# Tagged Questions

**3**

votes

**1**answer

122 views

### About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality
$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...

**4**

votes

**1**answer

115 views

### Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics
http://www.encyclopediaofmath.org/index.php/Differential_inequality
the following result is due to Chaplygin ...

**1**

vote

**0**answers

200 views

### Proof of an inequality for a linear combination of three trigonometric functions

Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$
where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ ...

**2**

votes

**1**answer

108 views

### Estimate of a ratio of two incomplete gamma functions

I would like to bound from above the expression
$$
\frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}
$$
for $x>y>0$. By plotting the above expression I have found that ...

**0**

votes

**1**answer

69 views

### An inequality involving multi-index

I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this:
For $x \in \mathbb{R}^{n}$ and $\alpha = ...

**11**

votes

**5**answers

702 views

### Understanding Gibbs's inequality

Short version
Gibbs's inequality is a simple inequality for real numbers, usually
understood information-theoretically. In the jargon, it states that
for two probability measures on a finite set, ...

**3**

votes

**0**answers

290 views

### A curious inequality

Let $r_k>0$ for $k = 1,\ldots, n$, let $\alpha_k, \beta_k\in \mathbb{R}$ be given such that
$|\alpha_k|\le \beta_k\le \frac{\pi}{2}$. Suppose further that
...

**0**

votes

**1**answer

204 views

### Inequality with even powers of trigonometric functions

For $m>0$,
$0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that
$$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...

**3**

votes

**3**answers

280 views

### Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?

What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$?
Or any good reference for tools to tackle this question?
...

**1**

vote

**2**answers

246 views

### how to solve a singular integral equation involving the kernel $1/x$

Dear all,
Suppose we know that $f(x)$ is nonnegative and HÃ¶lder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...

**0**

votes

**1**answer

140 views

### Strengthening an inequality

Let $k$ be an integer. The following inequality is standard.
$$
(a+b)^{k+1} - b^{k+1} \leq (k+1)a(a+b)^k
$$
for $a,b > 0$.
However, does the following inequality still hold
$$
(a+b)^{k+1} - ...

**1**

vote

**1**answer

288 views

### sobolev embedding theorem in the smooth metric measure space

we know the sobolev embedding theorem of Saloff-Coste
$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu
$
wtih $Ric\ge-(n-1)K$, for ...

**4**

votes

**2**answers

218 views

### Bounding the series of the geometric means of the terms of a given positive series

Let $ \{ a _ k \} _{k\in\mathbb{N} _ +} $ be a sequence of non-negative numbers, and let $MG(a_1,\dots,a_n)$ denote the geometric mean of the first $n$ terms. Then, the inequality
$$ \sum _ {n\ge ...

**3**

votes

**3**answers

866 views

### Poincare Metric on Hyperbolic Plane

as is well known, we can put a metric on the upper half plane $\mathbb{R}^+ \times \mathbb{R}$
by setting
$$
d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2},
$$
...

**7**

votes

**1**answer

392 views

### What would the best treatment of Gehring's lemma look like?

In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...

**0**

votes

**1**answer

181 views

### An inequality for a continuous non-smooth function

Hello,
I have a question about how to prove a lemma such as this one,
For any $0<\alpha<1$ and $M_{0}>0$, there exists a $M_{1}>0$ such that $\left|z\right|^{\alpha}\leq ...

**1**

vote

**0**answers

67 views

### Bound of polynomial on product space in terms of values on the diagonal

We work in the multivariate case so that $x$ stands for $(x_1,\ldots, x_n)$. Let $q(x,y)$ be a symmetric matrix representation of a homogeneous polynomial $f(x)$ of degree $d$. Explicitly,
($q$ is ...

**4**

votes

**5**answers

886 views

### An inequality on concave functions

Could somebody help me to answer the following question?
Let $f:R_+ \rightarrow R_+$
be a nonindentically zero, nondecreasing, continuous, concave function with $f(0)=0$. Do we have that for any ...

**1**

vote

**1**answer

340 views

### Is there a complex analog of this sharpened Cauchy Inequality?

Let $x$ and $y$ be two points on the unit sphere $S^{n-1}$ in Euclidean space ${\mathbb{R}}^n$. Suppose that the angle $\theta$ between the points $x$ and $y$ is acute, so that the dot product $x\cdot ...

**1**

vote

**1**answer

389 views

### Coefficient bounds of an inequality

Hello,
Given positive integers $k$ and $n$. Are there upper bounds on coefficients $A$ and $B$ such that they depends only on $k$ (eg., $2 k^k$) and for all non-negative integer sequences ...

**9**

votes

**2**answers

1k views

### Question on eigenvalue square root subadditivity

ORIGINAL QUESTION
Let $\lambda_{1}\left(\cdot\right)$ be the larger eigenvalue of a
$2\times2$ matrix and $\lambda_{2}\left(\cdot\right)$ the smaller
eigenvalue of a $2\times2$ matrix. Is it true ...

**5**

votes

**0**answers

441 views

### Any similar inequality in literature?

I got the following inequality:
$B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary.
$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$,
...

**1**

vote

**1**answer

1k views

### How to estimate derivatives of multivariate polynomial near a manifold

I have to provide a (Markov or Bernstein-based?) inequality that gives an upper bound for the partial derivatives of a multivariate polynomial calculated near a real smooth surface in terms of value ...

**6**

votes

**5**answers

516 views

### Binary operations compatible with the usual order on the reals

An officemate passes along the following natural-seeming question:
Say that a binary operation $\oplus$ is compatible with the usual order $\leq$ on $\mathbb{R}$ if for any $w, x, y, z$ in ...

**16**

votes

**6**answers

3k views

### Applications of Hardy's inequality

Every so often I would encounter Hardy's inequality:
Theorem 1 (Hardy's inequality). If $p>1$, $a_n \geq 0$, and $A_n=a_1+a_2+\cdots+a_n$, then
$$\sum_{n=1}^\infty ...

**1**

vote

**1**answer

319 views

### Statistical inequality

Let $X$ be a finite discrete variable and $X\ge0$. Is it true that
$$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$
where ...

**4**

votes

**2**answers

579 views

### Logarithm of AM/GM ratio: $\sqrt{\log((x+y)/(2\sqrt{xy}))}$

Recently, while playing around with infinite-divisibility, i arrived at the following metric:
$$d(x,y) := \sqrt{\log\left(\frac{x+y}{2\sqrt{xy}}\right)},$$
defined for positive reals $x$ and $y$. ...

**1**

vote

**0**answers

931 views

### Inequality concerning absolute value of a polynomial

Let
$$f(z) = (1-1/t) z^w + z/t - 1$$
with integers $t\geq2$ and $w\geq2$.Let $r=1+1/(tw^3)$. How do I show
$$\left\lvert f(r e^{i\varphi}) \right\rvert \geq \left\lvert f(r) \right\rvert$$
for any ...

**5**

votes

**1**answer

263 views

### Denominators in the solution to Hilbert's XVII

Hilbert's seventeenth problem asks to prove that every positive semidefinite form can be written as the sum of squares of rational functions. Currently we don't seem to have a good understanding of ...

**11**

votes

**2**answers

744 views

### Positivity of sequences via generating series

There are different ways of showing that a given sequence $a_0,a_1,a_2,\dots$
of integers, say, is nonnegative. For example, one can show that $a_n$ count
something, or express $a_n$ as a (multiple) ...

**4**

votes

**1**answer

322 views

### A plausible positivity

After getting stuck with the
previous positivity
(it probably sounds too complex),
I would like to give a version of the problem which is of most interest to me.
Consider a sequence of real numbers
...

**6**

votes

**1**answer

302 views

### Positivity of “harmonic” summation

The settings for the problem are as follows. Given
a real number $\alpha\in[0,1]$, consider
a sequence of real (positive, negative and zero) numbers
$a_1,a_2,\dots,a_n,\dots$ satisfying
(1) $a_1=1$,
...

**1**

vote

**3**answers

354 views

### monotonicity from 4 term-recursion.

In determining the monotonicity of coefficients in a series expansion (which appeared in one of my study), I come across the following problem.
Let $p\ge 2$ be an integer, and ...

**2**

votes

**3**answers

538 views

### l^p space inequality related to compressed sensing

I'm trying to read Donoho's 2004 paper Compressed Sensing and am having trouble with a supposedly trivial statement (equation 1.2 on page 3).
He makes the sparsity assumption on $\theta \in ...

**5**

votes

**3**answers

2k views

### When does a real polynomial have a pair of complex conjugate roots?

Suppose we have a polynomial function $f(z)=a_0+a_1z+a_2z^2+...+z^n$ with each $a_i$ between 0 and 1. Is there a method to determine if $f$ has a pair of complex conjugate roots?
There are many ...