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1
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1answer
324 views

sobolev embedding theorem in the smooth metric measure space

we know the sobolev embedding theorem of Saloff-Coste $\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu $ wtih $Ric\ge-(n-1)K$, for ...
10
votes
2answers
1k views

Proving a messy inequality

EDIT: After much work I was able to reduce the inequality to a single variable function which I need to show is non-positive. That function is (for $0\leq p\leq\frac{1}{2}$) $$\frac{p^2(\log(p))^2 - ...
2
votes
0answers
286 views

Does this inequality of negative relative entropy and quantum relative entropy hold?

Hello, everyone! Question I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector $|i\rangle$ and two Hermitian matrices ...
4
votes
1answer
339 views

Is it possible to extend this inequality about Euclidean distance &Frobenius norm to more general Bregman divergence such as relative entropy & von Neumann divergence?

Motivation- A Special Case Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...
1
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0answers
117 views

Boundedness of Integral

Suppose we operate on the unit simplex $\Delta \subset \mathbb{R}^d$ with $0$ as a corner point. Define the integral $$ Iu(x):=\int_0^1 t^{|\beta|-1}x^\beta u(tx)\mathrm{d}t,\quad x\in \Delta $$ and ...
1
vote
0answers
173 views

Given a jointly convex function $f$, what is the bound of $f\left(\sum_ip_i^2x_i,\sum_jq_j^2y_j\right)$if $\mathbf{p},\mathbf{q}$ are constrained in a manifold?

Suppose there is a jointly convex function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $\mathbf{x},\mathbf{y}\in\mathbb{R}^m$ and ...
4
votes
1answer
2k views

Determinant of a sum of two matrices (one dominating the other)

Let $A$ and $B$ be two $n \times n$ real matrices such that: $\forall i, j: a_{ij} \geq 0, b_{ij} \geq 0$ let $a_\max$ be the largest entry of $A$ and $b_\min$ be the smallest nonzero entry of $B$; ...
4
votes
1answer
340 views

A Hölder like inequality

If $0< a_1\le a_2\le \cdots \le a_n\le a_{n+1}$ and $p>1$, is it true that ...
11
votes
2answers
481 views

A property of unimodal sequences

It is well-known that $(-1)^j \sum_{i=0}^j (-1)^i\binom{n}{i} \geq 0$. This inequality can be used to prove Bonferroni's inequalities for example. Recently I noticed that a similar inequality applies ...
7
votes
2answers
326 views

Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...
1
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0answers
101 views

Inequality involving BV norm and a regularizing kernel

In the same article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# (related to this ...
10
votes
2answers
622 views

Bounding Euler products (or almost) by products of zeta functions

Let $s_1, s_2 \in (1/2,1\rbrack$. I would like to bound the product $$A=\prod_p \left(1 + \frac{p^{-s_1} p^{-s_2}}{(1-p^{-s_1}+p^{-1}) (1-p^{-s_2}+p^{-1})}\right)$$ Now, I am almost positive that ...
2
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0answers
191 views

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) = ...
3
votes
1answer
371 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: ...
3
votes
1answer
729 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 ...
4
votes
2answers
221 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 ...
1
vote
1answer
486 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 ...
0
votes
1answer
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 ...
2
votes
2answers
443 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 ...
0
votes
1answer
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 ...
7
votes
5answers
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 ...
0
votes
1answer
323 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 ...
0
votes
1answer
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}) ...
1
vote
1answer
718 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)| ...
-1
votes
1answer
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)| ...
4
votes
3answers
993 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? ...
3
votes
0answers
211 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 ...
3
votes
0answers
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} = ...
2
votes
2answers
927 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 ...
2
votes
1answer
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 ...
3
votes
4answers
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 \; ...
3
votes
3answers
898 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 $$ d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2}, $$ ...
10
votes
5answers
1k views

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 ...
6
votes
1answer
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: ...
4
votes
2answers
608 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 ...
4
votes
1answer
432 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 - ...
2
votes
2answers
600 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 ...
4
votes
2answers
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: ...
2
votes
1answer
517 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 ...
0
votes
2answers
558 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 ...
6
votes
0answers
516 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 $M$: ...
4
votes
1answer
226 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}) ...
7
votes
1answer
925 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 ...
3
votes
2answers
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 $\gamma_k$. -- ...
1
vote
1answer
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 ...
4
votes
1answer
312 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 $$ ...
7
votes
1answer
441 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 ...
4
votes
2answers
614 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 ...
12
votes
4answers
1k 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 ...