16
votes
Accepted
Higher order generalization of Cauchy-Schwarz?
Yes, because the OP stated that the ground field is $\mathbb{R}$, one can simply take the octic polynomial
$$
Q(v_1,v_2,\ldots,v_n) = \sum_{1\le i < j\le n} \bigl((v_i,v_i)(v_j,v_j)-(v_i,v_j)^2\...
- 108k
9
votes
How to prove that $1/ ((y+z) x^4) + 1/ ((z+x) y^4) + 1/ ((x+y) z^4) \geq 3/2$ for $x, y, z>0$ such that $xyz=1$?
Is the "cauchy-schwarz-inequality" tag a guess or a hint? . . .
At any rate, it turns out to be a good start. Let
$$
R := \frac1{(y+z) x^4} + \frac1{(z+x) y^4} + \frac1{(x+y) z^4}.
$$
We ...
- 79.9k
9
votes
How to prove that $1/ ((y+z) x^4) + 1/ ((z+x) y^4) + 1/ ((x+y) z^4) \geq 3/2$ for $x, y, z>0$ such that $xyz=1$?
$\newcommand\tH{\tilde H}$This problem is one of real algebraic geometry, which can be solved purely algorithmically. In Mathematica, such algorithms are implemented by ...
- 128k
8
votes
Accepted
A completely positive equivariant map $\varphi: A \to B$ induces a map $A \rtimes_r \Gamma \to B \rtimes_r \Gamma.$
This is Exercise 4.1.4 in the book by Brown+Ozawa (is that where this question ultimately comes from?) The "Hint" is "The reduced case is easy." Hmm.
Well, it might perhaps help ...
- 18.7k
6
votes
Accepted
An inequality involving a sum of power terms
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- 128k
6
votes
Accepted
A moment inequality
This inequality is false in general, by homogeneity considerations. Indeed, it can be rewritten as $L(x)\ge R(x)$, where $L(x):=\left(\chi(0)\chi(2)-\chi(1)^{2}\right)\left(\chi(4)\chi(2)-\chi(3)^{2}\...
- 128k
6
votes
Accepted
A probabilistic angle inequality
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- 128k
6
votes
A probabilistic angle inequality
Normalize $q$ such that $q^Tq=1$ and $q_i\geq 0$, for all $i=1,\ldots,n$. Let $X_i=q_ip_i$, $i=1,\ldots,n$. We must find an absolute constant $c>0$ such that
$$P\left(c\left(\sum_i X_i\right)^2\...
- 947
6
votes
Accepted
Good upper bound for a certain sum
$\newcommand{\ga}{\gamma}
$
Of course, without further assumptions on the $e_t$'s, no good bound can be given. However, looking at your examples, it appears that you are primarily interested in ...
- 128k
6
votes
The inequality $\int^\infty_0 (\sin(rt)r^3/\sinh^2(r)) dr\leq cte^{-At}$
$$\int^\infty_0 \frac{r^3\sin rt}{\sinh^2 r} \,dr=\tfrac{1}{8} \pi ^3 \frac{\pi t \cosh \pi t+2\pi t-3 \sinh \pi t}{\sinh^4(\pi t/2)}, $$
which decays as $e^{-\pi t}$ for large $t$, while it ...
- 188k
6
votes
Accepted
Prove inequality: $2\Big[ \sum_{k=0}^{n} (k+1) a_k \Big]^2 -\Big[1+ \sum_{k=0}^{n} a_k \Big]\Big[ \sum_{k=0}^{n} k(k+1)a_k \Big] \geq 0.$
A counterexample is given by the following conditions: $n=185$,
$$a_k=\sum_{j=k}^n b_j,\quad b_j:=\frac{c_j}{j+1},
\quad c_j:=\frac{1000}7\,1(j=47)+\frac{302}7\,1(j=185).$$
Indeed, then $a_0\ge\cdots\...
- 128k
5
votes
Accepted
The inequality $\int^\infty_0 (\sin(rt)r^3/\sinh^2(r)) dr\leq cte^{-At}$
This is to complement the answer by Carlo Beenakker by showing that
$$I(t)\le\pi^4 te^{-\pi t}\tag{1}\label{1}$$
for real $t\ge0$, where $I(t)$ is the integral in question.
Indeed, according to Carlo ...
- 128k
5
votes
Accepted
Is $N_\phi = \{x \in E: \phi(\langle x,x\rangle)=0\}$ a Hilbert submodule of $E$?
It is not true. Take $B= M_2(\mathbb C)$ (with standard matrix units $e_{i,j}$), $E= B$ as a Hilbert $B$-module in the usual way, and let $\phi \in B^\ast$ be compression to the $(1,1)$-corner. Then $...
- 2,471
5
votes
Accepted
Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$
By the main result of the paper Exact Rosenthal-type bounds, we have
$$E|X-np|^r\le c^r E|\Pi_\lambda-\lambda|^r
$$
for real $r\in(2,\infty)\setminus(3,4)\setminus(4,5)$, where real $c>0$ and $\...
- 128k
5
votes
A probabilistic angle inequality
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- 128k
4
votes
An alternative proof of Bayesian Cramer-Rao
In line with Deane's comment, this is an "answer" that also uses the Cauchy-Schwarz inequality but does so in a way that you might find more natural. I'll use different notation than yours (sorry; ...
- 27.7k
4
votes
Accepted
Is there a tight lower bound for the expectation of the product of two positive valued random variables?
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Let us present the exact lower bound on $EXY$ in terms of $\mu_1:=\mu_X$, $\mu_2:=\mu_Y$, $\si_1:=\si_X$, $\si_2:=\...
- 128k
4
votes
Accepted
Cauchy-Schwarz-like inequality with a power $p$ term
$\newcommand\th\theta\newcommand\R{\mathbb R}$This is false for any real $p>2$ (actually, this is false for any real $p>1$ such that $p\ne2$).
Indeed, if this were true, then, by continuity, we ...
- 128k
3
votes
Accepted
How is the Cauchy-Schwarz inequality used to bound this derivative?
You have a typo on the $\mathrm{Re}(Sv,v)$ term, the leading $A$ should be inside the integral. The formula from the paper reads
$$\mathrm{Re}\left(Sv,v\right) = \int -A |\nabla v|^2 + \left(A|\nabla \...
- 39k
3
votes
Good upper bound for a certain sum
Looks like we don't really need the log-convexity assumption in the accepted answer.
Indeed, define $\rho_N := e_{N} / e_{N-1}$ (with $\rho_1 := 1$), and suppose
Assumption. $\liminf_N\rho_N \ge \...
- 6,853
3
votes
Cauchy-Schwarz and pigeonhole
The relation between the Cauchy-Schwarz inequality and the measure theoretic version of the pigeon-hole principle can be illustrated by the following exercise.
${\bf Exercise.}$
Let $A_1$,...$A_n$ be ...
- 18.7k
3
votes
Is it possible to find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not $1/2$-Hölder
Indeed, disjoint tiny smooth spikes, of small heights and even much-much smaller widths, will do.
Let $K\in C^\infty(\mathbb R)$ be such that $K\ge0$, $K(x)=0$ if $|x|>1/2$, and $a:=K(1/3)-K(0)\...
- 128k
3
votes
On the Cauchy-Schwarz Inequality for trace function of random matrices
As Fedor said in the comments, the first inequality is correct by integrating the deterministic inequality over the probability space.
The second inequality, however, has no chance to be correct: ...
- 623
3
votes
Upper bound of $\frac{\sum_i c_ia_ie_i}{\sum_i d_ib_if_i}$?
There is no such upper bound in general. E.g., take $c_1=c_2=d_1=d_2=\frac12$ (or any other nonzero values), $b_1=f_2=1$ and $b_2=f_1=0$, which nullify the denominator of the fraction being bounded, ...
- 34.3k
3
votes
Accepted
Upper Bound for $\mathbb{E}\left[\max_{j \in \mathcal{N}} h_{j}\right]$
$\newcommand\si\sigma$The inequality in question fails to hold if e.g. $\mathcal N=[n]:=\{1,\dots,n\}$, the $h_j$'s are i.i.d. exponential random variables, and $n=4$. This follows because then
$$E\...
- 128k
2
votes
Cauchy-Schwarz and pigeonhole
The contrapositive of PH is: If you put at most one pigeon per hole, then you have at most $n$ pigeons. Letting $a = (1,\dots,1)$ and $b_i$ be the number of pigeons in hole $i$ with $b_i \in \{0,1\}$, ...
- 4,529
2
votes
Accepted
Geometric interpretation of a Grammian-like function
$\sqrt f$ equals $\sqrt 2$ times the area of a parallelogram with sides $|v|^{1/2}$ and $|w|^{1/2}$ and enclosed angle equal to $1/2$ the angle $\theta$ between $𝑣$ and $𝑤$.
- 188k
2
votes
Upper Bound for $\mathbb{E}\left[\max_{j \in \mathcal{N}} h_{j}\right]$
For $\{H_j\}_{j∈N}$ independent Gamma random variables as $H_j > 0$ then $max_{j\in N} H_j \leq \sum_{j\in N} H_j$. As $H$ are independent then $$\mathbb{E}[max_{j\in N} H_j] \leq \sum_{j\in N} \...
2
votes
Upper Bound for $\mathbb{E}\left[\max_{j \in \mathcal{N}} h_{j}\right]$
For an alternative bound (which I just saw was referred to by Sandeep in the comments while revising this), you can adapt the standard argument for the expectation of the maximum of Gaussians.
I do ...
- 2,183
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