1
vote
0answers
104 views

Uniform bound for an alternating series of functions

I have mainly two questions, the first one being motivated by the second one. 1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
3
votes
1answer
193 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) + ...
3
votes
1answer
273 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 ...
4
votes
1answer
129 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} ...
2
votes
1answer
144 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 < ...
30
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
75 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 = ...
13
votes
4answers
876 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 ...
9
votes
2answers
482 views

The fraction of the sphere a fixed distance from a subspace

The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a ...
12
votes
5answers
784 views

Understanding Gibbs's inequality

Short version Gibbs's inequality is a simple inequality for real numbers, usually understood information-theoretically. In the jargon, it states that for two probability measures on a finite set, ...
0
votes
1answer
243 views

An interpolation inequality.

For all $s>0$ define for $\epsilon\in(0,1)$ the function: \begin{equation} g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k. \end{equation} Prove that $\exists C>0$ and $\phi(s)$ such ...
4
votes
0answers
403 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 ...
1
vote
1answer
228 views

Does this variable have an upper bound?

Let $x$ be a positive scalar variable whose time derivative satisfies $$|\dot{x}(t)|\leq \exp \left\{\left(-\int_{0}^{t}\frac{1}{x(\tau)} \mathrm{d} \tau \right)\right\},$$ where $|\cdot|$ denotes the ...
3
votes
1answer
700 views

Ratio sum comparison on operators

It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$, where $s_i(S)$ is the $i$-th singular value of $S$. How would one prove that ...
1
vote
1answer
710 views

On an eigenvalue inequality

Let $\lambda_1 (\cdot)$ be the larger absolute value eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$ the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e. $|\lambda_1 (\cdot)| ...
2
votes
2answers
834 views

Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
2
votes
1answer
231 views

Quotients of perfect powers separated by an integer

Let $a_n=\frac{(n+1)^{n+2}}{n^n}$ and $b_n=\frac{(n+2)^{(n+1)}}{(n+1)^{n-1}}$. Then it is easy to see that $a_n \leq b_n$ for all integers $n\geq 1$ (because the sequence $(1+\frac{1}{n})^n$ is ...
0
votes
0answers
455 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 ...
2
votes
2answers
586 views

Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ . I recently discovered this result. I am sure it is known, but it is new to me. It is ...
2
votes
1answer
506 views

A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$, $s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the singular values of a $2\times2$ matrix. Is it true that ...
7
votes
1answer
485 views

Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$: $\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...
5
votes
2answers
407 views

Sturm chain analogue for exponential polynomials?

I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form $f(x) = \sum_{i=1}^k c_ie^{\alpha_ix}$ ($\alpha_i$s real). My first question is: is there an algorithm for ...
5
votes
1answer
266 views

Denominators in the solution to Hilbert's XVII

Hilbert's seventeenth problem asks to prove that every positive semidefinite form can be written as the sum of squares of rational functions. Currently we don't seem to have a good understanding of ...