Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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An inequality for polynomials

I have been thinking about the validity of the following inequality: if $P(z)=\sum_{k=0}^na_kz^k, a_n\neq 0$ and $P(z)$ is non-zero in $|z|<1, $ then for $\theta \in [0, 2\pi],$ and $p>0$ \...
user159888's user avatar
3 votes
0 answers
98 views

Convex minorants to convex functions, given partial Taylor expansion and smoothness estimate

Let $V$ denote a strictly convex function (in arbitrary dimension) whose Hessian is $L$-Lipschitz. Given only this knowledge, and the values of $\left\{ V \left( x \right), \nabla V \left( x \right), \...
πr8's user avatar
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0 answers
97 views

Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity

Problem: Consider the autonomous ODE system \begin{align*} \dot{x} &= (1-x) (z-xy)\\ \dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\ \dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z \end{...
Pavel Kocourek's user avatar
3 votes
2 answers
232 views

Extend an inequality on matrix norms

Let $A$ denote an $n \times n$ matrix, and $\sigma_i(\cdot)$ denote $i$-th largest singular value. Can we extend the following result to general $p \geq 1$? For all $k = 1, \dots, n$, $$ \sum_{i = 1}^...
Xiangxiang Xu's user avatar
1 vote
1 answer
220 views

What does Landau symbol mean in an inequality?

I'm reading about subdifferentiable function at page 232 of Villani's Optimal Transport: Old and New. Definition 10.5 (Subdifferentiability, superdifferentiability). Let $U$ be an open set of $\...
Akira's user avatar
  • 815
4 votes
0 answers
123 views

A "counterbalancing" trigonometric sum inequality

Is it true that $$s_{n,k}:=\sum_{j=1}^{n-1} r_{n,k,j} <0$$ for all natural $n\ge2$ and all natural $k\in\{1,\dots,n-1\}$, where $$\text{$r_{n,k,j}:=\frac{x_{n,2j}}{y_{n,k,j}\;y_{n,k+1,j}},\quad$ $...
Iosif Pinelis's user avatar
2 votes
1 answer
740 views

Can we get that $ P(N^{2/3}(\lambda_N-\lambda_{N-1})\le c)\ge 1-\epsilon$?

Following this question: Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy-Widom law?. We know that $$ \lim_{N\to\infty}P(N^{2/3}(\lambda_N-2)\le s_1,\dotsc,...
Hermi's user avatar
  • 274
1 vote
2 answers
179 views

What is the easiest way to prove the correctness of this inequality

I have the following inequality for some $0<x<0.1$: $$x^{1/10}-(1-(1-x)^{x^{-0.5}}) \geq 0$$ Is there an easy way to prove the correctness of such inequality? Thanks!
user496082's user avatar
0 votes
1 answer
84 views

Can we find the following $k$ so that the following inequality holds for asymptotic normal?

Following this question:Can we find such $k$ so that the following inequality holds?. Consider a sequence of independent $n-$dimensional random vectors $u, v_1, v_2,\dots, v_k$ uniformly distributed ...
Hermi's user avatar
  • 274
3 votes
0 answers
40 views

Bound of a regular function that cancels at some points

Let $K$ be a bounded convex set of $\mathbb{R}^n$ and $x_1,\ldots,x_k\in K$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function that cancels on points $x_1,\ldots,x_k$ . When $n=1$, ...
Pii_jhi's user avatar
  • 111
2 votes
3 answers
213 views

Can we find such $k$ so that the following inequality holds?

I found this question: Chernoff style concentration bound for ratio of variables. I want to ask if we get similar thing for the ratio of the sum and the one Gaussian variable. Given i.i.d. Gaussian ...
Hermi's user avatar
  • 274
2 votes
1 answer
132 views

Normalized concentration inequality for empirical CDF (iid sum)

Consider the empirical and population CDF, $$ F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad F(t) = \mathbb{E} [F_n(t)], $$ where above $X_1, \dots, X_n$ are iid, real-...
Drew Brady's user avatar
0 votes
0 answers
33 views

Inequalities for generalized variance

Let $(X, \mu)$ be a measured space with $\mu(X) = 1$. Given $\phi \in L^\infty(X, \mu)$, $\phi > 0$, let me define, for $\alpha \geq 1$, $\beta > 0$, the quantity $$ I(\alpha, \beta) = \left(\...
Romain Gicquaud's user avatar
4 votes
1 answer
146 views

An algebraic inequality in three real variables

Is it true that $$(v-u)^2+(w-u)^2+(w-v)^2 \\ +\left(\sqrt{\frac{1+u^2}{1+v^2}} +\sqrt{\frac{1+v^2}{1+u^2}}\right) (w-u)(w-v) \\ -\left(\sqrt{\frac{1+u^2}{1+w^2}}+\sqrt{\frac{1+w^2}{1+u^2}}\right) (w-...
Iosif Pinelis's user avatar
9 votes
2 answers
573 views

Integral inequality: Prove $\int_0^1 f\int_0^1 1/f \leq 1$ for a certain function $f$

Let $g$ be a piecewise smooth, zero average, function over $[0,1]$ such that $\min g^2>0$. I would like to show that $$ \int_0^1 g\sqrt{1-r/g^2}\int_0^1 \frac{1}{g\sqrt{1-r/g^2}} \leq 1 $$ for all $...
Hussein's user avatar
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10 votes
1 answer
457 views

For what kind of $C^*$ algebras does the inequality $\frac{(ab+ba)}{2}\leq\frac{ a^p}{p} +\frac{b^q }{q}$ hold for $a,b>0$?

Assume that $\frac{1}{p}+\frac{1}{q}=1$ for two positive real numbers $p,q$. For what kind of $C^*$ algebras $A$ does the following hold: $$\frac{ab+ba}{2}\leq \frac{a^p}{p}+\frac{b^q}{q}$$ $\...
Ali Taghavi's user avatar
0 votes
0 answers
238 views

Can we prove the following anti-concentration inequality of polynomials of square Gaussian variables?

Following this question Anti-concentration of Gaussian quadratic form. We have the following inequality: Let $X_1,\dots,X_n$ denote i.i.d. standard Gaussian random variables. For every $\epsilon>0$...
Hermi's user avatar
  • 274
4 votes
0 answers
141 views

Approximation by gaussian mollification in Sobolev spaces

I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify) $$\label{0}\tag{0} \|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
LL 3.14's user avatar
  • 220
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0 answers
122 views

Log of a truncated binomial

Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
Tom Solberg's user avatar
  • 3,929
6 votes
3 answers
919 views

Proof of a matrix implication

If $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$. I have observed by considering many examples of $x,y,z,...
BAYMAX's user avatar
  • 51
5 votes
1 answer
148 views

On existence of a concave function

Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that $$ f’’(x) \leq 0\quad \text{and} \...
Ali's user avatar
  • 4,077
4 votes
1 answer
331 views

Inequality with decreasing rearrangement function

Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$p'=\frac{pn}{n-p}.$$ Also, let $g$ be a positive ...
Shaq155's user avatar
  • 449
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2 answers
427 views

How to prove that $1/ ((y+z) x^4) + 1/ ((z+x) y^4) + 1/ ((x+y) z^4) \geq 3/2$ for $x, y, z>0$ such that $xyz=1$? [closed]

How to prove that $\dfrac{1}{(y+z) x^4} + \dfrac{1}{(x+z) y^4} + \dfrac{1}{(y+x) z^4}\geq3/2$ for $x, y, z>0$, such that $xyz=1$?
Jogn's user avatar
  • 53
9 votes
2 answers
611 views

How was Claim 5 in "A non-linear generalisation of the Loomis–Whitney inequality and applications" thought up?

In Bennett, Carbery and Wright's paper A non-linear generalisation of the Loomis–Whitney inequality and applications, Claim 5 was used to generalise the case from characteristic functions to simple ...
enihcamemit's user avatar
1 vote
1 answer
108 views

Finding an analytical upper bound on linear transform of matrix

Given a (real) positive semi-definite matrix $\underline M$ (with all elements of the principal diagonal equal to $1$) and a transformation $T:\underline A\to \underline A + \alpha \underline D$, with ...
kiyomi's user avatar
  • 111
6 votes
1 answer
447 views

a problem in complex-variable inequality

Let $n\ge2$ be a given positive integer, and $z_{1},z_{2},\cdots,z_{n}\in \mathbb{C}$,such $$|z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2\ge n.$$ Prove or disprove $$f_{n}=\sum_{j=1}^{n}\left|\sum_{I\subseteq ...
math110's user avatar
  • 4,230
1 vote
1 answer
67 views

Lower bound $L_{1}$-metric with $L_{2}$-metric for bounded pdfs, on common support

Setup To clarify, let constants $0 < a < b < \infty$, and $p \in \mathbb{N}$ be fixed. Further let $B \subset \mathbb{R}^{p}$ be a fixed compact support. We then define the space of bounded (...
user4687531's user avatar
2 votes
2 answers
183 views

Elementary convexity example

I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...
Joshua Isralowitz's user avatar
7 votes
2 answers
453 views

An elementary inequality of operators

Suppose $a,b$ are two positive-definite linear operators on (say) $\mathbb R^n$. For $p\in(0,1)$, do we then have $(a+b)^p\leq a^p+b^p$ (with respect to the Loewner order)?
Jim's user avatar
  • 71
2 votes
0 answers
238 views

Largest eigenvalue of a Laplacian matrix to lower bound the prime counting function?

Let $L_n$ be the Laplacian matrix of the undirected graph $G_n = (V_n, E_n)$ (which is defined here: Why is this bipartite graph a partial cube, if it is? ) with sorted spectrum: $$\lambda_1 (G_n) \ge ...
mathoverflowUser's user avatar
1 vote
1 answer
264 views

An inequality involving the Wasserstein distance and chi-squared distance

$\newcommand{\N}{\mathbb N}$Let $P$ be the set of all probability mass functions on $\N_0:=\{0\}\cup\N$, where $\N:=\{1,2,\dots\}$. Let $P_{>0}$ denote the set of all $q=(q_0,q_1,\dots)\in P$ such ...
Iosif Pinelis's user avatar
1 vote
0 answers
67 views

Shapiro inequality for divisor sets

The Shapiro inequality is the statement that if $x_1, x_2, \dots, x_n$ are positive, with $x_{n+1}=x_1, x_{n+2}=x_2$, then $$\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}.$$ This can be ...
JoshuaZ's user avatar
  • 6,090
4 votes
1 answer
302 views

When does the sine transform result in a positive function?

For each $x>0$ is, $$\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt > 0\ \ \ ?$$ In general, it is known that if the kernel is decreasing, then the sine transform is positive. Note that this ...
Bobby Ocean's user avatar
6 votes
2 answers
578 views

Doubts in first lemma in the paper of Adams regarding sharp Moser inequality

This question is on a point in D.R. Adams paper "A Sharp Inequality of J. Moser for Higher Order Derivatives". Precisely the lemma says: Given $a(s,t)$ be a non negative measureable function ...
User1723's user avatar
  • 307
0 votes
1 answer
91 views

Is set of integer solutions to these inequalities finite?

Consider the inequalities $$\frac{(2A-1)^2}{4A^2}xy\leq \Big(\frac{x+y}2\Big)^2\leq\frac{(2A-1)^2}{4(A-1)^2}xy$$ $$x,y\geq0$$ where $A>10^9$. Is the set of integer solutions to $x,y$ finite?
Turbo's user avatar
  • 13.7k
2 votes
1 answer
218 views

On the infimal convolution of two norms on $\mathbb R^n$

$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let \begin{equation*} K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1), \end{equation*} \begin{equation*} M:=M_{n,...
Iosif Pinelis's user avatar
7 votes
2 answers
353 views

On a von Bahr–Esseen-type inequality for pairwise independent zero-mean random variables

For $p\in(1,2)$, let $C_p$ be the smallest constant factor $C$ in the von Bahr–Esseen-type inequality \begin{equation}\label{eq:pair}\tag{1} E\Bigl\lvert\sum_{j=1}^n X_j\Bigr\rvert^p\le C\sum_{j=1}...
Iosif Pinelis's user avatar
3 votes
2 answers
242 views

Inequality for Gaussian polynomials III

Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials) $$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$ Given two ...
T. Amdeberhan's user avatar
5 votes
1 answer
235 views

Maximal inequality of iid random variables $\{X_{ij}\}_{1\leqslant i,j \leqslant n}$

Suppose that $\{X_{ij}\}_{1\leqslant i,j\leqslant n}$ are iid random variables with $\mathbb{E}(X_{11})=0$ and $\mathrm{Var}(X_{11})=1$, does the following convergence hold: $$ \max_{1\leqslant j\...
MHMH's user avatar
  • 71
1 vote
1 answer
54 views

Can the second-order difference control the first-order difference for nowhere differentiable functions?

Suppose that $f$ is a continuous, nonconstant function on $[0,1]$. Fix some $0<a<1$. Is it possible to establish the following inequality $$ |f(x+h)-f(x)| \leq C \left[ |h|^a + |2f(x)-f(x+h)-f(x-...
Watheophy's user avatar
  • 419
1 vote
0 answers
184 views

Schatten norm inequality

Let $A,B$ be two $n\times n$ matrices. Find a lower bound of the $p$-th Schatten norm $\|(A-B)(A-B)^\ast\|_{S_{p/2}}^{1/2}$ in terms of Schatten norm of $\|(AA^*+BB^*)\|_{S_q}$ for any relation ...
volond's user avatar
  • 97
5 votes
0 answers
238 views

Equality from the Grothendieck inequality

I asked the following question on math.stackexchange.com but have not received any response. So I would like to try my luck here. This question is related to the Grothendieck inequality. Let field $\...
Hans's user avatar
  • 2,169
1 vote
1 answer
142 views

Proof of lower bound on variance

I'm reading through the paper Poincaré type and spectral gap inequalities with fractional Laplacians on Hamming cube. However, I'm having a difficult time understanding the following proof: Lemma 2.1 ...
n3rl's user avatar
  • 13
3 votes
1 answer
142 views

Bounds on symmetric polynomials in power-sum form with bounded coefficients

Let $\boldsymbol{x}=(x_1,\ldots,x_n)$ be a real vector. Define the normalized power-sum symmetric polynomials by $\pi_j(\boldsymbol{x})=\frac 1n(x_1^j+\cdots+x_n^j)$. For a partition $\lambda= (j_1,\...
Brendan McKay's user avatar
-1 votes
1 answer
95 views

A proof of an interesting inequality [closed]

If $0<\beta<1$ and $0<x<1,$ how to prove that $$h(x)-2x+(4-2^{1+\beta})x^{1+\beta}<0,$$ where $$h(x)=(1+x)^{1+\beta}-x^{1+\beta}-1.$$The numerical simulation shows that it is true.
Renjun Qi's user avatar
2 votes
0 answers
80 views

Inequality on polynomials

Recall $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ and the Gaussian polynomial $\binom{a}{b}_q=\frac{[a]!_q}{[b]!_q[a-b]!_q}$ with $[0]!_q:=1$. Given two polynomials $U(q)=\sum_k\alpha_kq^...
T. Amdeberhan's user avatar
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,...
Roland Bacher's user avatar
4 votes
1 answer
164 views

Estimate an improper integral

Suppose that $f$ satisfies $a$-Hölder condition on $[0,1]$ ($0<a<1$). Fix $0<b<a$. For any $x\in [0,1]$ and sufficiently small $\delta>0$, is the following inequality established? $$ \...
Watheophy's user avatar
  • 419
2 votes
1 answer
73 views

An inequality about the second-order difference

Fix a continuously differentiable but nowhere twice differentiable function $f$ on $\mathbb{R}$ supported on $[0,1]$. Is it true that for all $x\in[0,1]$ and all $\delta$ sufficiently small \begin{...
Watheophy's user avatar
  • 419
0 votes
0 answers
67 views

Gagliardo-Nirenberg type inequality for fractional relativistic Laplacian operator?

In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality: $$\...
Jingeon An-Lacroix's user avatar

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