Questions tagged [inequalities]
for questions involving inequalities, upper and lower bounds.
1,736
questions
0
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0
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13
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Monotonicity of averages for positive-definite kernels
Let $\kappa:\mathbb{R}^d\times \mathbb{R}^d \to\mathbb{R}$ be a positive semidefinite kernel. If $A,B \subseteq \mathbb{R}^d$ are disjoint sets of equal, finite measure, then applying the definition ...
- 775
2
votes
0
answers
52
views
upper and lower bounds on rowlands sequence
rowlands sequence is defined as follows
\begin{equation}
a_{n}=a_{n-1} + b_{n}
\end{equation}
where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$
it originates from E. Rowlands 2008 paper "A Natural ...
6
votes
1
answer
118
views
More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let
$$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
- 123k
4
votes
1
answer
67
views
On the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$.
Can the mean of the squares of the off-diagonal entries of $G$ be $<1/5$?
Remark 1: A numerical experiment suggests that $...
- 123k
11
votes
2
answers
309
views
Maximization of a cubic form over the $14$-dimensional sphere
For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a nonnegative real number.
Is it true that, given the condition
$$\sum_{1\le i<j\le6}x_{ij}^2=1,$$
the sum
$$\sum_{1\le i<...
- 123k
6
votes
2
answers
398
views
Does this polynomial have a real zero less than or equal to $1/2$?
Is the smallest root $x$ of
$$
10x^{3}-30x^{2}+\left(30-2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}\right)x\\
+2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}-\sum_{1\le i<j<k\le6}\cos\alpha_{ij}\cos\...
- 61
9
votes
3
answers
2k
views
Smallest root of a degree 3 polynomial
Is it true that the smallest root $t$ of the polynomial
$$
20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
- 161
0
votes
0
answers
58
views
Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
4
votes
1
answer
241
views
Expectation comparison inequality for concave function of symmetric random variables
Suppose that $X_i$, $i\in[n]$ are
independent symmetric
random variables. I think the conjectured result holds in greater generality, but we can additionally assume that each $X_i$ takes the values $\...
- 6,415
0
votes
0
answers
49
views
Inequality involving random vectors and absolute values
Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
3
votes
1
answer
345
views
Dimensionality reduction for total variation
Let $P_i,Q_i$, $i\in[n]$,
be distributions on a finite set $\Omega$.
We will use $P^\otimes_{i\in[n]}$ to denote $n$-fold products of distributions.
For each $i\in[n]$, define the
dimensionally-...
- 6,415
3
votes
1
answer
200
views
A non-standard inequality for univalent functions
Related to my other question, here is an inequality from Rakhmanov's paper upon which the proof hinges.
Let $F(z) = z + a_0 + \mathcal O(z^{-1})$ be analytic and univalent on $|z|>1$, continuous up ...
- 319
2
votes
0
answers
47
views
Distance between a Hölder function and a Sobolev ball
Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively.
My question ...
- 422
2
votes
2
answers
114
views
Optimizing a matrix quadratic form with respect to Loewner order
Fix integers $1 \leq k \leq n$. Let $P \in \mathbb{R}^{n \times k}$ be such that $P^T P$ has full rank.
Let $\mathcal{X}$ denote the set of unit trace, real $n \times n$ symmetric positive ...
- 422
6
votes
1
answer
302
views
Question on a min inequality
Is it true that
$$
\min\left(a^2 + b^2 - \sqrt{a^4 + b^4 + 2a^2b^2\cos(x)}, b^2 + c^2 - \sqrt{b^4 + c^4 + 2b^2c^2\cos(x-y)}, a^2 + c^2 - \sqrt{a^4 + c^4 + 2a^2c^2\cos(y)}\right) \leq \frac{1}{3}
$$
...
- 161
2
votes
0
answers
92
views
System of quadratic inequations
I have variables $x_1,...,x_n>0$, and fixed parameters $a_1,a_2,...a_n >0$.
I compute quantities $s_1,s_2,...,s_n$, each with one of the following types of equation:
$s_i =a_i(x_i - x_j x_k)$
$...
- 1,321
15
votes
1
answer
591
views
On minimal eigenvalue
Is it true that $\min\left(\lambda_{\min}(M_{12}),\lambda_{\min}(M_{13}),\lambda_{\min}(M_{23})\right) \le \frac{7}{20}$ where $M_{ij}$ is the matrix obtained by selecting the entries at the ...
- 180
0
votes
1
answer
138
views
Inequality $(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$
Conjecture: Let $a_1, a_2, \cdots , a_n>0$ and $y \ge x $ then
$$(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$$
Equality iff $x=y$
Is the conjecture right? Have you ever seen this ...
- 1,734
2
votes
0
answers
93
views
A question from a proof of an inequality in Sobolev space $W^{1,1}$
I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot:
Here is what I did:
$$-u(x)=u(y)-u(x)=\...
- 1,759
3
votes
1
answer
317
views
Derivative norm estimates
Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...
- 42.3k
4
votes
0
answers
154
views
Szegő's inequality
I know Erdős-Lax's inequality and a couple of proofs. It states that:
If $P(z)=\sum_{v=0}^{n} a_{v} z^{v}$ is a complex polynomial of degree $n$ having no zeros in $|z|<1$, then
$$
\max _{|z|=1}\...
- 2,810
2
votes
1
answer
130
views
Concentration inequality for double sum
I am looking for a concentration inequality of a double sum….
Let $X_1,\dots, X_n$ be iid r.v. and also let $Y_1,\dots ,Y_n$ be iid such that even $X_i$ and $Y_j$ are independent.
I am looking for a ...
- 41
13
votes
2
answers
1k
views
Probability vector $p$ majorizes its normalized entropy vector $\small \frac{-p\log p}{H(p)}$
I guess the following inequality
$$ \sum_{i=1}^n g \left (\frac{-p_i \log p_i}{H(\boldsymbol{p})} \right ) \le \sum_{i=1}^n g (p_i)$$
holds for any continuous convex function $g$ and any probability ...
- 283
2
votes
1
answer
150
views
Maximizing a quadratic form involving a trace-bounded positive definite matrix?
$\newcommand{\tr}{\mathrm{tr}}$Suppose $P, Q$ are two real, symmetric positive definite matrices and $v$ a nonzero unit vector.
Consider
$$
f(X) = v^T(P + X^{-1})^{-1} v + v^T(Q + X^{-1})^{-1} v.
$$
...
- 422
1
vote
1
answer
143
views
How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?
Scenario
I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$.
Problem
I know that ${x}$ ...
2
votes
0
answers
90
views
A surprisingly simple and difficult problem on sums of upper bounds
Let $T$ be a large integer, and $C$ be a positive real constant.
Consider a sequence $\{p_t\}_{T\geq t\geq 1}$ of real numbers in $[0,1]$. The sequence $\{b_t\}_{T\geq t\geq 1}$ can be defined as ...
- 353
3
votes
0
answers
113
views
A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 560
17
votes
2
answers
2k
views
Boundedness of sum of sin(sin(n))
Playing with desmos I have accidentally noticed that the sequence of partial sums
$$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$
is bounded.
However, I did not succeed in proving this ...
1
vote
0
answers
42
views
Inequality Involving Concave Monotonic Function
Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
0
votes
1
answer
99
views
Techniques for bounding the operator norm of the expectation of random matrix?
Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix
$$
A = I_n - uu^T.
$$
Question: What techniques are available to provide (reasonably ...
- 422
1
vote
1
answer
139
views
An inequality about binomial distribution
Statement
Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that
$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$
...
- 35
5
votes
0
answers
152
views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,086
0
votes
0
answers
20
views
Explicit upper and lower bounds for a support function, with a different exponent
Let $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ be a sequence of nonincreasing nonnegative real numbers. Define the set, for $t > 1$,
$$
B_t = \Big\{b \in \mathbb{R}^n : b_i \geq 0, \sum_i b_i^2 \...
- 422
2
votes
1
answer
99
views
Inequality for Gaussian measures
Let $\mu$ denote a centered Gaussian measure on $\mathbb{R}^k$, $K=(-\infty, a] \times \mathbb{R}^{k-1}$ ($a\ge 0$) and $L=\mathbb{R}\times C$ where $C$ is a convex set in $\mathbb{R}^{k-1}$, ...
- 197
0
votes
0
answers
33
views
Support function of the intersection of two $\ell_p$ balls
Denote $\|\cdot \|_p$ for the norm in $\ell_p^n$, where $1 \leq p \leq \infty$, and $n \geq 1$.
Let $(x^\star_i)$ denote a nonincreasing arrangement of the sequence $(|x_i|) \in \mathbb{R}^n$.
We ...
- 422
9
votes
1
answer
825
views
Huygens' final unproved inequality
The analytic statement of Proposition XX in Huygens' "Inventa" is: If $x> 0$, and less than $\frac{\pi}{2}$, then
$$x>\sin x +\frac{10(4\sin^2\frac{x}{2}-\sin^2 x)}{12\sin\frac{x}{2}+9\...
- 371
6
votes
2
answers
196
views
On a trigonometric inequality by Huygens
The following inequality, ascribed to Huygens, appeared in this post:
\begin{equation*}
1-\frac43\,\frac{\sin^3\theta/2}{\theta-\sin\theta}
>(1-\cos\theta/2)\Big(\frac35-\frac3{1400}\frac{\...
- 123k
0
votes
0
answers
44
views
Upper bound on expectation of a convolution
Given probability densities $f, g\in L^p(\mathbb{R}^3), \ \forall p\geq 1$, with the same first and second moments
\begin{align} & \int_{\mathbb{R}^3} v f(v)\,dv = \int_{\mathbb{R}^3} v g(v)\,dv, \...
- 43
1
vote
3
answers
525
views
Huygens' trigonometric inequality
Prove that $$1-(4/3(\sin^3 \theta/2))/(\theta-\sin\theta)<(1-\cos\theta/2)(3/5-(3/1400)\pi^2/n^2)$$ holds for $0\le\theta\le\pi/2.$ Here $n$ is an integer greater than or equal to two.
This an ...
- 371
0
votes
0
answers
45
views
A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
- 6,182
3
votes
1
answer
169
views
A generalization of Barrow's inequality
More than seven years ago. I posted this problem in stackexchange:
Let $ABC$ be a triangle, $P$ be arbitrary point inside of $ABC$. Let $A_1B_1C_1$ be the tangential traingle of $ABC$. Let $A'$, $B'$,...
- 1,734
3
votes
1
answer
217
views
Bounds on relative entropy for MLE in Bernoulli coin tosses
In the context of estimating the parameter $p$ from a dataset of $n$ i.i.d Bernoulli coin tosses, we often use the relative entropy $D(p \parallel \hat{p})$ to measure the performance of an estimator $...
- 63
1
vote
0
answers
41
views
Moments on the Stiefel manifold
Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$.
Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
- 422
0
votes
0
answers
36
views
The reciprocal of the normalized tail of the Maclaurin power series expansion of the hyperbolic sinc function is a convex function
The classical Bernoulli numbers $B_j$ are generated by
\begin{equation}\label{Bernoulli-No-Generating}
\frac{x}{\operatorname{e}^x-1}=\sum_{j=0}^\infty B_j\frac{x^j}{j!}=1-\frac{x}2+\sum_{j=1}^\infty ...
- 985
3
votes
1
answer
64
views
Multiplicative approximation for a negative moment of the binomial distribution
Let $X$ be a binomial random variable with parameters $n,p$.
Define the function
$f(n, p, t) = E\frac{1}{1 + t X},
$
where $t > 0$.
Question: Can we find an elementary function $F(n, p, t)$ such ...
- 422
5
votes
1
answer
321
views
Lower bound for the rank of the sum of $n$ matrices
I found a mathematical note by George Marsaglia entitiled "Bounds for the rank of the sum of two matrices", where he proves the following result.
Let $A_1$ and $A_2$ be two complex matrices ...
- 5,118
1
vote
0
answers
69
views
inequality involving factorials
Let $p>1$ be a positive integer and let $a_0,a_1$ be non-negative integers with $0\le a_0\le p-1$ and $1\le p\le a_1$. Show that
$$\prod_{b=1}^{a_1}\left(\frac{(a_0+bp)!}{a_0!b!}\right)^{{a_1\...
- 914
6
votes
1
answer
280
views
Determinantal inequality for difference of substochastic matrices
Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if
$$
\forall i,\ \ \sum_j A_{ij}\le 1\ .
$$
In the course of my ...
- 22.5k
1
vote
2
answers
228
views
A real root of a cubic equation for a stationary point
Let us consider the quartic polynomial in $x$
\begin{equation}
F(x) = (2 a p +2)x^4+ (6a(1-a)p^2+(6-12a)p-6)x^3
+ p(2(a-2)(a-1)a p^2 + 3(5a^2-9a+2)p +12a-18)x^2
- p^2 ((a-2)(4a^2 ...
- 371
2
votes
1
answer
102
views
Sharp approximation to expectation of a ratio of a Gaussian vector
Let $g =(g_1, ..., g_n)$ denote a sequence of standard Gaussian variables. Let $p = (p_1, ..., p_n)$ denote a vector in the simplex $\mathcal{P}_n$, given by
$$
\mathcal{P}_n = \{p \in \mathbb{R}^n : ...
- 422