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0
votes
1answer
260 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
115 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
210 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
95 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
92 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
324 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 ...
4
votes
1answer
198 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 ...
6
votes
1answer
434 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
281 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
135 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
203 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
657 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
204 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
80 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 ...
0
votes
0answers
82 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} ...
3
votes
3answers
236 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
176 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
235 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 ...
1
vote
1answer
112 views

Estimate of a ratio of two incomplete gamma functions

I would like to bound from above the expression $$ \frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)} $$ for $x>y>0$. By plotting the above expression I have found that ...
4
votes
1answer
193 views

Given a correlation matrix $B$. What correlation matrix A (maximizes / minimizes) the following: det(A+B)

Given correlation matrix $B$ (positive semi-definite with ones in the diagonal). 1)Find the correlation matrix $A$ which maximizes $\det\left(A+B\right)$. 2)Find the correlation matrix $A$ which ...
7
votes
0answers
444 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} ...
2
votes
4answers
420 views

Sum over integer compositions

Sorry if the question is trivial - are there closed form expressions or good approximations for the sum of a symmetric function taken over all integer compositions (into given number of parts) of a ...
2
votes
1answer
223 views

What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2) < 2$?

Let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. Note that $\sigma(x)$ is the classical sum-of-divisors function. For example, $$\sigma(12) = 1 + 2 + 3 + 4 + ...
2
votes
0answers
155 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
1answer
122 views

Is vesica piscis a maximal length curve constrained to two points?

Let $r(t), t\in [0,1]$ be a continuous piecewise $C^1$ curve on the plane where $r(0)=(0,0)$ and $r(1)=(1,0)$. The distance $|r(t)|$ is a non-decreasing function and the distance $|r(t)-(1,0)|$ is a ...
3
votes
1answer
125 views

Dependence of the constant in Korn's inequality on the domain

Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and $$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} ...
3
votes
0answers
222 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} ...
2
votes
1answer
143 views

AM-GM interpolation in the limit

Given a sequence of positive numbers $X = (x_1, x_2, \dots)$, define the $k$th elementary symmetric mean of the first $n$ entries to be $$ S(X, n, k) := \frac{\displaystyle\sum_{1 \leq i_1 < ...
2
votes
1answer
143 views

Inequality for certain analytic functions

Suppose that $f(z)$ is analytic for $\Re(z)>0$ and it is positive on the real axis. Suppose that for some $y_0$ we have $$ \left| {f\left( {x + iy} \right)} \right| \le f\left( x \right) $$ for any ...
1
vote
0answers
88 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
0answers
286 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 ...
4
votes
1answer
195 views

Determinant inequality of square-product sum of diagonal matrix and upper-triangular matrix

Recently, I have seen a matrix inequality but don't know how to prove it. The inequality goes as follows. For an arbitrary $n\times n$ diagonal matrix $\mathbf{D}$ and an arbitrary upper-triangular ...
29
votes
1answer
2k views

An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
0
votes
1answer
161 views

upper bound on infinite series by exponentials

I'm interested in upper bounding the following infinite series by exponential functions for x>1 (I would be fine with x>2 as well). \begin{eqnarray*} \sum_{n=1}^{\infty}e^{-\frac{x^2n^2}{2}}\\ ...
1
vote
1answer
152 views

Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= ...
0
votes
1answer
73 views

An inequality involving multi-index

I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this: For $x \in \mathbb{R}^{n}$ and $\alpha = ...
7
votes
1answer
438 views

The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c} ...
4
votes
1answer
359 views

An inequality involving operator and trace norms

Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the ...
13
votes
4answers
862 views

A Normal Distribution Inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the following ...
23
votes
0answers
715 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 ...
33
votes
3answers
3k views

the following inequality is true,but I can't prove it

The inequality is \begin{equation*} \sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right) \end{equation*} for all integer $d\geq 1$. I use computer to verify ...
2
votes
3answers
656 views

General bound for the number of subgroups of a finite group

I am interested in the following: Let $G$ be a finite group of order $n$. Is there an explicit function $f$ such that $|s(G)| \leq f(n)$ for all $G$ and for all natural numbers $n$, where $s(G)$ ...
0
votes
1answer
94 views

What is the corresponding version in the complex space of this proposition got in the real space real

How can I transform the following proposition that is gotten in $real$ space into the corresponding one used in the $complex$ space,i.e.,$A\in C^{n\times n},x=(x_1,...,x_n)\in C^n$ ? suppose that ...
1
vote
3answers
244 views

how to proof this Stirling related equation

here is what I need to proof, have no idea were to start. I know there is some connection with the Stirling theorem. $$ \sum_{i=0}^{d}\binom{m}{i} \leq \left ( \frac{em}{d} \right )^{d} $$ I tried ...
1
vote
0answers
202 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
0answers
115 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 ...
5
votes
1answer
276 views

A question on the Mahler conjecture

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and $$ K^* := \{y \in \mathbb{R}^n : \langle y, x ...
7
votes
1answer
462 views

A spectral inequality for positive-definite matrices

Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues $$ \lambda_1 \leq \cdots \leq \lambda_n , $$ is there a sharp upper bound for the product $\lambda_2 \cdots ...
2
votes
0answers
50 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} &=& ...
0
votes
1answer
186 views

Inequality of Partial Taylor Series

Hi, For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k=0}^{N} ...