The inequalities tag has no wiki summary.

**3**

votes

**1**answer

133 views

### Inequality for a gamma function

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

**22**

votes

**0**answers

620 views

### Does a proof of Selberg's 3.2 inequality exist?

A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that
$$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi ...

**12**

votes

**0**answers

318 views

### 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 ...

**8**

votes

**0**answers

441 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}
...

**7**

votes

**0**answers

226 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 ...

**6**

votes

**0**answers

151 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, ...

**6**

votes

**0**answers

482 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 $M$:
...

**5**

votes

**0**answers

441 views

### Any similar inequality in literature?

I got the following inequality:
$B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary.
$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$,
...

**4**

votes

**0**answers

115 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:
...

**4**

votes

**0**answers

149 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 ...

**4**

votes

**0**answers

398 views

### System of Equations Upper Bound

I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here:
For ...

**4**

votes

**0**answers

200 views

### Applying the amplification trick + probabilistic method on connected graphs

First of all, I apologise if my question is not very specific. I am not really looking for a concrete answer but rather some hints and pointers what could be used/applied in this situation. A concrete ...

**3**

votes

**0**answers

148 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 ...

**3**

votes

**0**answers

88 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 ...

**3**

votes

**0**answers

71 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

**0**answers

209 views

### Inequalities in paper by Jean Bourgain

The question refers to the following paper by Jean Bourgain: http://arxiv.org/abs/math-ph/0011053
Specifically, I can't derive the following inequality in (1.20):
\begin{equation}
...

**3**

votes

**0**answers

132 views

### Upper bound on integrals of Legendre polynomials

Hi,
If $P_n(x) $ is unnormalized shifted Legendre polynomial, and $g_{n,m}(x) = \int_0^x P_n(x_1)x_1^m dx_1, n>m $ then what is the upper bound $ |g_{n,m} (x)|_{max} , x\in (0,1) $ as a function ...

**3**

votes

**0**answers

290 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
...

**2**

votes

**0**answers

60 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 ...

**2**

votes

**0**answers

202 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 $w\in\mathbb{R}$,
...

**2**

votes

**0**answers

145 views

### An inequality for Lp-functions

I am interested in the following inequality:
\begin{equation}
\int\limits_\Omega \left\vert f_1 - f_2 \right\vert^p ~ d\mu \le C_p \left[ \int\limits_\Omega \left\vert f_1 \right\vert^p ~ d\mu + ...

**2**

votes

**0**answers

43 views

### Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables

Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors:
\begin{eqnarray*}
\bar{a} &=& ...

**2**

votes

**0**answers

264 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 ...

**2**

votes

**0**answers

184 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) = ...

**2**

votes

**0**answers

196 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 ...

**1**

vote

**0**answers

130 views

### bound for $|E[\frac{X}{Y}]âˆ’\frac{E[X]}{E[Y]}|$

Is there some bound for $|E[X/Y]âˆ’E[X]/E[Y]|$ ? Here $X$ and $Y$ are both summation of a fixed number of Bernoulli random variables and a constant that is >0, which is to guarantee that the denominator ...

**1**

vote

**0**answers

76 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 ...

**1**

vote

**0**answers

200 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, $ ...

**1**

vote

**0**answers

74 views

### An inequality for elementary symmetric means of a periodic sequence

Given a tuple of positive numbers tuple $X = (x_1, x_2, \dots,)$, define the $k$th elementary symmetric mean of the first $n$ entries to be
$$ S(X, n, k) := \frac{\displaystyle\sum_{i_1 < \cdots ...

**1**

vote

**0**answers

277 views

### Inequality for complex Hankel function

Let $x>1$ and $0<\varphi<\frac{\pi}{2}$ be fixed. I would like to show that for any $s>0$, the following inequality holds:
$$
\left| H_{\frac{is}{e^{i\varphi } \cos \varphi}}^{\left( 1 ...

**1**

vote

**0**answers

191 views

### Bounding a sum of binomial coefficients in terms of 'the next one'

I need to bound a sum of a portion of binomial coefficients in terms of "the next one", and understand what is the best which can be said in this sense.
Given a real number $t \geq 2$, call $P(t)$ ...

**1**

vote

**0**answers

105 views

### Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable

I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is ...

**1**

vote

**0**answers

83 views

### Multivariate polynomial with positive coefficients

This question was originally asked at stack exchange (http://math.stackexchange.com/questions/292922/multivariate-polynomial-with-all-coefficients-positive), but did not receive any feedback for more ...

**1**

vote

**0**answers

131 views

### A Multiplicative version of McDiarmid's Inequality like the one of Chernoff-Hoeffding Bounds

McDiarmid's Inequality basically says the following:
Let $X_1, X_2, X_3, \ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the ...

**1**

vote

**0**answers

115 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

**0**answers

166 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 ...

**1**

vote

**0**answers

94 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 ...

**1**

vote

**0**answers

67 views

### 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 ...

**1**

vote

**0**answers

99 views

### Linear complementarity problem - Classification

For Linear complementarity problems (LCP) like
$\mathbf{Mz}+\mathbf{q} \ge \mathbf{0}$
$\mathbf{z} \ge \mathbf{0}$
$\mathbf{z}^{\mathrm{T}}(\mathbf{Mz}+\mathbf{q}) = 0$
there exists a vast amount ...

**1**

vote

**0**answers

931 views

### Inequality concerning absolute value of a polynomial

Let
$$f(z) = (1-1/t) z^w + z/t - 1$$
with integers $t\geq2$ and $w\geq2$.Let $r=1+1/(tw^3)$. How do I show
$$\left\lvert f(r e^{i\varphi}) \right\rvert \geq \left\lvert f(r) \right\rvert$$
for any ...

**0**

votes

**0**answers

45 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 ...

**0**

votes

**0**answers

79 views

### Lower-Upper bounds on the cardinality of a set

Let $S$ be a finite set which is a subset of $\{(\alpha ,\beta ):\alpha , \beta \in \mathbb{Z}, \alpha\geq 0, \beta \geq 0\}$ and $ T(x,y)=\sum_{(\alpha ,\beta ) \in S} h_{\alpha, \beta} ...

**0**

votes

**0**answers

57 views

### What is the closed-form or the deterministic form of a quadratic form probability inequality

Hello, everyone, I want to resolve one optimal problem, with the following probability inequality constraint.
$Pr(h^H(W_1 - W_2 -W_3 -U)h \geq \sigma^2) \leq \rho$
where
$h \sim CN(0,I) \ \text{is ...

**0**

votes

**0**answers

433 views

### A product sum inequality question

For any $x_{1},x_{2},\cdots x_{6}$ with $\sum_{i=1}^{6}x_{i}^{2}=1$
and $y_{1},y_{2},\cdots y_{6}$ in $\mathbb{R}$ with $\sum_{i=1}^{6}y_{i}^{2}=1$,
do there always exist $z_{1},z_{2},\cdots z_{6}$ in ...

**0**

votes

**0**answers

170 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 ...

**0**

votes

**0**answers

202 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 ...

**0**

votes

**0**answers

301 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 ...