Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
Suvrit's user avatar
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16 votes
0 answers
373 views

An inequality for matrix norms

Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically: Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
Mostafa's user avatar
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16 votes
1 answer
705 views

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 ...
David E Speyer's user avatar
16 votes
0 answers
780 views

Determinant inequality involving Hermitian, positive definite matrices

Let $A,B,C\in M_{n}(\mathbb C)$ 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 has been ...
Krokop's user avatar
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15 votes
0 answers
740 views

Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$

I would like to prove that $$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$ for any $\omega > 0$ and $...
Tanya Vladi's user avatar
14 votes
0 answers
1k views

Nice proof of inequality $(1-x^p)^{1/p}(1-x^q)^{1/q}\ge (1-x)(1+x^c)^{1/c}$ where $2^{1/c} = p^{1/p} q^{1/q}$?

Let $0\leq x < 1$, $1 \leq p < \infty$ and $q$ be the conjugate exponent defined by $$1/p + 1/q = 1.$$ I am looking for a nice proof that $$ \frac{(1-x^p)^{1/p}(1-x^q)^{1/q}}{(1-x)(1+x^c)^{1/...
George Shakan's user avatar
12 votes
0 answers
217 views

Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...
Wolfgang's user avatar
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11 votes
0 answers
290 views

$L_2$ minimizing makespan vs. $L_\infty$ minimizing makespan

There are $n$ positive real numbers. We partition these numbers into $m$ parts, the size of each part is the sum the numbers in this part. Maximum size of the parts is called a makespan of a partition....
kakia's user avatar
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9 votes
0 answers
278 views

Inequality for symmetric polynomial functions of log concave variables

Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$). ...
René Gy's user avatar
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9 votes
0 answers
673 views

Van der Pol's identity for the sum of divisors and a quartic polynomial equation for odd perfect numbers

In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following recurrence relation ($n>1$): $$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
Perfect Number's user avatar
9 votes
0 answers
4k views

Is the conjecture A+B=C following correct?

Is the conjecture on A+B=C following correct ? Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write: $...
Đào Thanh Oai's user avatar
9 votes
0 answers
354 views

How to count integer lattice points close to a subspace of $\mathbb R^n$?

Consider $m$ linearly independent vectors in $n$-dimensional Euclidean space, $v_1,...,v_m \in \mathbb R^n$ where $1\leq m<n$, and let $U := {\rm span}(v_1,...,v_m)$ denote the $m$-dimensional ...
Dierk Bormann's user avatar
9 votes
0 answers
772 views

Isoperimetric inequality, isodiametric inequality, hyperplane conjecture... what are the inequalities of this kind known or conjectured?

I duplicate here a question I asked on math.stackexchange. Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some ...
9 votes
0 answers
517 views

Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$, \begin{cases} &\sum\limits_{j=0}^{m-1}x_jx_{2k-j}=...
Binzhou Xia's user avatar
8 votes
0 answers
327 views

Bounding a sum of reciprocals of square-free integers

(Cross-posted from MSE, as the question did not get any clear answer) Fix positive integers $k$ and $n$. Let $N_1,\dots,N_r$ be all the integers less than or equal to $n$ that are squarefree and have ...
Juan Moreno's user avatar
8 votes
0 answers
374 views

When do we have $\|X - Y\| = \|\Sigma(X) - \Sigma(Y)\|$?

For any $X \in \mathbb{C}^{m\times n}$, let $\Sigma(X)$ be the "middle factor" in its SVD, so that $X = U\Sigma(X) V^H$ and the diagonal of $\Sigma(X)$ is arranged in descending order. ...
Nuno's user avatar
  • 213
8 votes
1 answer
449 views

How large can the dimension of a 'Span of powers of a finite field basis' be?

Let $p$ be a prime. For finite field $\mathbb{F}_{p^k}$ and $d\in\mathbb{Z}^+$, I am considering the following quantity, where we interpret the field $\mathbb{F}_{p^k}$ also as a $\mathbb{F}_p$-vector ...
actcon's user avatar
  • 89
8 votes
0 answers
394 views

Eigenvalues of cyclic stochastic matrices

Let's consider the following $n \times n$ cyclic stochastic matrix $$ M= \begin{pmatrix} 0 & a_2 & & & &b_n \\\ b_1 & 0& a_3& &&& \\\ & b_2 & ...
Hadrien's user avatar
  • 181
8 votes
0 answers
351 views

How localized can a polynomial be in the L1 norm?

Let $0<s<2$ be a parameter, $\Omega = [-1,1]$, and $\Omega_s\subset \Omega$ be a set of measure $s$. I would like to bound the following ratio from above: $$\sup_{p\in\mathcal{P}_n} \frac{\...
alext87's user avatar
  • 3,167
8 votes
0 answers
476 views

Strange determinant inequality $\det(C+ xA) \det(C-xA) \le (\det C)^2$

Let $A$ be an all-one $3$-by-$3$ matrix, let $C$ be a $3$-by-$3$ matrix, and let $x$ be a real number. How might one prove the following inequality? $$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
Martin's user avatar
  • 99
7 votes
0 answers
200 views

Li-Yau inequality on $\mathbb R^2$ for functions that are somewhat close to $1$

Let $u:\mathbb R^2\times \mathbb R_{>0}\to \mathbb R_{>0}$ be a positive solution to the heat equation on $\mathbb R^2$ ($u_{xx}+u_{yy}=u_t$, no constants). The Li-Yau inequality in this case ...
Alexander Kalmynin's user avatar
7 votes
0 answers
219 views

Loomis-Whitney versus Gagliardo inequalities

When searching for a reference, I discovered a curious fact about the Wikipedia page concerning the Loomis-Whitney Inequality (LWI).This page, which exists only in an English version, states that the ...
Denis Serre's user avatar
  • 51.5k
7 votes
0 answers
563 views

A rank inequality

Suppose $$M := \begin{bmatrix} M_{11} & \cdots &M_{1d} \\ \vdots & \ddots & \vdots \\ M_{d1} & \cdots & M_{dd} \end{bmatrix}$$ is a $d \times d$ block matrix such that $$M_{...
SMD's user avatar
  • 480
7 votes
0 answers
311 views

An inequality which involves a sum of integrals

Please help me to prove $$ \sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \...
Mikhail_K's user avatar
7 votes
0 answers
716 views

Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson

Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites. Let the ...
Luis Mendo's user avatar
7 votes
0 answers
483 views

A curious inequality

Let $r_k>0$ for $k = 1,\ldots, n$, let $\alpha_k, \beta_k\in \mathbb{R}$ be given such that $|\alpha_k|\le \beta_k\le \frac{\pi}{2}$. Suppose further that $\left|\sum\limits_{k=1}^nr_ke^{i(\...
Betrand's user avatar
  • 478
6 votes
0 answers
651 views

Sophomore dream, sine, and the golden ratio

It's an inequality I found nice let me propose it : $$\int_0^\infty \sin\left(x^{-x}\right) \, dx<\varphi=\frac{1+\sqrt{5}}{2} \tag I$$ My attempt : First of all I recall two facts : Fact 1 $$\...
DesmosTutu's user avatar
6 votes
0 answers
116 views

Weak-type inequality for the partial Fourier sum operator

I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark: For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
pureorapplied's user avatar
6 votes
1 answer
430 views

What inequalities for convex sets are known since the work of Scott and Awyong?

In 2000, Paul R. Scott and Poh Way Awyong published the paper Inequalities for Convex Sets, which nicely collates the known results relating various natural geometric functionals (diameter, area, etc.)...
RavenclawPrefect's user avatar
6 votes
0 answers
316 views

An inequality related to the numbers of faces of polytopes with d+2 facets

I would like to prove an inequality related to the number of $k$-faces of two $d$-polytopes with $d+2$ facets; see (1) below. Let $r>0$, $s>0$, $t\ge 0$, and $d\ge 2$ be such that $d=r+s+t$. We ...
Guillermo Pineda-Villavicencio's user avatar
6 votes
0 answers
128 views

Q-analogue of an inequality

Pick integers $b\geq a \geq 0$ and $k\geq j\geq 0$. It is not super-difficult to prove the inequality $$ \binom{kb}{ka}^j \geq \binom{jb}{ja}^k. $$ This is actually quite a nice inequality that was ...
Per Alexandersson's user avatar
6 votes
0 answers
126 views

A reference for an integrability property?

In a recent paper of mine (Compensated integrability), I established a functional inequality which has nice consequences. For instance, it contains the isoperimetric inequality, and it gives a new ...
Denis Serre's user avatar
  • 51.5k
6 votes
0 answers
222 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 ...
Oai Thanh Đào's user avatar
6 votes
0 answers
563 views

Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
Bullmoose's user avatar
  • 897
6 votes
0 answers
238 views

Operator arithmetic-harmonic mean inequality with operator-valued weights

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
Alexander Shamov's user avatar
6 votes
0 answers
626 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 $M$: ...
Zur Luria's user avatar
  • 1,613
5 votes
0 answers
163 views

Monotonicity of ratio of symmetric polynomials

The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by \begin{equation*} h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
Rachid Ait-Haddou's user avatar
5 votes
0 answers
159 views

Is there a sharper Golden–Thompson inequality?

For any two Hermitian matrices $A$ and $B$, the Golden–Thompson inequality $$\mathrm{Tr} (e^A e^B) \geq \mathrm{Tr} \, e^{A + B}$$ holds, and it is known to be a strict inequality whenever $[A, B] \...
Karen H.'s user avatar
5 votes
0 answers
340 views

Extending Gromov's inequality

In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound $$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol} \frac{\stsys_2^n}{...
Mikhail Katz's user avatar
  • 15.1k
5 votes
0 answers
139 views

Log Sobolev inequality uniform in parameters

Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...
Matt Rosenzweig's user avatar
5 votes
0 answers
232 views

On the weighted Hilbert inequality a la Montgomery and Vaughan

The weighted Hilbert inequality estimates the norm of the skew-hermitian matrix $$B :=\left[ \frac{\delta_m^{1/2} \delta_n^{1/2}}{x_m-x_n} \right]_{m,n}$$ on $\ell_2$, where $x_1,\ldots$ is a finite ...
Narutaka OZAWA's user avatar
5 votes
0 answers
238 views

Equality from the Grothendieck inequality

I asked the following question on math.stackexchange.com but have not received any response. So I would like to try my luck here. This question is related to the Grothendieck inequality. Let field $\...
Hans's user avatar
  • 2,169
5 votes
0 answers
683 views

Gershgorin's 2nd theorem (disjoint circles): elementary proof?

Let $A \in \mathbb{C}^{n\times n}$ be a complex matrix. We let $a_{i,j}$ be the $\left(i,j\right)$-th entry of $A$ for all $i, j \in \left[n\right]$ (where $\left[n\right]$ denotes $\left\{1,2,\ldots,...
darij grinberg's user avatar
5 votes
0 answers
152 views

Inequality for functions on $[0,\infty)$

Let $0<q<1$ and $\varphi(q;x)=\displaystyle \prod_{j=0}^\infty (1+q^jx),\;x\geqslant 0.$ Consider the following functions: $$l_k(x;q):=\frac{q^{k(k-1)/2} x^k}{(1-q)(1-q^2)\dots (1-q^k)\varphi(x;...
Deepti's user avatar
  • 743
5 votes
0 answers
266 views

An integral trigonometric inequality

Problem 1. Suppose that $\xi>0$ and $\sin(2\xi)<0$. Let $$b_\nu=(N-v+1)\tfrac{\pi}{\xi}\quad\mbox{for}\quad\nu=1,\dots,N:=\big[\tfrac{\xi}{\pi}\big].$$ Prove that $$\mathrm{sgn}(\sin \xi)\...
Lviv Scottish Book's user avatar
5 votes
0 answers
298 views

In arbitrary cyclic polygon then $\sum_{i<j} x_{ij}^\alpha \ge \sum_{i<j} y_{ij}^\alpha $

I am looking for a proof of the inequality as follows: Conjecture: Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$. Let $B_1B_2....B_n$ be a polygon incribed the circle $(O)...
Đào Thanh Oai's user avatar
5 votes
0 answers
558 views

A minimal eigenvalue inequality

Suppose $A$ is an $n\times n$ real symmetric positive definite matrix. Let $A^{(-1)}_{i,j}$ be the $n\times n$ matrix the entries $(i,i),\,(i,j),\,(j,i),\,(j,j)$ of which equal to the corresponding ...
Hans's user avatar
  • 2,169
5 votes
0 answers
834 views

Anti-concentration inequality for Gaussian random vector

I am trying to obtain an explicit expression for $C$ in terms of $b$ in the following inequality. Suppose that $Y$ is a centred Gaussian random vector in $\mathbb R^p$ such that $\operatorname EY_j^...
Cm7F7Bb's user avatar
  • 403
5 votes
0 answers
168 views

Inequality about moments of a random variable and of its conditional expectation

This is a follow-up to a question I asked earlier: Moments of a random variable and of its conditional expectation My claim turned out to be false. Here is a new claim. Let $X$ be a bounded random ...
Hedonist's user avatar
  • 1,269
5 votes
0 answers
289 views

inequality in a shape of inclusion exclusion formula

I have two inequalities to show, both of which describe some probabilities. First I know how to handle, and it follows from applying arithmetic-harmonic mean inequality: consider 9 numbers $a_1,a_2,...
Marek Adamczyk's user avatar

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