The inequalities tag has no wiki summary.

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### Proving that an increasing iterative sequence increases at a decreasing rate

In this question
Proving a sequence of integrals increases (iterated minimax distributions)
Pietro Majer proved that
$$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = ...

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**1**answer

360 views

### Inequality involving perimeter and area

I am studying an article: The parametric problem of capillarity: the case of two and three fluids, by U. Massari. In one of his proofs, he uses an inequality I can't manage to prove. It is like this:
...

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**1**answer

722 views

### Ratio sum comparison on operators

It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$,
where $s_i(S)$ is the $i$-th singular value of $S$.
How would one prove that
...

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219 views

### Bounding the series of the geometric means of the terms of a given positive series

Let $ \{ a _ k \} _{k\in\mathbb{N} _ +} $ be a sequence of non-negative numbers, and let $MG(a_1,\dots,a_n)$ denote the geometric mean of the first $n$ terms. Then, the inequality
$$ \sum _ {n\ge ...

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**1**answer

483 views

### A question on gauge functions

In the second paragraph on Page 71 of the book Matrix Analysis by
Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem
III 4.4''. How can one get the inequality in Theorem III 4.4 from
...

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**1**answer

2k views

### Why does sample standard deviation underestimate population standard deviation? [closed]

I am aware of Jensen's inequality where, given the concave square root function, the mean of the square root is lesser than the square root of the mean.
However, I cannot figure out why the square ...

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441 views

### Poincare inequality for the annulus

Assume that $A=A(r,1)=\{x: r<||x||<1\} \subset R^n$ is an annulus.
Whether is known the constant of Poincare inequality for A or some its estimation (w.r.t. $L^2$): the constant $C$ in the ...

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**1**answer

94 views

### Problem with making an estimate when values of many variables are unknown?

Hi all, I was wondering whether you could help me understand a part in a proof that is assumed to be an "obvious detail" by the author, but that I can't wrap my head around. Essentially, as far as I ...

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**5**answers

830 views

### Rearrangement-style inequality with lots of terms and little evidence

This is another of the problems I designed for contests but wasn't able to solve on my own for years. Maybe AoPS is a better place for it, but let me try it here.
[UPDATE: I have streamlined the ...

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**1**answer

317 views

### Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains
$10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$,
$x_{1}\ge\cdots\ge x_{10}>0$ and ...

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**1**answer

359 views

### Help prove a maximal inequality

Let $X_1,…,X_n$ are exchangeable of random variables, and $n$ is an even number.
$S_k=X_1+\dots+X_k$. $M_k=X_{n/2}+\dots+X_{n/2+k}$.
I want to prove:
$$\Pr(\max_{1 \le k \le n}{|S_k|>\epsilon}) ...

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**1**answer

716 views

### On an eigenvalue inequality

Let $\lambda_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda_1 (\cdot)| ...

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**1**answer

727 views

### On an eigenvalue inequality [closed]

Let $\lambda_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda_1 (\cdot)| ...

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989 views

### Factorial-inequalities

Let $n>15$ be an integer. Suppose also $n=\sum_{i=1}^n ic_i$, where $c_i$ are non-negative integers. Assume further that $c_1<4$.
Is the following inequality true?
...

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209 views

### Where can I find interpolation inequalities for derivatives of the following form?

Here they are:
$$||f||_{\infty} \leq C ||f||_q^{\frac {qk} {n+kq}} \left( \sum_{|\mu|=k} ||D^\mu f||_{BMO} \right)^{ \frac n {n+kq}}$$
and
$$||f||_{Lip_\alpha} \leq C ||f||_q^{\frac {qk} {n+kq} \frac ...

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222 views

### Maximise $L^q$ norm of a vector, for fixed $L^1$ and fixed $L^p$ norms [closed]

Consider a vector $x \in \mathbb R_+^n$ and $p,q \in \mathbb R$ such that
$1 < p < q$
We fix $\sum \limits_{i=1}^{n}|x_i| = 1$ and $ \left(\sum \limits_{i=1}^{n}|x_i|^p \right)^\frac{1}{p} = ...

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900 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 ...

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**1**answer

232 views

### Quotients of perfect powers separated by an integer

Let $a_n=\frac{(n+1)^{n+2}}{n^n}$ and $b_n=\frac{(n+2)^{(n+1)}}{(n+1)^{n-1}}$. Then it is easy to see that $a_n \leq b_n$ for all integers $n\geq 1$ (because the sequence $(1+\frac{1}{n})^n$ is ...

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473 views

### Equidistribution Theorem: distance between solutions

Can please someone help me with the following problem.
Say we have a sequence $nx \; \mathrm{mod} \; 1$, where $n$ is a whole number and $x$ is irrational.
Now I need to solve the inequality
$nx \; ...

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893 views

### Poincare Metric on Hyperbolic Plane

as is well known, we can put a metric on the upper half plane $\mathbb{R}^+ \times \mathbb{R}$
by setting
$$
d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2},
$$
...

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### Bounding the absolute sum of entries of the inverse of a 0-1 matrix

I have a non-singular square 0-1 matrix and I want to bound the sum of absolute values of its inverse as a function of n (or the vector 1-norm).
Asymptotic results are also useful.
Does anyone know ...

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**1**answer

1k views

### Stochastic Matrix: Second largest eigenvalue and second largest absolute value of eigen value

Setup
Let $A$ be a stochastic matrix.
Let the eigenvalues of $A$ be $1 = \lambda_1 \geq \lambda_2 \geq \lambda_3 ... \geq -1$.
Let $\lambda = \max_{x: x \perp 1} \frac{||Ax||}{|| x ||}$
Question:
...

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605 views

### Proving a sequence of integrals increases (iterated minimax distributions)

I am trying to show that $$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = (1-(1-F_{n-1}(x))^c)^c$$ and $F_0(x) = x$ and $n$ and $c$ are integers, $n\geq 1$ and $c \geq 2$
Note that ...

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425 views

### total variation distance of product of measures

Let $f, \hat{f}, g,$ and $\hat{g}$ be continuous probability densities. Define probability densities $p \propto fg$ and $\hat{p} \propto \hat{f}\hat{g}$. Is it true that
\begin{align*}
||p - ...

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598 views

### Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ .
I recently discovered this result. I am sure it is known, but it is new to me. It is ...

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1k views

### bounding the trace of a matrix product by the operator norms; generalized Hölder inequality?

Let $A_i$ with $i=1,\dots,N$ and $p$ be real $M\times M$ matrices.
Further, let $p$ be positive definite, i.e., $p\succ 0$, with $Tr(p)=1$. Let $0< a_i<1$ and $\sum_{i=1}^N a_i = 1$.
Claim:
...

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**1**answer

514 views

### A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
...

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556 views

### Do you know this form of an uncertainty principle?

I hope this question is focused enough - it's not about real problem I have, but to find out if anyone knows about a similar thing.
You probably know the Heisenberg uncertainty principle: For any ...

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514 views

### Counting permutation matrices in 0,1,2 matrices

Let $M$ be a matrix with entries equal to 0, 1 or 2, such that all of the row and column sums are equal to $c$.
The Van der Waerden bound gives (roughly) the following bound on the permanent of $M$:
...

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223 views

### Extensions to the Golden-Thompson inequality?

Let $A$ and $B$ be two Hermitian matrices. The famous Golden-Thompson inequality states that
$$\text{tr}(e^{A+B}) \le \text{tr}(e^Ae^B)$$
However, for determinants we have equality
$$\det(e^{A+B}) ...

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911 views

### Multilinear generalization of Cauchy-Schwarz inequality

Let $V$ be a real vector space, and let $(\cdot,\cdot;\cdot,\cdot) : V^4 \to \mathbb{R}$ be a multilinear form with the following properties:
$(x,y;z,w) = (y,x;z,w) = (x,y;w,z)$ (symmetry in the ...

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212 views

### How to find a proper decay rate from an iterative inequality

Suppose we have the iterative inequality $\gamma_{k+1} \leq \gamma_k(1 - c \gamma_k^\alpha)$ with $c, \alpha \in (0, 1)$ and $(1 - c \gamma_k^\alpha)>0$ for all non-negative terms $\gamma_k$.
-- ...

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505 views

### Conjecture:if $i<j$,then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$

p[i] is the i-th prime. $\pi(x)$ is prime counting function.
Firstly, I think that this Prime gap inequality holds true,
$ p[i+1] - p[i] <= i $
Prove:for any i>0, we can always find distinct ...

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311 views

### Ask the validity of a scalar inequality

Let $a_i>0$, $x_i, y_i\in \mathbb{R}$ $i=1,\cdots, n$, such that
$\sum\limits_{i=1}^nx_iy_i=0$, $\sum\limits_{i=1}^nx_i^2=\sum\limits_{i=1}^ny_i^2=1$. Is it true $$
...

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434 views

### What would the best treatment of Gehring's lemma look like?

In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...

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610 views

### Asymptotics for the number of ways to sum primes such that the sum is <= n

Hello!
Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq ...

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### 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 ...

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177 views

### Two-sample t-test variant?

I have a situation that is the opposite of the usual two-sample t-test for equal means: I'm trying to show that it's statistically unlikely that two samples are different.
Is there a statistical test ...

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208 views

### Square variation norm and non-negative, non-increasing sequences

I am trying to understand the properties of square variation, namely, the possibility of preserving it under certain operations. I am following Albiac & Kalton's book: Let $J$ stand for the usual ...

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880 views

### Sufficient and necessary conditions on equality of Cauchy-Schwarz inequality of determinants

Cauchy-Schwarz inequality of determinants:
for $A_{n\times k}$, $B_{n\times k}$, and $B'B$ non-singular, we have
$|A'B|^2\leq |A'A||B'B|$
I was wondering what's the sufficient and necessary ...

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190 views

### An inequality for a continuous non-smooth function

Hello,
I have a question about how to prove a lemma such as this one,
For any $0<\alpha<1$ and $M_{0}>0$, there exists a $M_{1}>0$ such that $\left|z\right|^{\alpha}\leq ...

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### Bound of polynomial on product space in terms of values on the diagonal

We work in the multivariate case so that $x$ stands for $(x_1,\ldots, x_n)$. Let $q(x,y)$ be a symmetric matrix representation of a homogeneous polynomial $f(x)$ of degree $d$. Explicitly,
($q$ is ...

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954 views

### An inequality on concave functions

Could somebody help me to answer the following question?
Let $f:R_+ \rightarrow R_+$
be a nonindentically zero, nondecreasing, continuous, concave function with $f(0)=0$. Do we have that for any ...

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**1**answer

748 views

### Probability inequalities

Hi everyone,
I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts.
My problem is to find an ...

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807 views

### How covariance-variance inequality implies Kantorovich inequality

Hi, I read this article OPERATOR INEQUALITIES RELATED TO CAUCHY-SCHWARZ
AND H\"OLDER-McCARTHY INEQUALITIES, Nihonkia Math. J. 1997,117-122. In the introduction part, it states covariance-variance ...

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310 views

### Divisor function inequality

I have been reading a paper on the Goldbach conjecture found at
http://people.exeter.ac.uk/pt224/Goldbach.pdf.
At one point, the author (Paul Truman), states: Let $z=N^{1/8}$, then
$$\sum_{w\leq ...

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**6**answers

11k views

### Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices.
I've proved that $\det(A+B) \ge \det(A) + \det(B)$ in the case that $A$ and $B$ are two dimensional.
Is this true in general for ...

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**1**answer

345 views

### Is there a complex analog of this sharpened Cauchy Inequality?

Let $x$ and $y$ be two points on the unit sphere $S^{n-1}$ in Euclidean space ${\mathbb{R}}^n$. Suppose that the angle $\theta$ between the points $x$ and $y$ is acute, so that the dot product $x\cdot ...

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545 views

### Is this ergodic inequality true?

Is anything similar to the following inequality true,
$\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$
where $A_n f = ...

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3k views

### Upper bounds for the sum of primes <= n

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...