The inequalities tag has no wiki summary.

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### An inequality concerning non-negative integer matrices with constant row and column sums

[I posted this question on math.stackexchange a few weeks back, but no luck there so far: ...

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83 views

### Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...

**-1**

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28 views

### How to prove an inequality $\left| {g(j + 1)} \right| \le 5/4$ in Stein's method for Poisson approximation [migrated]

The following is a lemma in
Barbour, A. D., Holst, L., & Janson, S. (1992). Poisson approximation. Oxford: Clarendon Press,p7.
For $j=1,2,...$ and $\lambda > 0$, we have
$\left| {g(j + ...

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votes

**2**answers

134 views

### Rademacher average based Hoeffding Inequality

I am following these lecture notes:
Given the i.i.d. $\mathcal{Z}$-valued random variables $Z_1,\dotsc,Z_m$ and $\mathcal{G}$ is a set of bounded functions $g\colon \mathcal{Z}\to[a,b]$.
Corollary ...

**5**

votes

**1**answer

83 views

### Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?

Is it true that
$$
||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ?
$$
where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...

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42 views

### Show that $(\frac{d}{dt}||S(t)||_{\infty})_{t=0}=0$ where $S(t)$ is the Contraction semigroup for Laplacian [closed]

My Try:
I was able to prove one side of inequality using
$$
||S(t)\phi||_p\leq (4 \pi t)^{-N/2(\frac{1}{q}-\frac{1}{p})}||\phi||_q
$$
take $p=q=\infty$(as inequality is valid as long as $1\leq ...

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votes

**1**answer

116 views

### Proving a complicated inequality with powers of logarithms

I am currently formalising some results from complexity theory with a theorem prover. For that, I have to prove the following statement:
Let $p, b, \varepsilon \in \mathbb R$ with $\varepsilon>0$ ...

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votes

**1**answer

38 views

### Characterisation of a matrix ordering property

Let $n$ be a positive integer; we consider all matrices mentioned henceforth to be $n$-by-$n$ matrices. Let $A$ and $B$ be matrices wherein all entries are nonnegative (such matrices will be called ...

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**1**answer

188 views

### Proving a certain $ C^{*} $-algebraic inequality

Let $ A $ be a non-unital $ C^{*} $-algebra. Is there an ‘elementary’ way to prove, for all $ (a,\lambda) \in A \times \mathbb{C} $, the inequality
$$
|\lambda| \leq \sup_{b \in A, ~ \| b \| \leq 1} ...

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vote

**1**answer

95 views

### Matrix Submodular Inequality

Given $a,b,x > 0$ I know following the submodularity property holds:
\begin{align}
\frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x}
\end{align}
My question is, does this property ...

**7**

votes

**2**answers

228 views

### An upper bound for the difference between arithmetic and harmonic mean

Let $a_i\gt0$ for all $1\le i\le n$. It is well known that
$$
\frac{a_1+a_2+\cdots+a_n}{n}-\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}\ge0,
$$
with the equality when all $a_i$ are ...

**3**

votes

**1**answer

111 views

### Lower bound for the $p$-th absolute moment of a sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $\operatorname E X_k=0$ and $\operatorname E |X_k|^p<\infty$ with $1<p<2$ for each $1\le k\le n$. I am interested in the ...

**1**

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30 views

### Bound of spectral radius of polynomial of a complex matrix

I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and ...

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**1**answer

229 views

### Is the conditional expectation a contraction in weak $\mathbb L^p$ spaces?

Let $(\Omega,\mathcal F,\mu)$ be a probability space. It is well-known that if $\mathcal A$ is a sub-$\sigma$-algebra of $\mathcal F$, $p\geqslant 1$ and $X$ is an element of $\mathbb L^p$ which takes ...

**3**

votes

**1**answer

154 views

### Is this parametric inequality true?

Puzzled by this still open question, I tried comparing the arithmetic mean $A(x,y)=(x+y)/2$ with a mean intermediate between a geometric-type mean $G(X)=(x^a y^{1-a}+x^{1-a} y^a)/2\;$ for $0\le a \le ...

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774 views

### This inequality why can't solve it by now (Only four variables inequality)?

I asked a question at Math.SE last year and later offered a bounty for it, only johannesvalks give Part of the answer; A few months ago, I asked the author(Pham kim Hung) in Facebook, he said that ...

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57 views

### A “gradient” weak Harnack inequality for quasilinear elliptic equations

Suppose we are in the following loosely described setting:
we have a non-negative supersolution $h$ of the following elliptic equation:
\begin{equation}
\Delta h + \|\nabla h\|^2 + f(x) \geq 0
...

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35 views

### Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...

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**1**answer

221 views

### Nepero game (by Yacov Perelman)

I have already posted this question time before on stackexchange, but didn't receive a definitive solution.
So this is the game: consider a positive integer number $n$ and divide it in a finite ...

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31 views

### An inequality concerning restricted isometry property

Let $A\in \mathbb{R}^{m\times n}$ be a matrix and let us denote by $A_S$ the submatrix of $A$ with the columns restricted to a set $S\subset [n]:=\{1,2,\cdots, \ n\}$. Then one says that the matrix ...

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164 views

### this complex inequality have some background?and we can find stronger than this inequality?

Somedays ago,Chinese mathematical olympiad(2014) have this following problem:
Let $z_1,z_2,...,z_n$ be complex numbers satisfying $|z_i - 1| \leq r$ for some $r$ in $(0,1)$.
show that
...

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17 views

### Looking to derive bound for modulus of harmonic eigenfunction on weighted graph

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus:
$\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$
and wish to lower ...

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439 views

### Question on a proof by Solonnikov,Ladyzhenskaya,Ural'tseva

I have already asked this question on Mathematics SE, because I suppose that it is not research level. But I haven't got an answer, possibly here someone can answer.
Let $G(t,x)$ be the fundamental ...

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161 views

### Operator arithmetic-harmonic mean inequality with operator-valued weights

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...

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553 views

### Conjecture on maximum of symmetric combinatoric function

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture).
(The question was first asked at math.SE, ...

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**1**answer

173 views

### using jensen's inequality

Suppose we have an expression
f(x, h(x,y)), for some function f and h, and x, y are random variables,
now we know that the function f(a, b) is concave w.r.t. a for given b. Can we use Jensen's ...

**2**

votes

**1**answer

62 views

### Local fractional Sobolev inequality

If $\mathcal{X}$ is a smooth cutoff near 0 in $\mathbb{R}^n$, then $M_0 = \mathcal{X}(-\Delta+Id)\mathcal{X}$ is a self-adjoint operator in $L^2(\mathbb{R}^n)$. Because $M_0$ is semi-positive and the ...

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65 views

### How to prove it is uniformly bounded?

Let $\Omega$ be a bounded domain with smooth boundary. Say $\theta\in(0, 1]$. Let $u(x, \theta)$ be a solution to the problem $\Delta u-\theta u=g(x)$ subject to Neumann boundary condition. Suppose ...

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343 views

### Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)

Is there a "Cauchy-Schwarz proof" of the following inequality?
Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has
$$
\int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) ...

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**1**answer

82 views

### $2$-D Hlawka inequality

The classical counter-example to Hlawka inequality
$$|a+b|+|b+c|+|c+a|\le|a+b+c|+|a|+|b|+|c|$$
is the $l^\infty$-norm in dimension $3$, with vectors
$$a=\begin{pmatrix} 1 \\ 1 \\0 \end{pmatrix},\quad ...

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31 views

### Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions
Let
$L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric),
$a,b$ be arbitrary $n$-dimensional points,
$c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...

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205 views

### Characterizing orthants with polynomials

Let $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$. Can one find a polynomial $p$ (of arbitrary degree) in the coordinates of $x$ such that $p(x)\geq 0$ if and only if $x$ is an element of the positive orthant ...

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248 views

### Poincare-like inequality on compact Riemannian manifolds

I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact Riemannian manifold without boundary. The inequality I am looking for is the equivalent of
$$ ...

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104 views

### Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added.
This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...

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81 views

### Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?

By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the ...

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69 views

### Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...

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**1**answer

146 views

### How does Azuma's Inequality result from Pinelis Inequality?

According to [1]
Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let
$S(\mathcal{X})$ denote the class of all sequences
$f=(f_j)=(f_0,f_1,...)$ of Bochner-integrable random ...

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**1**answer

368 views

### Is the heat kernel more spread out with a smaller metric?

Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the ...

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552 views

### Is $\lceil \frac{n}{\sqrt{3}} \rceil > \frac{n^2}{\sqrt{3n^2-5}}$ for all $n > 1$?

An equivalent inequality for integers follows:
$$(3n^2-5)\left\lceil n/\sqrt{3} \right\rceil^2 > n^4.$$
This has been checked for n = 2 to 60000. Perhaps there is some connection to the ...

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votes

**1**answer

109 views

### Estimate for Levy metric

In the Encyclopedia of Mathematics there is an inequality for Levy metric ($d_L$):
$$d_L(E,F) \leq \{\beta_r(F)\}^{r/(r+1)},$$
where $E$ is a a distribution that is degenerate at zero, $\beta_r(F)$, ...

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**4**answers

185 views

### Simple bound for generalized geometric series

Let $b \in (0,1)$, $m\in \mathbb{N}$ and $a>0$. I want to bound
$$\sum_{k=m+1}^\infty b^{k^a} \leq c \; b^{m^a}, $$
where $c>0$ is independent from $m$.
Is there a simple way of proving this ...

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**1**answer

170 views

### Is this inequality true?

Let $\Omega$ be a bounded domain with smooth boundary. Let $$ S=\{u\in C^2(\overline \Omega): \frac{\partial u}{\partial n}=0 \text{ on } \partial\Omega \}.$$ Fix $\Phi\in S$ with $\Phi(x)>0$ for ...

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**0**answers

69 views

### Behavior of elementary symmetric polynomials near zero sets

It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with ...

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**1**answer

140 views

### trigonometric sum and inequalities

let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq ...

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votes

**2**answers

231 views

### Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $

Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold?
$$
\int_{k + 1/2}^{k + 3/2}
\frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}}
...

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votes

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211 views

### Operator norm versus Hlawka inequality

Let $E$ be a finite dimensional normed vector space. If $E$ is $\ell^1$-embeddable, then the norm satisfies Hlawka inequality
...

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201 views

### Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, ...

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**1**answer

196 views

### Generalized Hlawka inequality

Let $E$ be a vector space over the real (the complex case is interesting too). We consider functions $f:E\rightarrow\mathbb R$ which satisfy the homogeneity property
$$f(\lambda x)=|\lambda|\,f(x).$$
...

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votes

**1**answer

75 views

### Summability of ratios of moments a weight

Recently, I encounter the following problem:
Let $w$ be a probability density on $[0,1]$. Let mk be the $k$-th moment, i.e.,
$$m_k=\int_0^1t^kw(t)dt.$$
Under what condition can we have
...

**1**

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**0**answers

73 views

### Hlawka inequality for Lorentz quadratic form

Let $K$ be a convex cone in ${\mathbb R}^n$. A continuous function $f:K\rightarrow\mathbb R$ satisfies a Hlawka inequality if
$$f(0)+f(x+y)+f(y+z)+f(z+x)\le f(x)+f(y)+f(z)+f(x+y+z),\qquad\forall ...