Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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44 views

Some estimates on tensor norms

Denote $M_n$ to be $n\times n$ matrix. For $X\in M_n$ define $\|X\|_1:=\max\limits_{1\leq j\leq n}\sum_{i=1}^n|x_{ij}|$ and $\|B\|_2:=\max\{|\sum_{i,j=1}^nb_{ij}x_iy_j|:|x_i|=|y_j|=1,\ 1\leq i,j\leq n\...
2
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0answers
51 views

Dense property of intersection of Sobolev space

I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim: Pick an arbitrary real number $s$, we have that the ...
2
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1answer
72 views

Estimate of Hölder Norms (Littlewood–Paley theory)

I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem: Recall that ...
3
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1answer
68 views

Inequality in a triangle associated with Golden ratio

Let $ABC$ be arbitrary triangle, $D$, $E$, $F$ are the midpoints of $BC$, $CA$, $AB$ respectively. Define points, segments in the figure belows. I am looking for a proof that: $$DE+EF+FD \le (DG+DH+...
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81 views

Lower bounds on norm of convolution operator

This seems like too easy of a problem to not have an answer, but I've been stymied so far: Given $\phi\ge 0$, $f\ge 0$, I am interested in lower bounds of the form $\Vert\phi*f\Vert \ge C\Vert f\Vert$,...
6
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1answer
107 views

Mittag-Leffler function

Let the Mittaq-Leffler function be defined by the expression $$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$ Now let $n\in \mathbb ...
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1answer
76 views

Is the Jaccard distance between probability vectors a metric?

Let X and Y be probability vectors, meaning that X = $[x_1, x_2, ..., x_n]^T$, where $x_i\leq 1$ and $\sum_{i=1}^{n}x_i=1$ (Y is defined similarly). Define the Jaccard distance as \begin{equation} ...
5
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1answer
111 views

Kullback–Leibler chains

The following question was asked and then deleted by the post author: Let $P$ and $Q$ be two probability distributions defined over the same space, with $KL(P \parallel Q) < \infty$. For $\epsilon ...
5
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1answer
158 views

Anti-concentration of Gaussian when conditioning on event

Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector ...
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1answer
59 views

Interpolated Sobolev norm inequality

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz set, and let $W^{k,p}(\Omega)$ denote the usual Sobolev space with $k \in \mathbb{N}$ being the order of the derivatives and $p \in [1, \infty)$...
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2answers
240 views

High degree differences in bipartite graphs

Consider a finite, simple and undirected graph $G=(V,E)$ with $V=\{v_1,\dots, v_n\}$. Let us define the quantity: $$\mathcal{I}_k(G) := \sum_{1\le i,j \le n} \mathbb{1}{\Big\{|\mathrm{deg}(v_i)-\...
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0answers
40 views

An inequality for a recursively defined sequence of numbers

Consider an arbitrary sequence $(x_n)_{n \in \mathbb{N}} \subseteq \mathbb{R}$ and $r \in \mathbb{R}$ with $r > 2$. Set $y_0 = 1$ and $z_0 = 0$ and for $n \in \mathbb{N}$ recursively define $$y_n = ...
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1answer
83 views

Finding the optimal size of $|E|$ for specific inequality

Suppose that we have the following expression $$p^{-2}|E|^6+16p^{-1}|E|^4+p^{2}|E|^2-8p^{-\frac{3}{2}}|E|^5-2|E|^4+8p^{\frac{1}{2}}|E|^3,$$ where $p$ is a prime number. I was wondering is it possible ...
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197 views

Is this function concave?

Let $$h(u):=u^3 \left|\int_u^\infty \frac{e^{-i t}}{t^3} \, dt\right|$$ for $u>0$. Is the function $h$ concave on $(0,\infty)$? (For context, see Proposition 4.4.4 and formula (4.4.21) in this ...
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32 views

Tail probability for the sum of laplace RIVs

Suppose we have $n$ RIVs $X_1,X_2,..,X_n$ where $\forall i. X_i \sim Lap(\frac{1}{\epsilon})$. For convenience, we denote $Y=\sum_{i=1}^{n}X_i$. Also, it is known that - $$ Pr[Lap(\frac{1}{\epsilon}) \...
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1answer
107 views

Solving the inequality between a and b [closed]

I run into this inequality $$ (a + b)^{1 - \epsilon} \;a < b $$ where $a \in \mathbb{Z}^+$ and $\epsilon \in (0, 1)$. What value (w.r.t $a$ and $\epsilon$) should I set $b$ equal to such that this ...
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1answer
32 views

A non-polynomial Young function satisfying a power-like condition

This post asked, essentially, for an example of a "non-polynomial" invertible increasing function $f\colon[0,\infty)\to[0,\infty)$ such that $f(0)=0$ and \begin{equation} f(cu)f(t)\le f(...
2
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1answer
109 views

Looking for non-polynomial functions: with the growth condition: $\phi\big(\theta \frac{s}{t}\big) \leq \frac{\phi(s)}{\phi(t)}$

I am for example(s) of an invertible Convex or concave function $\phi: [0,\infty)\to [0, \infty)$ such that $\phi(0)=0$ and there exists $\theta>0$ and for all $s\leq t$ we have \begin{align}\label{...
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1answer
213 views

Generalization of Bernstein’s inequality

I'm using Muscalu and Schlag's textbook to study harmonic analysis and I encountered the following claim: Given some function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ ...
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93 views

$L^p$ estimate of a multiplier operator

I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies: $$\sum_{n \in \...
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86 views

Weak-type inequality for the partial Fourier sum operator

I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark: For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
2
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1answer
88 views

Analogous form of Hardy-Littlewood maximal inequality (weak/strong type) on affine subspaces

I'm using some online notes (Professor Schlag, Yale University) to study harmonic analysis by myself. He introduced the following claim as an exercise: For any function $f \in L^{1}(\mathbb{R}^{d})$ ...
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1answer
47 views

Expectation of the sum of the squares of the cardinal of an inverse function

I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as: $$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$ where $\oplus$ is the bitwise XOR. ...
2
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1answer
194 views

Operator norm of triangular truncation on symmetric matrices

Inspired by this question. It is known that for the matrix $T_n \in \mathcal{M}_n$ (the space of real-valued $n \times n$ matrices) defined by \begin{equation*} (T_n)_{ij} = \begin{cases} 1 & i \...
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0answers
54 views

Formulas involving traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices

While working on the Atiyah problem on configurations of points, I came across formulas involving products of traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices. To ...
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49 views

Bounds on the lengths of circuits in a metric space

Given a collection $V$ of $N$ points in a metric space $(M, d)$, I define a circuit as a sequence $w = (w_1, w_2, ..., w_N)$ which visits each point in $V$ and has a length given by $$|w| = \sum_{i=1}^...
2
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1answer
63 views

Upper bound for $|y^TAyx^TAx - (x^TAy)^2|$ where A is PSD?

Let $x, y$ be vectors in $\mathbb{R}^n$ and $A \in \mathbb{R}^{n \times n}$ is a PSD matrix. I would like to bound $$|y^TAyx^TAx - (x^TAy)^2| \leq ?$$ for a fixed $A, x$ with a varying $y$. For ...
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2answers
264 views

show this inequality with $\frac{d^i}{dx^i}\left(1-\left(\frac{-x}{\ln(1-x)}\right)^{1/K}\right) \Bigg|_{x=0}>0, ~~~\forall i\in N^{+}$

I am trying to solve this Komal problem 661: Let $K$ be a fixed positive integer. Let $(a_{0},a_{1},\cdots )$ be the sequence of real numbers that satisfies $a_{0}=-1$ and $$\sum_{i_{0},i_{1},\cdots,...
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3answers
413 views

Pairs of vertices with high degree difference

Consider a finite, simple and undirected graph $G=(V,E)$ with $V=\{v_1,\dots, v_n\}$. Let us also fix an integer $k> n/2$. What are we able to say about the following quantity: $$\mathcal{I}_k(G) :=...
1
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1answer
102 views

Riesz rearrangement inequality

In the Lieb-Loss's book Analysis, they present the Riesz rearrangement in Section 3, Theorem 3.9 (page 93). Note that the functions $f, g, h,$ are all nonnegative. I want to ask whether the ...
3
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1answer
159 views

“Reversed” Bernstein Inequality

I'm studying harmonic analysis by myself, and I read some online notes that introduce the Bernstein inequality. One of them mention a reversed form of the Bernstein inequality, which is stated below: ...
3
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0answers
82 views

A connection between the Bell numbers and Bell polynomial

Let $B(n,x) = \sum_{k=0}^n {n\brace k}x^k$ be the Bell polynomials and $B_n = B(n,1)$ be the Bell numbers. I recently proved a nice relation between the two: $$ B(n,x)^{1/n}/x \ge B_{n/x}^{x/n}, $$ ...
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0answers
114 views

Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and $f(...
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1answer
97 views

How can I check this volume comparison?

I am reading the paper Ricci Curvature and Volume Convergence written by Professor Colding. In section 2, they define Lipschitz functions $b_j^+:M\to\mathbb R$ with $|\nabla b_j^+|=1$ and set $$\Phi=(...
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0answers
185 views

Conjecture about arcsin and $\sqrt{\quad}$

Let $r(a,b)$ be a rational number depending on $a,b$ and nonnegative. For every $b$ there is an $a$ such that $r(a,b)$ is not $0$. Let $C(a,b)$ be a squarefree positive integer depending on $b$ and ...
3
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0answers
31 views

Stability of matrix equation

Let $M=I+A\in \mathbb{R}^{n\times n}$ for a skew-symmetric matrix $A$ with $\|A\|<1$ in the spectral norm. Using the $LU$-decomposition of $M$, it is easy to construct a solution $L,U\in \mathbb{R}^...
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0answers
148 views

Does there exist a type of discriminant not only for irreducible polynomials but also for exponential functions, logarithm functions?

I think discriminant is the strongest tool that I've used_ https://math.stackexchange.com/q/4035405/822157, however, does there exist a type of discriminant not only for irreducible polynomials but ...
4
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1answer
284 views

Inequality of two variables

Let $a$ and $b$ be positive numbers. Prove that: $$\ln\frac{(a+1)^2}{4a}\ln\frac{(b+1)^2}{4b}\geq\ln^2\frac{(a+1)(b+1)}{2(a+b)}.$$ Since the inequality is not changed after replacing $a$ on $\frac{1}{...
5
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0answers
115 views

Inequality for functions on $[0,\infty)$

Let $0<q<1$ and $\varphi(q;x)=\displaystyle \prod_{j=0}^\infty (1+q^jx),\;x\geqslant 0.$ Consider the following functions: $$l_k(x;q):=\frac{q^{k(k-1)/2} x^k}{(1-q)(1-q^2)\dots (1-q^k)\varphi(x;...
5
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1answer
262 views

Function of two sets

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
0
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1answer
44 views

Reverse Hölder type inequality for the Laplacian raised to a power

I am studying integrals of the form $\int (\Delta \rho)^{\alpha} f^{\beta}$ where $0<\alpha < 1, \beta \geq 0$ and $\rho, f \in C_c^{\infty}(\mathbb{R}^n).$ My goal is to obtain lower bound on ...
3
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6answers
392 views

how to prove this high degree inequality

Let $x$,$y$,$z$ be positive real numbers which satisfy $xyz=1$. Prove that: $(x^{10}+y^{10}+z^{10})^2 \geq 3(x^{13}+y^{13}+z^{13})$. And there is a similar question: Let $x$,$y$,$z$ be positive real ...
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0answers
109 views

An inequality involving a polynomial and its first and second derivative

Given a real polynomial $P(x)$ all whose roots are real, it is not hard to show that $$P(x)P''(x) \leq P'(x)^2 \, \, \, \, (1).$$ Proof sketch: Assume that $P(x) = \prod_{i=1}^n (x-r_i)$. Look at $\...
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0answers
57 views

Interpolation inequality involving negative Sobolev space

$\newcommand\norm[1]{\left\|#1\right\|}\newcommand\inner[2]{\langle #1,#2\rangle}$ Let $u\in \dot{H}^1(\mathbb{R}^n)$ for $n\geq 3$ where $\dot{H}^{1}$ denotes the homogeneous Sobolev space that is ...
4
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1answer
178 views

Smallest regular $m$-gon covering a regular $n$-gon

I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question. Let us fix a regular $n$-gon with area $1$. What is the smallest ...
1
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0answers
57 views

Monotone rearrangements of function, constrained optimization in $\mathcal{L}^p$

By $\mathcal{S}$ let us denote the set of such step functions $f:[0,1]\to [0,1]$ that additionally satisfy: $$\forall_{ x>\frac{1}{2}} \ \ \lambda\Big(f=x\Big) \ = \ x\cdot \Big[\lambda\Big(f=x\Big)...
3
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2answers
437 views

The direction that gets me closest to a given point in $\mathbb{R}^n$

Let $p \in \mathbb{R}^{n}$ and $p=\lambda_1 e_1+...+\lambda_n e_n$ where $e_i$ are standard basis vectors then if I want to find the component along which I can get closest to the point $p$ then it ...
3
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1answer
203 views

Showing the positivity of the determinant of $\mathfrak{sp}(n)$ without making use of diagonalization

Let $\mathfrak{sp}(n)$ be the lie algebra of compact symplectic group $\mathrm{SP}(n)$, regarded as a compact form of $\mathfrak{sp}(2n,\mathbb{C})$, so we can talk about its (complex) determinant. ...
3
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2answers
203 views

On some inequality (upper bound) on a function of two variables

There is a problem (of physical origin) which needs an analytical solution or a hint. Let us consider the following real-valued function of two variables $y (t,a) = 4 \left(1 + \frac{t}{x(t,a)}\right)...
2
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1answer
159 views

How did the story of Kim-Vu type inequalities continue?

I am interested in the concentration of polynomials of random variables. I have been reading Boucheron, Lugosi, and Massart's "Concentration inequalities" and they give some references. ...

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