# Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

1,526
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### How to estimate an integral by the variation and upper buond 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 ...

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### A one-sided/monotone version of min/max-stable distributions -- does this have a name?

In a couple of papers I am working on (in random graph theory) I have encountered the following property of certain probability distributions, which I will describe shortly, and I am wondering if this ...

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### Cauchy-Schwarz-like inequality with a power $p$ term

We set :
$\phi_1, \phi_2 : \mathbb{R} \to \mathbb{R}^M$ with compact support
$\theta \in \mathbb{R}^M$ non-zero, $\theta_{p-1} = (\operatorname{sign}(\theta_i)|\theta_i|^{p-1})_{1 \leq i \leq M} \in \...

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### Is there any identity for the expression : a^2 + b^2 + c^2 +d^2? [closed]

Please state any and all conditions for it to be true.

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### An inequality problem for certain positive-definite matrices

Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $<0$. Let $a$ be a column $n\times1$ matrix such ...

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1
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### Deducing norm concentration from MGF bounds

Suppose that $X$ is a centered, $\mathbf{R}^d$-valued random variable such that for all $t \in \mathbf{R}^d$, there holds the bound $$\log \mathbf{E} \left[ \exp \langle t, X \rangle \right] \leqslant ...

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### An inequality for certain positive-definite matrices

Suppose that $G=(G_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Let $a$ be a column $n\times1$ matrix with ...

4
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328
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### An inequality for certain positive-semidefinite matrices

Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...

1
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1
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96
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### 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 + \...

3
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1
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### Gronwall lemma for a $2$-dimensional system of linear differential inequalities

Let $$z(t)=\begin{pmatrix}x(t) \\ y(t)\end{pmatrix}~,~ A=\begin{pmatrix}0 & -\beta \\ \alpha & -(\alpha + \beta)\end{pmatrix}$$ satisfies the following system of linear differential ...

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### On $[0,1]$ with the Lesbegue measure, is it possible to have $\lVert \cdot \rVert^{1/n}_p \leq \lVert \cdot \rVert_q$ for $p>q$ and $n$ large?

The question is as in the above.
In all literature, I only find that on $[0,1]$ with the Lebesgue measure, $\lVert \cdot \rVert_q \leq \lVert \cdot \rVert_p$ for $p>q$.
(I deleted the last question ...

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### Proof of the inequality $\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x}$ when $x,y \in (0,1]$

I am trying to prove the following inequality:
$$\frac{y}{x}-1-\log\left(\frac{y}{x}\right)\geq \frac{1}{2}\frac{(x-y)^2}{x} \quad \forall x,y \in (0,1]$$
This inequality appears in the paper "...

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### Question on a mixed-norm estimate

I am currently reading the paper Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $\mathbb{R}$ by Colliander, Holmer, Visan, Zhang. In this article,...

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### Simple anticoncentration bound for binomially distributed variable

The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument.
For a binomially distributed variable $X \sim \text{Bin} \...

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### Does the Gaussian Poincare inequality hold for infinite dimensional measure metric spaces?

This is a question subsequent to the one:
Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
There, I received a very helpful answer that the Gaussian poincare inequality for any ...

2
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2
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580
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### An inequality related to Catalan's constant and $\zeta(3)$

Problem :
Show that :
$$\frac{1}{\zeta(3)}<2C-1$$
Where we can see the zeta function and the Catalan's constant .
After a bounty on Maths Stack Exchange there is no satisfying answer .
See https://...

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1
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113
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### Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?

Let $X$ be a real-valued standard normal variable. Then, for any differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $E[f(X)^2] < \infty$ and $E[\bigl( f'(X) \bigr)^2] < \infty$, it ...

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### Show $\langle \log(R), \log(R^{-1}S) \rangle \geq \langle \log(R), \log(S) - \log(R) \rangle $ for all $R,S \in \mathrm{SO}(3)$

$\DeclareMathOperator\SO{SO}$I have a similar question to one I asked a few days ago. Lately, I've been researching Lie groups equipped with bi-invariant Riemannian metrics. One common object is $\SO(...

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### How to upper bound the difference between these two Gaussian-like densities?

$
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\cov}{cov}
\DeclareMathOperator*{\supp}{supp}
\DeclareMathOperator*{\dom}{dom}
\newcommand{\...

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### Inequality for log-likelihood ratio

Let $ p, q $ be two probability densities on $ [0,1] $, strictly positive over $ (0,1) $. Let $ P $ be the cumulative function of $ p $, i.e., $ P(x) = \int_0^x p(x') \, \mathrm{d}x' $, $ x \in [0,1] $...

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2
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### What are the bounds of $xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a$ for $0 \le x \le 1$ and $a > 0$?

Posting from MSE since it was unanswered in MSE.
Let $0 \le x,y \le 1$ and $a$ be a real and let
$$
f(x,y,a) = xy^{y^a/x^a} + yx^{x^a/y^a} - x^a - y^a \tag 1
$$
For a fixed $a$, the graph of the ...

2
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0
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### Inequality for a weighted bilinear form in Fourier variables

Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$.
Consider the ...

10
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1
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988
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### Is every positive polynomial the ratio of 2 positive coefficient polynomials?

Define $P(x)$ to be positive if $P(x)>0$ for $x>0$.
I can prove that a quadratic positive polynomial is the ratio of 2 polynomials with non negative coefficients, for example $\displaystyle x^2-...

5
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1
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### Maclaurin's inequality on elementary symmetric polynomials of arbitrary real numbers

Is there a universal constant $C$ such that the following statement holds? For concreteness, you may assume $C=10000$.
Let $a = (a_1, \ldots, a_n)$ be $n$ arbitrary real numbers. For an integer $k$, ...

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2
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### Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm?

$\DeclareMathOperator\SO{SO}$I asked this initially in math stack exchange, but thought to ask it here since it is more advanced and related to my research topic. I study optimization on Lie groups ...

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49
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### Sobolev trace inequality with exterior domains

Let $x_1\in \mathbb{R}^n$, $n\geq 3$, $\Omega=\mathbb{R}^n\backslash B_1(x_1)$, define $D_{\Omega}$ by taking the closure of $C_{c}^{\infty}(\overline{\Omega})$ under the norm
\begin{align*}
\|u\|_{D_{...

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### Proving Geometric Inequality Using Equation Discriminant

I met this question before:
An acute $\triangle ABC$ (you can imagine $BC$ below) has a point $D$ on side $AC$. The line parallel to BC through $D$ meets $AB$ at $E$, and the parallel line $BD$ ...

2
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1
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### Given $P\in \mathbb{Z}[x]$ there is a nonzero $Q\in \mathbb{Z}[x]$ such that $H(PQ)\leq M(P)$

I am looking for a proof (or references) for the following result:
If $P\in \mathbb{Z}[x]$ then there exists a nonzero polynomial $Q\in \mathbb{Z}[x]$
such that
$$H(PQ)\leq M(P)$$
where
$H(R)=\max\{|...

0
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0
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### Bound on integral of rational functions over [0,1]

I have a question about integrals of rational functions over $[0,1]$. Suppose $f(x),g(x)$ are two nonzero polynomials with non-negative (integer, in fact) coefficients. Under what non-trivial ...

1
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1
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34
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### Covariance inequality for left skewed distributions

Consider a left skewed random variable $X$ with mean $1$, median $>1$ and support on $[0,2)$. Suppose we have a class of functions $\mathbf{G}$ and each of it's members satisfy $G(x): [0,\infty) ...

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### Upper-bound on energy of nonlinear boundary-value problem

The problem:
Consider the following boundary-value problem for the function $\rho : \mathbb{R}^{+} \to \mathbb{R}$ with boundary conditions $\lim_{x\to \infty}\rho(x) \to 1$ and $\lim_{x\to 0}\rho(x)...

7
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### When does equality hold in a specific triangle inequality?

I'd like to know when the equality holds in the following inequality
$$
| x - y |^a \le | x - z |^a + | y - z |^a.
$$
More precisely, for which points $x = (x_1, x_2)$, $y = (y_1, y_2)$ and $z = (z_1, ...

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### Given optimality of L1 norm, prove that absolute value of sum of a vector with proper sign is less than 1?

Problem:
Given a domain $\mathcal{D}\subset\mathbb{R}^{l}$, we can find $l$
points $\boldsymbol{v}_{i}\in\mathcal{D}$, $i=1,\cdots,l$. Each
point is a column vector with dimension $l\times1$. They ...

0
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0
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54
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### Upper bound Variational Representation of $\chi^2$?

The $\chi^2$ divergence is an $f$-divergence given by
$$\chi^2(P\parallel Q) = \mathbb{E}_Q\left[\left(\frac{dP}{dQ}-1\right)^2\right].$$
It has a variational representation,
$$\chi^2(P\parallel Q) = \...

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### On the monotonicity of the ratio of two logarithmic expressions

According to this comment and this comment, a positive answer to this recent question (about Bernoulli numbers) would be sufficient to prove the following:
$r:=f/g$ is increasing on $(0,\pi/2)$ from $...

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### Ask for a proof of an inequality involving the Bernoulli numbers

Let $B_k$ be the Bernoulli numbers and let
\begin{equation}
T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1.
\end{equation}
Prove the inequality
\begin{equation*}
\frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...

5
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0
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119
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### On the weighted Hilbert inequality a la Montgomery and Vaughan

The weighted Hilbert inequality estimates the norm of the skew-hermitian matrix
$$B :=\left[ \frac{\delta_m^{1/2} \delta_n^{1/2}}{x_m-x_n} \right]_{m,n}$$
on $\ell_2$, where $x_1,\ldots$ is a finite ...

1
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2
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### Lemma about the weighted interpolation inequality

In this article Interpolation inequalities with weights
Chang Shou Lin the following lemma is stated and proved.
Lemma: Suppose $\dfrac{1}{p}+\dfrac{\alpha}{n} > 0$, then there exists a constant $C$...

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### Lower bound $\int_0^1 \frac{|f'(x)|^2}{f} \,\mathrm{d} x$ by $\int_0^1 |f-1|^2\, \mathrm{d} x$

Assume that $f$ is a probability density on $x \in (0,1)$, I want to obtain a bound of the following form (if it is possible at all): $$ \int_0^1 \frac{|f'|^2}{f} \,\mathrm{d} x \geq C\,\int_0^1 |f-1|^...

5
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1
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### Are truncated even degree binomial polynomials psd?

For integers $n$ and $k$, let $P(n,k)(x) = \sum_{i=0}^k \binom ni x^i$ be the truncated binomial polynomial. There has been work on whether $P(n,k)$ is irreducible, but this is a different question. ...

1
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2
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### Prove inequality [closed]

I have to prove the following inequality (See Appendix A of Classical Fourier Analysis of Grafakos).
$$
\left(\frac{(1+s/y)^y}{e^s}\right)^{2y} \leq (1+s)^2 / \exp(s), \quad s\geq 0
$$
and
$$
\left(\...

3
votes

2
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438
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### Short sequence beats long sequence

I have encountered some comparison between two binomial sums. It was amusing how the one with "fewer" summands exceeds (in value) than the other which consists of many more terms. In fact, ...

1
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0
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### Bound an integral with parameter

Let us define, for $x_0 > 0$ and $x_0 \ll 1$,
$$K(x) = \int_{-x_0}^{x - g(x)^2} \frac{f(y)}{x - y}dy, \quad \text{for } x \in [-x_0/4, x_0],$$
and
$$g(x) = \frac{(x_0 - x)|\log(x_0)|}{|\log(x_0 - x)...

8
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1
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### Big triples in a matrix

Consider an $n\times n$ real matrix $A=(a_{ij})$ with non negative entries. Assume that
- the sum of the three largest entries in each row is a constant $R$ (the same for all rows),
- the sum of the ...

0
votes

2
answers

116
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### Integer solutions of system of inequalities

I am trying to solve a problem in combinatorics and I came up with the following system of inequalities:
$0\leq x<y<z\leq n$ and $x+y<n$ and I am trying to count the number of integer ...

5
votes

1
answer

294
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### The decay rate of a degenerate heat equation in torus $\mathbb{T}^2$

Consider the degenerate heat equation on torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$:
$$ \frac{\partial}{\partial t} u(x,t)= \left( \sin^2(\pi x_1) \frac{\partial^2}{\partial^2 x_2} + \sin^2(\pi ...

2
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0
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### Techniques for solving linear inequalities

For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...

2
votes

1
answer

110
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### The lower bound of bivariate normal distribution

Suppose $(Z_1, Z_2)$ is the zero-mean bivariate normal distribution with covariance $\left( \begin{matrix} 1 & \rho; \\ \rho & 1\end{matrix} \right)$ with positive $\rho > 0$. What I want ...

0
votes

1
answer

126
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### BMY inequality for surfaces of general type in characteristic 0

Let $X$ be a smooth, complex, projective, minimal surface of general type, i.e. the canonical (line) bundle $K_X$ is big and nef.
It is known that $3c_2\geq c_1^2$ (the Bogomolov-Miyaoka-Yau ...

5
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0
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135
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### 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. $x_i^2 \le x_{i-1}x_{i+1}$).
...