Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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22 views

Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables

Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
0 votes
1 answer
108 views

Approximation for an expectation expression

Let $\mathbf{x} \in \mathbb{C}^M$ is an unknown distributed random vector (certainly not gaussian), and matrix $\mathbf{A}\in \mathbb{C}^{M \times M}$ which is fix (known). Also, assume we know the ...
0 votes
1 answer
37 views

Norm of a $2$-tuple of operators

Let $E$ be a complex Hilbert space and $K_1,K_2$ are bounded linear operators on $E$. Let $\omega(K_1)$ and $\omega(K_2)$ be the numerical radius of $K_1$ and $K_2$ respectively. That is \begin{align*}...
2 votes
1 answer
291 views

Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?

For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it. Problem definition: Let $f(\xi) \in \...
3 votes
1 answer
234 views

Does $f(t) \le \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$ imply $f=0$?

Let $\beta \in (0, 1)$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that $$ f(t) \le \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s, \quad \...
2 votes
1 answer
109 views

Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$

Let $\alpha \in (0, \infty)$ and $\beta \in (0, 1]$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that $$ f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(...
12 votes
2 answers
638 views

Polynomial inequalities of the form $\int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \,dx$

Let $P_n$ denote all (real or complex) polynomials $f(x)=\sum_{k=0}^n a_k x^k$. I'm interested in inequalities of the form $$ \int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \, dx, \quad \text{for ...
2 votes
1 answer
528 views

Another functional inequality

Is there some general solution to the functional inequality: $$ f(xy) \leq y f(x) + x f(y)$$ Where $x,y\in[0,1]$? I can find many particular solutions but I just wonder if there is a more general ...
3 votes
0 answers
314 views

Some specific case of the abc-conjecture related to the Mason-Stothers proof of Serge Lang for polynomials?

I stumbled upon a version of the Mason-Stothers theorem, which can be modified to make a similar statement for natural numbers, which I think I can prove here. It is not the abc-conjecture, but it has ...
5 votes
1 answer
274 views

Does the Poincaré inequality hold on annular domains?

Does the following Poincaré inequality hold $$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$ where $B_r$ denotes a ball of radius ...
-1 votes
2 answers
199 views

Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$

Suppose $f \colon [0, 1] \to \mathbb{R}$ is continuously differentiable, and satisfies $f(0) = f(1) = 0$ and $\|f'\|_2 \leq 1$. I am wondering if it there is a constant $C > 0$ such that for all ...
-2 votes
0 answers
67 views

On the proof of Gagliardo–Nirenberg inequality

Is there anyone checked on the paper Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks by A. Fiorenza, etc. carefully? In part 3.2 Proof of Theorem 1.2 ...
2 votes
1 answer
167 views

Tighter lower bound of the lower triangular sum of an arbitrary Latin square

In this math.stackexchange.com question I seek a tighter bound than the one I presented in there in the question. Rob Pratt puts forth a conjecture in his answer motivated by the dual problem of the ...
5 votes
0 answers
164 views

Monotonicity of ratio of symmetric polynomials

The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by \begin{equation*} h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
0 votes
0 answers
54 views

Reference request for equivalent Lipschitz smoothness conditions

For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
2 votes
1 answer
256 views

A result about sub-exponential random variables

I am reading the proof of Theorem 1(a) in the paper that proposed the CLIME method for estimating precision matrix. I am puzzled by an inequality on Page 605 three lines above formula (29). I isolate ...
1 vote
1 answer
153 views

Inequalities involving entropy: quantum discord and mutual information

My question is inspired by the following paper of Olivier and Żurek but for this question to be self-contained I will recall all the necessary definitions: for a quantum state $\rho$ we define the ...
1 vote
1 answer
97 views

Size of sets associated to Gaussian integers

Given a non-zero Gaussian integer $z$, we define the set $\mathcal S(z)$ containing all solutions of $ab+cd=z$ satisfying $\min(\vert a\vert,\vert b\vert)>\max(\vert c\vert,\vert d\vert)$ with $a,b,...
2 votes
2 answers
192 views

$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?

Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has $$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
0 votes
0 answers
62 views

Maximizing the trace of the resolvent of a Wishart matrix over positive unit trace matrices?

Let $G$ be a standard $d \times d$ Wishart random matrix and consider the problem of maximizing the function $$ f(M) = \mathbb{E}\Big[\mathrm{tr}((G + M^{-1})^{-1})\Big], $$ over the class of real ...
0 votes
1 answer
55 views

Does point process ordering ever imply conditional intensity ordering?

Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
8 votes
0 answers
327 views

Bounding a sum of reciprocals of square-free integers

(Cross-posted from MSE, as the question did not get any clear answer) Fix positive integers $k$ and $n$. Let $N_1,\dots,N_r$ be all the integers less than or equal to $n$ that are squarefree and have ...
2 votes
1 answer
211 views

How to estimate an integral by the variation and upper bound of the integrand?

Suppose that $f$ is a continuous function on $\mathbb{R}$. I want to estimate the definite integral $$ I:= \int_{0}^a [f(x)-f(0)]dx $$ by the upper bound $M = \sup_{x\in[0,a]}|f(x)|$ and the variation ...
-1 votes
0 answers
47 views

Does a constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $\sup_{z\in rD^2}|p(z_1,z_2)|\le C\sup_{z\in D^2}|p(z_1,z_2)|$?

Question: Does a finite constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $$\sup_{z\in r \mathbb D^2}|p(z_1,z_2)|\le C\sup_{z\in \mathbb D^2}|p(z_1,z_2)|$$ where $r>...
2 votes
0 answers
133 views

Taylor coefficients of the integral of the ordered exponential

Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of $$ X_A'(t) = A(t) X_A(t), \qquad X(0) = I. $$ In other words $X_A$ is the ordered exponential of $...
2 votes
1 answer
136 views

Upper bound of a product of sines

Consider the function $$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$ I wonder whether it is possible to compute some nontrivial upper ...
0 votes
3 answers
266 views

A generalisation of Tchebychev inequality

Let $f,g \in C(\mathbb R)$ with $\exists M \in \mathbb R^*, \forall (x,y) \in \mathbb R^2, M\times (f(x)-f(y))(g(x)-g(y)) \geq 0$. Is it true that exists $ u$ any real function, and $a,b$ monotone ...
0 votes
0 answers
73 views

Some new questions on Rademacher complexity

For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable. Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...
3 votes
0 answers
78 views

Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set

Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE \begin{equation} dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,. \end{equation} Let $f(x,t|x_0,0)$ denote its transition density function. ...
1 vote
1 answer
143 views

Interpolation of scalars

For $a,b$ and $\alpha_i, \beta_i $ where $ i \in \{ 1,2 \} $, are non-negative real numbers, is it possible to find a constant $C$ such that $$(\alpha_1 a + \beta_1 b) ^{(1-\theta)} (\alpha_2 a + \...
-1 votes
0 answers
89 views

Does domination of stochastic processes imply the domination can always be realized by the coupling temporally/incrementally?

Suppose we have two stochastic processes $X=(X_t)_{t\in[0,\infty)}$ and $Y=(Y_t)_{t\in[0,\infty)}$. Assume that we have all the necessary structure to make sense of stochastic domination in the ...
2 votes
2 answers
179 views

$L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has $$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
0 votes
0 answers
87 views

On polynomial equation of fourth order depending on two parameters and bound on a maximal root

I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$: \begin{eqnarray} F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p^...
2 votes
1 answer
165 views

Estimating ${\left(\sum_{i=j}^k {x_i}\right)^2} \times \left\lvert\sum_{i=j}^k {a_i}\right\rvert$

Given two sets; $X = \{x_i : x_i \geq 0; i \in [\sqrt{n}]\}$ and $A = \{a_i : |a_i| \leq 1; i \in [\sqrt{n}]\}$ of size $n^{\frac{1}{2}}$ each, with the following properties \begin{equation}\label{...
1 vote
0 answers
69 views

Integral inequality related to the (mixed?) moments of two functions

For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set $$ S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}. $$ Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce $$ ...
5 votes
1 answer
139 views

Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$

Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let $$ h = \frac{f}{f+g}. $$ I want to prove that the $n$-th derivative of $h$ satisfies: There exists $C > 0$ such that $$ |h^{(...
0 votes
1 answer
98 views

Demonstrating bound on integral

I found this question on another forum and it got me interested because of how tight the bound is: prove that $$\int_0^\infty \frac{\arctan(x)}{x^2 + 4}\,dx > \frac{\pi}{4}.$$ The difference ...
1 vote
1 answer
604 views

Hanson-Wright inequality (quadratic form concentration inequality) for bounded random vectors

Is there a concentration inequality for quadratic forms of bounded random vectors $X \in [-1, 1]^n$ with zero mean and given covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ but otherwise ...
4 votes
1 answer
706 views

Bound the operator norm of the Fréchet derivative of a Lipschitz function in this setting

I want to find a bound for the operator norm of the Fréchet derivative of a Lipschitz continuous function in the following setting: Let $E$ be a $\mathbb R$-Banach space; $v:E\to[1,\infty)$ be ...
0 votes
0 answers
58 views

Upper bound for an additional Product formula

We have three sequences of positive integers $l$, $p$ and $q$ such that: $$ p_1 \geq p_2 \geq \cdots \geq p_k\text{ and } q_1 \geq q_2 \geq \cdots \geq q_k \geq \cdots \geq q_h \text{ where: } k < ...
0 votes
1 answer
134 views

An inequality of Huygens

I am preparing a paper on Huygens' approximations to $\pi$ and I have to prove the following inequality: if $0 \le x\le \pi/2$ then $$ \pi\ge \frac{\pi}{x}\sin(x)\bigl(20+51\cos(x)-2\cos^2(x)+6\cos^3(...
0 votes
0 answers
208 views

Gauss transformation in fractional Sobolev space

Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that $$ \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
3 votes
1 answer
126 views

Bounding distance to an intersection of polyhedra

Let $P$ and $Q$ be polyhedra in ${\mathbb R}^m$ with a non-empty intersection. I believe there should exist a constant $C_{PQ}>0$ such that for any point $x\in {\mathbb R}^m$ the following ...
1 vote
1 answer
88 views

Bound on $L^1$ norm of solution of two-point boundary value problem

This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
2 votes
0 answers
96 views

Poincare inequality on the hemisphere

Background: Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
3 votes
2 answers
164 views

Bounding distance to a polyhedron

I need to estimate the Euclidean distance from a point $x\in {\mathbb R}^m$ to a polyhedron $P\subset {\mathbb R}^m$ in terms of distances from $x$ to the tangent hyperplanes which define $P$. By a ...
9 votes
0 answers
278 views

Inequality for symmetric polynomial functions of log concave variables

Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$). ...
1 vote
1 answer
294 views

Finding a connection between two types of convergence

Please, help me find connections between two types of convergence: Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences: 1) $X_n \...
0 votes
0 answers
17 views

Solve $S_{nm} \sum_{j}x_{nj}\operatorname*{sign}\left( \sum_{k}x_{kj} S_{km} \right) \geq 0$ for $x_{kj}$

Let $S_{nm} \in \{\pm1\}^{N\times M}$ be a matrix of signs. Can we find real numbers $x_{kj} \in \mathbb{R}^{N\times K}$, such that: $$S_{nm} \sum_{j}x_{nj}\operatorname*{sign}\left( \sum_{k}x_{kj} S_{...
28 votes
3 answers
1k views

Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$, define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...

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