did the $\operatorname{div}_x$ “disappear”?

Given $E(x,t) := \nabla_x (\frac 1{|x|} * \rho)$ where $\rho : \mathbb R^3 \times [0,\infty) \to \mathbb R$ is defined by $\rho(x,t):=\int_{\mathbb R^3} f(x,v,t) dv$, I have understood that
$$\rho(x,t)...

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**2**answers

266 views

### A conjecture generalization of Karamata inequality

Fist I observe function $f(x)=x^2$ in the figure as following
I found that when $x_1 \ge y_1$ and $x_2 \le y_2$ $\Rightarrow$ $AB \ge CD$
$\Rightarrow$ $$\frac{f(x_1)+f(x_2)}{2}-f(\frac{x_1+...

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

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### estimate the error term in CLT

Let $X_m = \frac{1}{\sqrt{m}}\sum_{k=1}^m Z_k$ where $Z_k$ are iid equally likely on $\{\pm 1\}$. Then $X_m$ convergens to $X \sim \mathcal{N}(0,1)$ in distribution by CLT.
Let $f$ be a smooth ...

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88 views

### If $N = q^k n^2$ is an odd perfect number, and $n < q^{k+1}$, does it follow that $k > 1$?

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. According to Dickson (as pointed out recently by Beasley), Descartes conjectured $k=1$ in a letter to Mersenne in 1638, with Frenicle'...

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58 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 ...

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105 views

### An inequality in product space $V$ [on hold]

I found an inequality as following: Let $x, y, z$ be three complex numbers then:
\begin{equation*} \frac{1}{2}(|y+z-x|+|x+z-y| + |y+x-z|) \le |x| + |y|+|z|+\frac{1}{2}|x+y+z| \end{equation*} (1)
The ...

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**1**answer

76 views

### Positive semidefinite ordering for covariance matrices

Suppose that X and Z are matrices with the same number of rows. Let
$$ D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & ...

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**1**answer

84 views

### Can this equality hold for a nonzero $b$?

Please may you kindly assist me on this integration exercise: For real $a, b$ with $a \neq 0$, consider the equality
$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\...

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161 views

### A two-point inequality

Let $M(p,q) = (2p-\sqrt{p^{2}+q^{2}})\sqrt{p+\sqrt{p^{2}+q^{2}}}$ and set $B(t) = M(x+t, \sqrt{t^{2}+(y+bt)^{2}})$. Given any real $x,y,b$ is it true that $\varphi(t) = B(t)+B(-t)$ is decreasing in $...

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49 views

### prove that $\min\{|z_{j} - w_{1}|,|z_{j} - w_{2}|\}\leq 1$ holds [on hold]

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $$ (z-z_{1})(z-z_{2})+(z-z_{2})(z-z_{3})+(z-z_{3})(z-z_{1})=0$$ ...

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83 views

### An inequality of real continuous function with f'>0 and f''>0

I proposed my conjecture as follows:
Let $f(x)$ is a real continuous function on $[m, M]$ and $f'>0, f''>0$ on $[m, M]$, let $m \le x_i \le M$, for $i=1, 2,..., n$. Then
$$\frac{f(x_1)+f(x_2)+...

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140 views

### An inequality in cyclic polygon and tangential polygon

I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following:
Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. Let P be ...

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**1**answer

234 views

### A generalization of Erdős–Mordell inequality [closed]

I proposed my conjecture generalization of Erdős–Mordell inequality as following:
Let $A_1A_2....A_n$ be a polygon in a plane, $P$ be the point in $A_1A_2....A_n$. Let $d_i$ be the distances from $P$ ...

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39 views

### Concentration of functional of Gaussian random variable

Suppose I have two Gaussian distributions
$p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...

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123 views

### L1 analog of Bernstein's inequality

Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that
$$
\|q\|_1 \leq O(n) \|p\|_1
$$
where we define $\|f\|_p := \left(\int_{-1}^1 |f(x)|^pdx\...

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68 views

### uniform one-sided van der Corput inequality

Is the following true (and if yes, where the best proof is written?)?
For any $c>0$ for large enough positive integers $N$ we have $\sum_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$?
Hm,...

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548 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 \...

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111 views

### Global Poincaré type estimate

For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x ...

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55 views

### How to compute bounding coefficients for McDiarmid's inequality?

I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question.
Given a ...

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### Interpolation inequality for fractional Sobolev spaces

In Theorem 5.2 of the book
Robert A. Adams and John J. F. Fournier, MR 2424078 Sobolev spaces, ISBN: 0-12-044143-8.
is stated that, for any integers $0 \leq j \leq m$ and $u \in W^{m,p}(\Omega)$ (...

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414 views

### Find the best constant to this bounded inequality

Let $n$ be postive integer number, and $x_{i}\ge 0$, such
$$x_{i}x_{j}\le 4^{-|i-j|},1\le i,j\le n$$
then I have prove
$$x_{1}+x_{2}+\cdots+x_{n}<\dfrac{5}{3}$$
Edit Add Proof:since $x^2_{i}\le 1,...

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239 views

### Negative probabilities - what are two ordinary pgfs that correspond to the gf of a half-coin?

In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$:
$$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...

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### maybe this conjecture also hold to this complex inequality

I have solve this following
Question: Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{...

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### deriving concave upper bounds of a domain constrained nonconvex function over a simplex

Consider a nonconvex function $h(X)=f(X^\dagger AX)$, where $X\in C^{r \times n}$, $A$ is a positive semidefinite matrix, and $f$ satisfies the following two properties:
\begin{align}
&f(W): H_{+}^...

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266 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 ...

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100 views

### An inequality for real numbers [closed]

I need to prove the following inequality.
For every $a,b,x,y \in [0,1]$, it holds that:
$abx + (1-a)(1-b)y \leq \left( a^2b^2x^4 + (1-a)^2(1-b)^2y^4 \right)^{1/4}$
I don't know if this is correct ...

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531 views

### Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$, define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...

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86 views

### Differential equation with inequality constraints

Does there exist a $\gamma > 0$ and functions $g$ and $f$ s.t. we have the following for all $x \in [0, 1]$:
$$
g(x) \in [0, 1] \\
f(x) \in [0, x] \\
x \frac{d}{dx}g(x) - \frac{d}{dx}(g(x)f(x)) = 0 ...

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105 views

### estimation of a vector-function

Let $x(t)\in C^1(\mathbb{R}_+,\mathbb{R}^m)$ be a vector-function such that
1) $\|x(t)\|+\|\dot x(t)\|\to 0$ as $t\to\infty$ and
2) for some real $c_1>0$ and all $t>0$ one has $\|x(t)\|\le c_1\...

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188 views

### If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general? [closed]

(Note: This has been cross-posted to MSE. However, I feel that it is more likely to receive a good answer here, because I believe that it is a research-level question. For the mathematicians who ...

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### Symmetry of concentration bounds on mean

Question summary:
If I have a two-sided bound, can I immediately get a one-sided bound with tighter constants?
Question details:
Let $\mathbf X = X_1,...,X_n$ be $n$ i.i.d. real-valued random ...

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### Probabilistic Modeling Parameters Request

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let us restrict our attention to the vectors $\overrightarrow{x} = (x_{1},x_{2},\ldots,...

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### On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers [closed]

(Note: This question has been cross-posted to MSE.)
Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$.
A number $M$ is called almost perfect if $\sigma(M) = 2M -...

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### 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,...

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661 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-...

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### 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 $\lambda(M_2) \leq \lambda(M_1)$,...

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### 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):
$$
...

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176 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 ...

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**1**answer

90 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 ...

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### 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 ...

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### 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|^...

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### 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 $(1)$...

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398 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 ...

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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>...

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**1**answer

69 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 $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, }j}a_{ij}$...

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205 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 \...

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**1**answer

222 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,...

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**2**answers

221 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 ...