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2
votes
1answer
328 views

A functional inequality

$g:[0,1]\to[0,1]$ continuously differentiable and increasing such that for all integers $t>0$ and for all $r\in(0,1)$, $g(r^{t+1})>g(r)\cdot g(r^t)$. Does this imply that for all ...
4
votes
2answers
211 views

is 1/max(i,j) a bounded matrix on hilbert spaces?

I would like to know if the infinite matrix $[\frac{1}{\max(i,j)}]_{i,j\geq 1}$ represents a bounded operator on $\ell^2(\mathbb{N}^\star)$. It would be sufficient to know if the Lehmer matrix ...
30
votes
3answers
2k views

Absolute value inequality for complex numbers

I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution. Does the inequality $$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$ ...
0
votes
0answers
62 views

On the product of relatively prime number $< N$ [duplicate]

Let $FI(N)$ denote the product of all $\phi(N)$ [relatively prime numbers $<N$] . And define $SFI(N)$ as the product of remaining $N-\phi(N)$ numbers $\le N$ (Which are not relatively prime to $N$) ...
0
votes
0answers
91 views

Estimating when does a certain binomial sum exceed an upper bound

Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers $$f_{n,m} = \sum_{i=\lfloor m/2 \rfloor}^m {n-2i \choose n - m -i}{i+1 \choose m - i +1}.$$ For example $f_{n,0} = ...
1
vote
0answers
105 views

about an inequality which looks like a Hardy inequality

I would like to know if the following inequality can be true : consider a double sequence $(u_{i,j})_{(i,j)\in (\mathbb{N}^\star)^2}$ of real numbers and a real $p\geq 1$, do we have $$ ...
3
votes
1answer
166 views

An inequality involving Bessel functions of imaginary order

The following inequality: $$\frac{\pi k}{\sinh{(\pi k)}}\;|J_{ik}(\tau)|^2\le 1,\;\;\;k,\tau\ge 0,$$ for Bessel function $J_{ik}(\tau)$, I found in http://link.springer.com/article/10.1134%2F1.558677 ...
2
votes
2answers
129 views

Bounds for the fat tail after trimming the mean?

I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$. 1) Does this quantity $f(X,t)$ have a name? ...
2
votes
0answers
85 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 ...
4
votes
0answers
139 views

Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all $n\geq3$, the function: ...
12
votes
2answers
266 views

Subset of vectors whose sum has a large norm

In Rudin - Real & Complex Analysis we have the following Lemma 6.3. If $z_1, \ldots, z_n \in \mathbb{C}$ then there is a subset $S \subseteq \{1,\ldots,n\}$ for which $$\left|\sum_{k \in S} ...
3
votes
1answer
168 views

Is this bounded from below?

Let $u_1, u_2, u_3 \in \mathbb{Z}$ such that $u_1^2 + u_2^2 = u_3^2$. Is $(u_3 + \frac{u_1 + u_2}{\sqrt{2}})^2$ bounded from below? The irrationality of $\sqrt{2}$ certainly precludes zero, but can ...
8
votes
1answer
277 views

An inequality for positive definite matrices

Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with nonnegative coordinates we have $$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...
7
votes
2answers
424 views

Inequality for a gamma function

Let $s=\sigma+it$ and $\Gamma(s)$ be the Euler gamma function. Does the inequality hold? $$ \left|\frac{\Gamma(s)}{\Gamma(2-s)}\right|\leq |s|^{2(\sigma-1)},\, 1<\sigma<2,\, t>t_0>0. $$ ...
7
votes
1answer
162 views

Inequality for Laguerre polynomials

Let $L_n$ be the $n$-th Laguerre polynomial defined by $\quad L_n (x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad $ I want to prove that $$ \forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le ...
7
votes
2answers
176 views

Local-to-global inequalities for measures: Brunn-Minkowski, Ahlswede-Daykin, what else?

This question is motivated by an obvious formal analogy between two well-known inequalities: Log-concavity and Brunn-Minkowski inequality Let $\mu(dx) := m(x) dx$ be an absolutely continuous ...
7
votes
0answers
210 views

Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$ Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...
0
votes
0answers
58 views

Bounds or approximations for the conditional probability of an event involving correlated random variables

Let $\tilde{\gamma_1}, \tilde{\gamma_2}, \ldots, \tilde{\gamma_N}$ be exponential random variables (RVs) that are correlated with each other. Let $\gamma_n$ be another exponential RV that is ...
20
votes
3answers
556 views

A sumset inequality

A friend asked me the following problem: Is it true that for every $X\subset A\subset \mathbb{Z}$, where $A$ is finite and $X$ is non-empty, that $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$ ...
3
votes
1answer
218 views

About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality $f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...
10
votes
1answer
410 views

Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if $$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$ But ...
0
votes
1answer
210 views

Is this Holder-type inequality true $\;(\int fg )\,(\int f^2 + g^2)\leq \int f^2g+fg^2\;$? [closed]

Suppose $f$ and $g$ are non-negative functions such that $\int f(x)dx = \int g(x)dx=1$. Is it true that $$\Big(\int fg \Big)\,\Big(\int f^2 + g^2\Big)\leq \int f^2g+fg^2\quad?$$
3
votes
1answer
284 views

An elementary inequality: reference request

Consider the problem of minimizing $\sum_{i=1}^{n}{x_{i}}$ under the constraints $\sum_{i=1}^{n}{x_{i}^{2}}=1$ and $x_{i} \geq 0$. Obviously the solution is given by the vector $(1,0,\ldots,0)$. Now ...
-3
votes
2answers
96 views

a inequality of $L^p$ [closed]

I want to prove the inequality $$ |x-y|^p \le \frac{p}{2}\big|x-y\big|\;\big(x^{p-1}+y^{p-1}\big) $$ if $p \ge 1$. For the case $p$ is an integer, it's easy to do, but I have no idea when $p$ is not ...
7
votes
3answers
315 views

Rosenthal like inequality for weak $\mathbb L^p$-norms

Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if ...
-3
votes
1answer
84 views

Inequality with four variables [closed]

Prove or find a counterexample. Let $a_1, a_2, b_1, b_2 \in \mathbb{R}$ satisfy $a_1+a_2=1$ and $b_1+b_2=1$. Then, $$ \frac{1}{a_1^2+a_2^2} + \frac{1}{b_1^2 + b_2^2} \geq (\frac{1}{(a_1 b_2)^2 + (a_1 ...
4
votes
1answer
164 views

A homogeneous but slightly asymmetric inequality

I need to prove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$ \begin{equation} \left|\left|\sum_{j=1}^l ...
0
votes
1answer
140 views

A Poincaré-type inequality with logarithmic function

For any function $f(x)$ we denote $\bar{f}:=\frac{1}{\Omega}\int_\Omega f(x)\,dx$. Let $\Omega\subset \mathbb{R}^n$ be a bounded smooth domain and $u(x)> 0$ be a smooth function defined on ...
4
votes
1answer
165 views

Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics http://www.encyclopediaofmath.org/index.php/Differential_inequality the following result is due to Chaplygin ...
3
votes
0answers
182 views

Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
5
votes
0answers
186 views

Maximizing Renyi entropy for a certain channel

The channel under consideration is $T = A + B$, where $A$ and $B$ take on values in $\{0, 1\}$ according to a probability mass function. Let (joint) random vector $(A_1, A_2,\ldots, A_n)$ be denoted ...
5
votes
1answer
106 views

Deviation bound for the maximum of the norm of Wiener process

Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof: $$ ...
1
vote
1answer
339 views

Prove or disprove $ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > \int_{0}^{\infty} \int_{-\infty}^{-x} f(x)f(y)dydx. $

Consider a symmetric, unimodal distribution $f(x)$ such that $\int_{0}^{\infty} f(x) > 1/2$. I want to prove or disprove the following: $$ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > ...
0
votes
1answer
298 views

Asymptotic growth for $\sum_{i=1}^{n-1}(n-i)\binom{k}{i}$ [closed]

For positive integers $n,k$, define $$f(n,k):=\sum_{i=1}^{n-1}(n-i)\binom{k}{i}.$$ What are upper and lower bounds of $f(n,k)$ by simpler terms? (e.g. finding bounds which are not a summation like ...
3
votes
1answer
118 views

An interpolation type inequality

Let $u(x),x\in R_+$ be a non-negative decreasing smooth function with compact support $[0,L]$, I want to know the following inequality is true? $a\in (0,1)$ $$\int_0^\infty \frac{1}{1+x}u^{1+a}dx \le ...
2
votes
1answer
223 views

How to prove that the odd continued fraction approximants of ln(1+X) are upper bounds?

The odd order continued fraction approximants for $\ln(1+X)$ are $$X,\quad \frac{X^2+6X}{4X+6,}\quad \frac{X^3+21X^2+30X}{9X^2+36X+30,}\quad \dots.$$ In "Some bounds for the logarithmic function", ...
3
votes
0answers
105 views

Comparison of Hardy's inequality and Sobolev's inequality

Both, Hardy's inequality and Sobolev's inequality, are estimates that compare the Laplacian of a function and to the function itself, admittedly in a slightly different fashion. Still they seem to be ...
1
vote
0answers
101 views

Maximizing an integral over a convex region

Let $C$ denote a compact, convex region in the plane containing the origin with unit area, and let $f$ be a probability distribution on $C$. Let $f^\ast$ denote the distribution that maximizes the ...
10
votes
2answers
339 views

Inequalities for averaging over partially ordered sets

Let's start from a classical inequality: If $0\le a_1\le\cdots\le a_k$ and $0\le b_1\le\cdots\le b_k$ then $(a_1+\cdots+a_k)(b_1+\cdots+b_k)\le k(a_1b_1+\cdots+a_k b_k)$. It can be written also in ...
5
votes
1answer
270 views

Inequality involving the side lengths of a quadrilateral

If $a$, $b$, $c$ and $d$ are the four sides of a quadrilateral, the problem is to show that $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. I've verified it to be true for quite a large number of ...
7
votes
1answer
492 views

Can the Brun-Titchmarsh theorem be improved when the modulus is smooth?

For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The Brun-Titchmarsh theorem states that $$\pi(x,q,a)\leq ...
7
votes
3answers
288 views

An inequality related to the number of binary strings with no fixed substring

This is basically a repost of this math.se question. At the time I was writing this I thought it has to have a straightforward solution so I posted it there. Now I am not so sure about it being so ...
2
votes
1answer
144 views

Equivalent metrics on symmetric positive definite matrices

By similar arguments as for the proof of the golden-thompson inequality (see "Log majorization and complementary Golden-Thompson type inequalities" by T.Ando and F.Hiai) we can show that for all A,B ...
1
vote
0answers
213 views

Proof of an inequality for a linear combination of three trigonometric functions

Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$ where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ ...
5
votes
3answers
806 views

Set of Positive Definite matrices with determinant > 1 forms a convex set

While reading a paper An Arithmetic Proof of John’s Ellipsoid Theorem by Gruber and Schuster, I have a question on their proof. Consider an $n\times n$ real symmetric and positive definite matrix ...
2
votes
0answers
207 views

An integral inequality

Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be bounded with derivative $g'$. I have shown that the following inequality holds for all $w\in\mathbb{R}$, ...
3
votes
0answers
101 views

A challenging non homogenous fractional inequality

I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in ...
3
votes
3answers
246 views

An inequality about the sum of some unit fractions with a property

Question : Is the following true for any $n,N\in\mathbb N$? $$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$$ Motivation : I've known the $N=3$ case : ...
0
votes
1answer
219 views

Bounding the positive semi-definite matrix with its block diagonal matrix [closed]

Can we bound $\mathbf{A}$ with $\mathbf{A^*}$ as ${\bf{A}} \preceq {{\bf{A}}^*}$ where \begin{equation} {\bf{A}} = \left[ {\begin{array}{*{20}{c}} {{{\bf{A}}_{11}}}&{...}&{{{\bf{A}}_{1N}}}\\ ...
7
votes
0answers
274 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 ...