# Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

1,736
questions

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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 ...

2
votes

0
answers

52
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### 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
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### 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
$$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...

4
votes

1
answer

67
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### 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 $...

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<...

6
votes

2
answers

398
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### 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\...

9
votes

3
answers

2k
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### 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 \...

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} \\\
& ...

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 $\...

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-...

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 ...

2
votes

0
answers

47
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### 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 ...

2
votes

2
answers

114
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### 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 ...

6
votes

1
answer

302
views

### Question on a min inequality

Is it true that
$$
\min\left(a^2 + b^2 - \sqrt{a^4 + b^4 + 2a^2b^2\cos(x)}, b^2 + c^2 - \sqrt{b^4 + c^4 + 2b^2c^2\cos(x-y)}, a^2 + c^2 - \sqrt{a^4 + c^4 + 2a^2c^2\cos(y)}\right) \leq \frac{1}{3}
$$
...

2
votes

0
answers

92
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### 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:
$s_i =a_i(x_i - x_j x_k)$
$...

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 ...

0
votes

1
answer

138
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### 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 ...

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)=\...

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? ...

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
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 ...

13
votes

2
answers

1k
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### 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 ...

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
$$
f(X) = v^T(P + X^{-1})^{-1} v + v^T(Q + X^{-1})^{-1} v.
$$
...

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 ...

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 - ...

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 ...

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
$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$
...

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 ...

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 \...

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 $\mathbb{R}^{k-1}$, ...

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 ...

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\...

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{\...

0
votes

0
answers

44
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### 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, \...

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 ...

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}=\...

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 $A'$, $B'$,...

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 $...

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, ...

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 ...

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 ...

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 ...

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\...

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 ...

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 ...

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 : ...