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2
votes
1answer
207 views

An elementary functional inequality

Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold $\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...
2
votes
0answers
136 views

Why hexagons? The maximal minimum of a sum of cosines on the plane with frequencies on the unit circle

We wish to maximize the minimum of a weighted sum of cosines in the plane, when the frequency components are on the unit circle. Formally: $$\max_{\{ a_i,\theta_i,\phi_i \}_{i=1}^{N} } \min_{(x,y) ...
0
votes
1answer
62 views

Lower bound on Spectral Gap of Rank one + Diagonal

For some $x\in\mathbb{R}^n, \|x\|_2^2=1$ and $\alpha\geq 0$, consider the positive semi-definite matrix $$ X_\alpha := xx^T + \alpha\sum_{k=1}^nx_k^2e_ke_k^T. $$ Suppose for simplicity that the ...
2
votes
2answers
212 views

If $A>B>0$, can we always find a positive real number $\alpha$, $0<\alpha < 1$ such that $\alpha A \geq B $?

Suppose we have positive-definite matrices $A$, $B$, if $A>B>0$, can we always find a positive real number $\alpha$, $0<\alpha < 1$ such that $ \alpha A \geq B $? If it has, then what ...
15
votes
1answer
263 views

Zinn's “doubling” conjecture on weighted sums of independent Rademacher random variables

Let $a_1,\dots,a_n$ be real numbers such that $a_1^2+\dots+a_n^2=1$. Let $\eta_1,\dots,\eta_n$ be independent Rademacher random variables (r.v.'s), so that $P(\eta_i=\pm1)=\frac12$ for all $i$. Let ...
15
votes
3answers
467 views

An inequality for two independent identically distributed random vectors in a normed space

Suppose that $X$ and $Y$ are independent identically distributed random vectors in a separable Banach space $B$. Does it always follow that $E\|X-Y\|\le E\|X+Y\|$? Some background information on ...
25
votes
0answers
531 views

Finding a path through real rooted polynomials

This is a lemma I wanted in order to solve Patrizio Neff's conjecture. It turned out to be the wrong way to think about it, but I am still curious if it is true. Let $z^n+a_{n-1} z^{n-1} + \cdots + ...
36
votes
4answers
2k views

The sum of squared logarithms conjecture

I am searching for the first proof of (or counterexample to) the following conjecture. (The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
3
votes
1answer
108 views

Concentration and Correlation for Magnitudes of Gaussian Vectors

Suppose we have a large collection of standard normal random variables $a_i\in\mathbb{R}^n$. We know by standard concentration results that if we take $m \geq C\left(t/\epsilon\right)^2n$ samples, ...
1
vote
0answers
46 views

Bounds on the spherical measure of sub-level sets of quadratic forms

I'm wondering if there are any bounds on the spherical measure of sets of the form $$ \mu_n\left(\{y\in S^{n-1} : \frac{y_1^2}{y_2^2} < \alpha\}\right) \leq f(\alpha) $$ where $\alpha$ is some ...
0
votes
0answers
45 views

Solving nonlinear inequality that involves norm2 operator

I have an equation of the form $$ \newcommand{\norm}[1]{\left\lVert#1\right\rVert} \norm{\mathbf{Z} \left[ \sum_{n = 0}^{N - 1} (-1)^n \psi^n \mathbf{C}^n \right] \mathbf{q} }^2 \leq |p|^2, $$ where ...
7
votes
3answers
490 views

Derivatives of radial functions can be bounded by derivatives in terms of radial distance?

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
5
votes
2answers
326 views

Minimum of squared sum minus sum of squares

I know that $$ \min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2 $$ with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates. I'm ...
2
votes
0answers
74 views

Finding an explicit constant in finite element error estimates

Background: In a finite element approximation to the solution of a linear PDEs, estimates on the order of convergence of the approximation to the solution rely on a theorem of Bramble and Hilbert ...
19
votes
1answer
353 views

Maximum height of intersection of triangles

I'd like some advice regarding the following question, which I have been struggling with for long time. Let's call the shaded region in the below $S_3$. It is the union of three congruent isosceles ...
3
votes
0answers
107 views

Maximizing the discrepancy in Jensen's inequality for a certain function

Let $\underline{b}=\{b_1,\dots,b_n\}$ be a fixed sequence of positive numbers, and let $a>0$ be a parameter. Define $$ D(a;\underline{b}):=\frac{1}{\frac{1}{na}+\frac{1}{\sum_{i=1}^n b_i}} ...
1
vote
1answer
58 views

local bernstein type inequality for multivariate polynomials

Let's say $p(x_1,...,x_n)$ is an n-variate degree d homogenous polynomial. Assume $U \subset S^{n-1}$ and $ vol(U) > 0 $ is there any Bernstein type inequality saying $$ \max_{x \in U , y \in ...
4
votes
1answer
192 views

Hardy-Littlewood-Sobolev inequality using fractional sobolev norm on the RHS

Using Hardy-Littlewood-Sobolev inequality, we can prove that: $$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|_{L^{4/3}(0,1)}^2 \leq C ...
2
votes
2answers
273 views

Entropy inequality

Let $P,Q$ be probabilities on a finite set $A$ with $Q(a)\gt 0$, for all $a\in A$, and let $H(P)$, $H(Q)$ denote their entropies and $D(P\,\|\,Q)$ denote their Kullback-Leibler distance. Is it always ...
3
votes
1answer
131 views

generalized mean inequality extension

from generalized inequality, we now that for $p>q$, we have $M_p(\mathbf{x})\ge M_q(\mathbf{x})$. now I am curious to know if we can find a constant $\alpha(p,q)$ which is only function of $p,q$ ...
3
votes
1answer
218 views

A Poincare-Type Inequality and its generalization

Let $f(\theta)$ be a fixed positive $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$ Does for any ...
9
votes
2answers
290 views

A “quadratic” triangular inequality

In a Euclidian space (Hermitian as well), say $\ell^2_n$, the following inequality holds true $$(QI)\qquad |b|\cdot|c-a|\le|c|\cdot|a-b|+|a|\cdot|b-c|,\qquad\forall a,b,c\in\ell^2_n.$$ In other words, ...
1
vote
1answer
98 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...
1
vote
1answer
197 views

Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s

This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s? So far I only figured out that I can do Monte ...
2
votes
2answers
215 views

Estimating a Selberg-type integral (or a Fredholm determinant)

I am concerned with the asymptotical behavior of integrals like this for large $n$ $$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq n}(x_{j}-x_{i})^{2}\,\prod_{j=1}^{n}e^{-x_{j}^{2}}dx_{j},$$ ...
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: ...
6
votes
1answer
179 views

Upper bound for a Selberg-type integral over a rectangular region

(Cross-posted from math-SE). I am trying to estimate the values of the following integral for large $n$, $$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq ...
1
vote
2answers
516 views

Variance of truncated normal distribution

Let $ X \sim \mathcal{N} ( \mu, \sigma^2 ) $, $ - \infty \leqslant a < b \leqslant +\infty $ ($ a, b \ne \infty $ simultaneously) and $ Y $ has a truncated normal distribution on $ (a, b )$, i.e. ...
1
vote
1answer
90 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
0answers
62 views

What are good bounds on ratios of subdeterminants?

Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ? Using the ...
1
vote
2answers
375 views

More recently published comprehensive reference on inequalities in the spirit of Hardy-Littlewood-Pólya

Is there a comprehensive reference book on inequalities in the spirit of the one written by G.H. Hardy, J.E. Littlewood, and G. Pólya(*), but more up-to-date (i.e., published in more recent years and ...
5
votes
1answer
170 views

Symmetric inequality on generalized means

Do there exist two functions $f$ and $g$ continuous and strictly increasing $[0,1] \to \mathbf{R}$ such that $$ f^{-1}\left(\frac{1}{3} f(x) + \frac{2}{3} f(y)\right)<g^{-1}\left(\frac{1}{3} g(x) + ...
2
votes
1answer
116 views

Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in). I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and ...
5
votes
0answers
140 views

An inequality concerning non-negative integer matrices with constant row and column sums

[I posted this question on math.stackexchange a few weeks back, but no luck there so far: ...
6
votes
1answer
484 views

Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has $$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)} +\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi, $$ where $f$ is the ...
5
votes
2answers
259 views

Rademacher average based Hoeffding Inequality

I am following these lecture notes: Given the i.i.d. $\mathcal{Z}$-valued random variables $Z_1,\dotsc,Z_m$ and $\mathcal{G}$ is a set of bounded functions $g\colon \mathcal{Z}\to[a,b]$. Corollary ...
6
votes
1answer
133 views

Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?

Is it true that $$ ||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ? $$ where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...
0
votes
1answer
188 views

Proving a complicated inequality with powers of logarithms

I am currently formalising some results from complexity theory with a theorem prover. For that, I have to prove the following statement: Let $p, b, \varepsilon \in \mathbb R$ with $\varepsilon>0$ ...
0
votes
1answer
48 views

Characterisation of a matrix ordering property

Let $n$ be a positive integer; we consider all matrices mentioned henceforth to be $n$-by-$n$ matrices. Let $A$ and $B$ be matrices wherein all entries are nonnegative (such matrices will be called ...
7
votes
1answer
242 views

Proving a certain $ C^{*} $-algebraic inequality

Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality $$ |\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} ...
1
vote
1answer
144 views

Matrix Submodular Inequality

Given $a,b,x > 0$ I know following the submodularity property holds: \begin{align} \frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x} \end{align} My question is, does this property ...
7
votes
2answers
481 views

An upper bound for the difference between arithmetic and harmonic mean

Let $a_i\gt0$ for all $1\le i\le n$. It is well known that $$ \frac{a_1+a_2+\cdots+a_n}{n}-\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}\ge0, $$ with the equality when all $a_i$ are ...
3
votes
1answer
241 views

Lower bound for the $p$-th absolute moment of a sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $\operatorname E X_k=0$ and $\operatorname E |X_k|^p<\infty$ with $1<p<2$ for each $1\le k\le n$. I am interested in the ...
1
vote
0answers
134 views

Bound of spectral radius of polynomial of a complex matrix

I am trying to prove or disprove the following inequality. $$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$ where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and ...
6
votes
1answer
387 views

Is the conditional expectation a contraction in weak $\mathbb L^p$ spaces?

Let $(\Omega,\mathcal F,\mu)$ be a probability space. It is well-known that if $\mathcal A$ is a sub-$\sigma$-algebra of $\mathcal F$, $p\geqslant 1$ and $X$ is an element of $\mathbb L^p$ which takes ...
3
votes
1answer
224 views

Is this parametric inequality true?

Puzzled by this still open question, I tried comparing the arithmetic mean $A(x,y)=(x+y)/2$ with a mean intermediate between a geometric-type mean $G(X)=(x^a y^{1-a}+x^{1-a} y^a)/2\;$ for $0\le a \le ...
16
votes
2answers
987 views

This inequality why can't solve it by now (Only four variables inequality)?

I asked a question at Math.SE last year and later offered a bounty for it, only johannesvalks give Part of the answer; A few months ago, I asked the author(Pham kim Hung) in Facebook, he said that ...
4
votes
0answers
87 views

A “gradient” weak Harnack inequality for quasilinear elliptic equations

Suppose we are in the following loosely described setting: we have a non-negative supersolution $h$ of the following elliptic equation: \begin{equation} \Delta h + \|\nabla h\|^2 + f(x) \geq 0 ...
1
vote
0answers
63 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
1
vote
1answer
236 views

Nepero game (by Yacov Perelman)

I have already posted this question time before on stackexchange, but didn't receive a definitive solution. So this is the game: consider a positive integer number $n$ and divide it in a finite ...