Questions tagged [mixing-variables]
37 questions
15
votes
4 answers
How prove this $x^3+y^3+z^3+3\ge 2(x^2+y^2+z^2)$
Question:
let $x,y,z>0$ and such $xyz=1$, show that
$$x^3+y^3+z^3+3\ge 2(x^2+y^2+z^2)$$
My idea: use AM-GM inequality
$$x^3+x^3+1\ge 3x^2$$
$$y^3+y^3+1\ge 3y^2$$
$$z^3+z^3+1\ge 3z^2$$
so
$$2(x^3+y^3+z^3)+3\ge 3(x^2+y^2+z^2)$$
But this is not my…
math110
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- 17
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- 519
11
votes
1 answer
prove this inequality with $63$
Let $x,y,z,w>0$, and such $x^2+y^2+z^2+w^2=1$. show that
$$x+y+z+w+\dfrac{1}{63xyzw}\ge\dfrac{142}{63}\tag{1}$$
I know
$$x^2+y^2+z^2+w^2\ge 4\sqrt[4]{x^2y^2z^2w^2}\Longrightarrow xyzw\le \dfrac{1}{16}$$
so we…
math110
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- 17
- 148
- 519
8
votes
2 answers
For $a,b,c,d > 0$ and $abcd = 1$, show that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{12}{a + b + c + d} \geq 7$
Question:
For $a,b,c,d > 0$ and $abcd = 1$, show that
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{12}{a + b + c + d} \geq 7$$
My Attempts:
Trivial to see that equality occurs for $a = b= c = d$ - so this indicates to use…
Anon
- 2,727
6
votes
1 answer
Finding a parametre that satisfies an inequality
For what values of $k>0$ does
$$a^2+b^2+c^2+d^2+4(\sqrt3 -1)(abcd)^k\geq\sqrt{12(abc+abd+acd+bcd)}$$
hold for all $a,b,c,d\geq0$ satisfying $a+b+c+d=4$?
On the one hand, I found a lower bound for the LHS by using an equivalent form of Turkevich's…
Math Guy
- 531
4
votes
3 answers
If $a+b+c=x+y+z=ax+by+cz=0$, then find value of $\frac{a^2}{a^2+b^2+c^2}+\frac{x^2}{x^2+y^2+z^2}$
If $a+b+c=x+y+z=ax+by+cz=0$, then find value of $\frac{a^2}{a^2+b^2+c^2}+\frac{x^2}{x^2+y^2+z^2}$
From the $a+b+c=0$ equality I tried taking systems of equations into…
user1291864
4
votes
1 answer
For positive $a$, $b$, $c$ with $abc=1$, show that $\sum_{cyc}\sqrt{a^2-a+1}\geq a+b+c$
Let $a,b,c$ are positive number such that $abc=1$. Prove that:
$$\sqrt{a^2-a+1}+\sqrt{b^2-b+1}+\sqrt{c^2-c+1}\;\geq\; a+b+c$$
This problem froms my Math teacher. I have attempted to let $$(a,b,c)=(\frac{x}{y}, \frac{y}{z}, \frac{z}{x})$$. The…
Analyn_a
- 309
4
votes
4 answers
Find all the natural solutions of (a+b+c)a-3bc=0
I was trying to solve a geometry puzzle when I came across a simple algebraic problem that I couldn't solve.
Given the expression $(a+b+c)a - 3bc = 0$, find all natural solutions for $a$, $b$ and $c$.
I've tried to isolate one variable, like…
Lucas Cleto
- 393
3
votes
1 answer
Example of a distribution that is ergodic but not $\phi$-mixing?
The book "asymptotic theory for econometricians" ststes the theory that if a stationary sequence is alpha or phi mixing, it is also ergodic, but not the other way around. However, when I look at the definitions they seem intuititely to me to capture…
user56834
- 12,323
2
votes
3 answers
Prove that: $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{2}{abc}\geq1$
Let $a,b,c>0$ with $a^2+b^2+c^2=3$.
Prove that $$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{2}{abc}\geq1.$$
I can prove it by using "mixing variables":
\begin{align*}
P&=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{2}{a b c} \\
&\geq…
trungbk
- 177
- 8
2
votes
0 answers
Does the empirical c.d.f. converge to the population c.d.f. when the data is drawn from a mixing stochastic process
For i.i.d. observations, the Dvoretzky–Kiefer–Wolfowitz inequality tells us that the empirical c.d.f. almost surely converges to the population c.d.f. (i.e., $\underset{x \in \mathbb{R}}{\sup} | F_n(x) - F(x)| \to 0$ almost surely), but I wonder…
ZouX
- 21
2
votes
1 answer
Is Gower's Distance a metric?
A novice here
My previous question was closed due to inadequate details
So here I've added more details
A metric should basically satisfy 3 properties
Distance is equal to zero if and only if $x$ is equal to $y$ ($d(x,y)=0 ⇔ x=y$))
Distance from…
2
votes
1 answer
Overview of the different types of mixing sequences
I have come across the terms strong mixing, $\alpha$-mixing, $\beta$-mixing, $\phi$-mixing, $\rho$-mixing. Could somebody please compile an answer that would summarize their definition in some clear and unified way, the relationships between these…
ippiki-ookami
- 1,322
1
vote
1 answer
Prove $\sum \sqrt{a+5bc+4}\ge 2\sqrt{6}\sqrt{ab+bc+ca}, \forall a,b,c\ge 0:a+b+c=4.$
I'm looking for some ideas to prove the following inequality.
Problem. Let $a,b,c$ be three non-negative real numbers with $a+b+c=4.$ Prove that$$\sqrt{a+5bc+4}+\sqrt{b+5ca+4}+\sqrt{c+5ab+4}\ge 2\sqrt{6}\sqrt{ab+bc+ca}.$$When does equality…
30 Anh Ti 711
- 647
1
vote
4 answers
For $x+y+z=3,$ prove $\frac{1}{x^2+4}+\frac{1}{y^2+4}+\frac{1}{z^2+4}\le \frac{3}{5}.$
Let $x,y,z\ge 0: x+y+z=3.$ Prove that$$\frac{1}{x^2+4}+\frac{1}{y^2+4}+\frac{1}{z^2+4}\le \frac{3}{5}.$$
Here is just my thought progress.
I set $0\le xy+yz+zx=q\le 3; 0\le r=xyz\le 1.$ After full expanding, I get $$7q^2-16q+3r^2-42r+24\ge 0.$$
From…
user1071608
1
vote
1 answer
Follow-up question on Abel Summation
This is a follow-up to a much simpler question I asked here, which @PrincessEev answered promptly and perfectly. She showed me how to rewrite the sum $\sum _{i=1}^x \phi (x-i)$ in such a way that Abel Summation could be applied, to obtain
$$\sum…
Richard Burke
- 925