for questions involving inequalities.

**2**

votes

**2**answers

147 views

### Is it possible to find the maximum value of S?

Let $a_1$, $a_2$, …, $a_n$ and $b_1$, $b_2$, …, $b_n$ be $2n$ strictly positive integers not greater than $M$, with $M$ is a given positive integer, such that $$a_1+ a_2+ \dotsb+ a_n=b_1+ b_2+ \dotsb+ ...

**1**

vote

**1**answer

111 views

### Upper bound of the fraction of gamma functions

Is there a simple upper bound of the following fraction of gamma functions for any $a,b\geq1/2$:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$
An upper bound in the following form is ...

**4**

votes

**1**answer

146 views

### Intuitive proof of Golden-Thompson inequality

Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality:
For any hermitian matrices $A,B$:
$$
\text{tr}(\exp{(A+B)}) \...

**8**

votes

**3**answers

242 views

### Matrix determinant inequality proof without using information theory

Let $A$ be a $k \times n$ orthogonal matrix; i.e., $AA^T = I_{k \times k}$. For $1 \leq j \leq n$, let the squared norm of the $j$-th column of $A$ be denoted by $\alpha_j^2$; i.e.,
$$\sum_{i=1}^k a_{...

**6**

votes

**4**answers

230 views

### Improvement of Chernoff bound in Binomial case

We know from Chernoff bound
$P\bigg(X \leq (\frac{1}{2}-\epsilon)N\bigg)\leq e^{-2\epsilon^2 N}$ where
$X$ follows Binomial($N, \frac{1}{2}$).
If I take $N=1000, \epsilon=0.01$, the upper bound is ...

**9**

votes

**1**answer

176 views

### The discrete Hardy-Littlewood-Sobolev inequality

Let $p>1$, $q>1$, $0<\lambda<1$ be such that
$\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that
$(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$.
It is known ([1,2,3]...

**2**

votes

**1**answer

99 views

### it's convex sequence inequality

Sequence of real numbers $a_0,a_1,\dots,a_{n}$ are called concave if for each $0<i<n$, $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$.$a_{0}=0$ Find the largest $c(n)$ such that for every concave sequence ...

**0**

votes

**1**answer

33 views

### Right tail decay of F distribution [closed]

Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$?
$$\mathbb{P}(X\geq x)$$
what is the order of the above probability as $x\to+\infty$?

**23**

votes

**4**answers

898 views

### show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$

I saw the following results in a book. The author said it was not difficult to prove how I felt it was difficult to prove, so I asked here. The result comes from a book that has no electronic version....

**1**

vote

**2**answers

101 views

### lower bound the probability of at least L collisions

Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$.
If we now ask ...

**13**

votes

**1**answer

906 views

### A conjecture of Littlewood

The following is a conjecture due to Littlewood.
For any set of distinct non-zero integers $n_1,\ldots,n_k$ the inequality
$$\int_0^{2\pi}|1+e^{in_1x}+\cdots+e^{in_kx}| \, dx\geq C\log k$$ holds....

**14**

votes

**3**answers

589 views

### What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$?

My question is to find the minimum of the following expression:
$$A(x,y) = \sum_{1\leq i<j\leq n} |x_i-x_j|\ |y_i-y_j|,$$
over the set of pairs of real vectors $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ ...

**4**

votes

**1**answer

170 views

### An inequality of T. Carleman

I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given.
Let $f(z)$ be an analytic function on a subdomain $...

**0**

votes

**2**answers

243 views

### Strange inequality with $\zeta(5)$ [closed]

$$\frac{\pi^2}{1+\exp(-1/\pi^2)}<\sum\limits_{k=1}^{\infty}\frac{5}{k^5}<\frac{\pi^2}{1+\exp(-\pi/31)}$$
How can I prove it (not only with computation)?

**3**

votes

**1**answer

103 views

### Log concavity of the maximum of dependent Gaussians

Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in ...

**3**

votes

**1**answer

141 views

### Why does the assumption $|U_t| \le \frac1{p_{\min}}$ work in this paper?

I am reading a 2009 paper right now "Importance Weighted Active Learning" and on page 5, there is a theorem that uses the inequality $|U_t| \leq \frac{1}{p_{\min}}$. I am not sure how the paper found ...

**1**

vote

**1**answer

54 views

### additive discrepancy under a multiplicative constraint

Consider four sequences of numbers, $0 \le a_i, b_i, c_i, d_i \le 1$, suppose they satisfy the following constraints:
(1). $\sum_{i=1}^K a_i, \sum_{i=1}^K b_i, \sum_{i=1}^K c_i \ge 1/2 + \epsilon$;
(...

**2**

votes

**0**answers

49 views

### How to solve such integer program problem?

Consider a $3$-tuple $(a,b,s)$ with $a,b\in\mathbb{Z}_+,s\in\mathbb{Q}_+$. Denote $ab-s$ by $\Delta$. Let $A$ be a positive number. What are the values of $A$ such that for any $(a,b,s)$ with $\Delta\...

**1**

vote

**1**answer

90 views

### Any inequalities / estimates for a lower bound of the $L^2$ inner product of a quantity and its derivative?

For some numerical analysis of a fluid, I am wondering if there is any inequality that provides a lower bound (in the $L^2$ norm), for the $L^2$ inner product of a quantity with its derivative. In ...

**10**

votes

**3**answers

472 views

### Entropy and total variation distance

Let $X$, $Y$ be discrete random variables taking values within the same set of $N$ elements. Let the total variation distance $|P-Q|$ (which is half the $L_1$ distance between the distributions of $P$ ...

**5**

votes

**2**answers

390 views

### On the upper bound of $\sum_{i=1}^{n}x^m_{i}$ subject to the conditions $\sum_{i=1}^{n}x_{i}=0$ and $\sum_{i=1}^{n}x^2_{i}=n$

The following question has been posted on mathematics stackexchange: inequalities problem, perhaps arising from a question on expectations.
Let $x_{1},x_{2},\cdots,x_{n}$ are real numbers, and such ...

**2**

votes

**2**answers

109 views

### How to estimate a recursive inequality with an upper bound

The below is a simplification of part of a proof I'm working on, in numerical analysis. It is similar to a paper that I studied some months ago, for which I got some advice here on MathOverflow.
I ...

**0**

votes

**1**answer

97 views

### Need for Reference and Name Of Inequality

Some time ago, I ran across the following inequality (if I remember rightly): $\bigg(\sum_{i=1}^{n}v_{i}^{k_{1}}\bigg)\geq n\bigg(\frac{\sum_{i=1}^{n}v_{i}^{k_{2}}}{n}\bigg)^{\frac{k_{1}}{k_{2}}}$, ...

**12**

votes

**2**answers

501 views

### Show that there exist $k\in\{1,2,\cdots,n\}$ such that $\frac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\frac{1}{2}\right)^2>\frac{1}{12}-\frac{1}{6n}$

The following question has post mathsatck :Nice problem, perhaps from a question of expectation.The problem seemed so elementary, but I had tried to solve it for a long time and eventually failed.I 'd ...

**0**

votes

**0**answers

40 views

### Function in two variables inequalities

Let $A$ be the set of all subintervals of $[0,1]$. For any function $f:[0,1]\times [0,1]\rightarrow A$, define $g(a,b)=[0,1]\setminus f(a,b)$. What are all functions $f$ such that for any $a,b\in[0,1]$...

**3**

votes

**2**answers

284 views

### Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$

The question is to prove:
$$
\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1.
$$
Numerically it seems to hold true. So I have made some attempts to ...

**1**

vote

**1**answer

79 views

### A distribution which is Wasserstein-close to a compactly supported distribution is almost compactly supported

I wonder whether this is true: If a distribution is very close (in the Wassertein sense) to another distribution with compact support, then the former must put only tiny amount of mass outside a ...

**1**

vote

**0**answers

57 views

### Show that the norm's bound is an exponent

Let $f$ be a continuous function on $S^2$. Consider $g\in C^{\infty}(R)$, such that $g(x)=1$ for $|x|\leq 1$ and for $|x|\geq 2$. Let $h(x)=g(x)-g(2x)$. The notation $proj_k$ denotes the orthogonal ...

**1**

vote

**0**answers

100 views

### Locally Lipschitz sufficiently implies a Gronwall inequality?

In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...

**1**

vote

**1**answer

233 views

### A simple two variable analytic inequality, inspired by probability

I'm trying to prove the following inequality:
$$
bf_1g_1 + (x-b)f_1g_0 + (y-b)f_0g_1 + (1-x-y+b)g_0f_0
\le
(|f_1|^p x + |f_0|^p (1-x))^{1/p} (|g_1|^p y + |g_0|^p (1-y))^{1/p}
$$
where $0\le xy\le b\le ...

**4**

votes

**0**answers

229 views

### Maximize product of sums

Let $n,k\geq 2$ be positive integers. For each $1\leq i\leq n$, let $I_i$ be a nonempty subset of $\{1,2,\dots,k\}$. Let $P_i=\sum_{j\in I_i}x_j$, and let $P=P_1\cdot P_2\cdot\dots\cdot P_n$. (For ...

**1**

vote

**2**answers

88 views

### One inequality connected with the linear second order ODE

Is the following statement true?
Let $ a>0, b>0, h>0 $, $x(t)$ be the solution of the differential equation
$ \ddot{x}+a \dot{x}+bx=h$
with initial conditions $x(0)=u<0 , \dot{x}(0)...

**0**

votes

**0**answers

50 views

### Bounding a particular ratio of sums

Let $p_i = \frac{1}{w_i + x}$ for $w_i >0$, $1\leq i \leq N$, and $x \in [\min_i w_i^{2/3},\max_i w_i]$. We wish to bound the ratio of the expected value of $w/(w+x)$ under the probability ...

**4**

votes

**2**answers

123 views

### A kind of exponential concavity for polynomials?

Is there $C > 0$ such that the inequality
$$
\prod_{n\in\mathbb{N}} p(n)^{a_n} \leq p\left(C\prod_{n\in\mathbb{N}} n^{a_n}\right)
$$
holds for all finitely supported sequences $(a_n)$ with $a_n\geq ...

**3**

votes

**1**answer

184 views

### Inequality for exponential sum in Dvoretzky 1972

I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has ...

**2**

votes

**1**answer

181 views

### Computing minimum / maximum of strange two variable funcion

I want to compute $$\max_{\frac{1}{5 \theta }\leq \alpha \leq \frac{1}{2}} \left(\frac{\alpha\log (\alpha)}{1-\alpha} + \log \left( 1 - \alpha\right) + \frac{1}{1-\alpha} \cdot \left( f\left(\frac{...

**-2**

votes

**1**answer

143 views

### A generalization of Chebyshev's sum inequality

From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference.
Inequality: Let $y=f(x,y)$ is ...

**2**

votes

**1**answer

100 views

### Parabolic variational inequality: regularity of the time derivative in $L^2(0,T;H)$?

Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$u(t) \leq \psi(t) ...

**0**

votes

**1**answer

117 views

### A rearrangement inequality for exponentiation function

Inequality: If $n$ be positive integer $n \ge 2$ and $a_1 \ge a_2 \ge \cdots \ge a_n \ge 0$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 0$ then
$${\left(\sum_{i=1}^{n}{a_i^{\...

**2**

votes

**1**answer

193 views

### Some inequalities on chain of circle packing

By my computation, I pose a conjecture as follows and I am looking for a proof:
Conjecture: Let $(O)$ be a circle with radius $R$, and $n$ be positive integer $n\ge 3$. Construct $n$ circles $(...

**2**

votes

**0**answers

111 views

### A generalization of Bernoulli's inequality and what does it application for?

Let $a_1 \ge a_2 \ge \cdots \ge a_n \ge 1$ or $0 \le a_1 \le a_2 \le \cdots \le a_n \le 1$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 1$ then
$$\left(\sum_{i=1}^{n}{\alpha_i} \right)\...

**8**

votes

**1**answer

215 views

### Expectation inequality for sampling without replacement

Is the following proposition correct?
$X_1, X_2, X_3$ are uniformly at random sampled from a finite set $\mathcal X$ without replacement.
$f : \mathcal X^2 \rightarrow \mathbb R_{\ge0}$ is symmetric:...

**0**

votes

**2**answers

175 views

### A symmetric polynomial inequality

I improve my previous question. Because this conjecture is exactly natural development of A Muirhead Like Inequality and Muirhead's Inequality so I think the conjecture is true. But I can not prove it....

**1**

vote

**1**answer

93 views

### Proof of an inequality $s_m(n) \le f_m(n)$

For fixed $m = 0, 1, 2, ...$
$$f_m(k) = \prod_{j=1}^{m}(k+j).$$
Some examples of $f_m(k)$ are as following:
$$f_0(k) = 1, \quad f_1(k) = (k+1), \quad f_2(k) = (k+1)(k+2).$$
The $s_m(n)$ is defined as ...

**3**

votes

**2**answers

196 views

### Inductive proof of $s(n)≤n+1$

I was able to conclude, numerically, the following:
$$s(n) = \sum_{j=0}^n\frac{(-4)^j}{(2j+1)!}\left(\sum_{k=j}^n\frac{(k+1)(2k+1)(k+j)!}{(k-j)!}\right)x^{2j+2}\le n+1$$
for $x\in[0,1]$. For example
...

**0**

votes

**2**answers

137 views

### An inequality on length of two curves [closed]

I am looking for a proof, reference, comment of an inequality as follows:
If $f(x)$ and $g(x)$ be two continuous derivative funcions in interval $[a, b]$. Such that:
$f(a)=g(a)$ and $f(b)=g(b)$
$(...

**4**

votes

**2**answers

239 views

### Combination power elementary symmetric polynomial inequality

Combine my first previous question and second previous question with the Muirhead inequality. I have posed conjectures of two inequalities as follows:
Inequality 1: Let $n>2$ and $1 \le m \le n$...

**2**

votes

**0**answers

155 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)...

**7**

votes

**1**answer

149 views

### Distributing $N$ points on the sphere so that the sum of their mutual distances is maximized?

Generaliation the result in our paper for sum and similarly my previous question for product. I have a question:
My question: Distributing $N$ points on the sphere so that the sum of their mutual ...

**5**

votes

**2**answers

122 views

### An inequality related to area and sidelengths of a polygon $Area(A_1A_2…A_n) \le \frac{1}{n}cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$

I am looking for a proof (or a reference) of an inequality related to a rea and the sidelengths of a polygon as follows:
Let $A_1A_2...A_n$ be arbitrary polygon, then:
$$Area(A_1A_2....A_n) \...