Mathematics

Questions tagged [mixing-variables]

40 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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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 have…
math110
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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 some…
Anon
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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
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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 consideration: $$\begin{cases} c=-(a+b)\\ z=-(x+y)\\ ax+by+(a+b)(x+y)=0…
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
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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
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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
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2
votes
4 answers

Prove $\sum_{cyc}\sqrt{25a^2+144bc} \ge 5(a+b+c)+\frac{24(ab+bc+ca)}{a+b+c}.$

Problem. Let $a,b,c\ge 0: a+b+c>0$ then prove that $$\sqrt{25a^2+144bc}+\sqrt{25b^2+144ca}+\sqrt{25c^2+144ab} \ge 5(a+b+c)+\frac{24(ab+bc+ca)}{a+b+c}.$$ Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ or $(a,b,c)\sim(0,0,t)$ where…
user1656827
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
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2
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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
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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…
Shaveen De Silva
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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
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1
vote
0 answers

Prove that $\sum\limits_{cyc}\sqrt{8a(b+c)+(b-c)^{2}}\ge 12\cdot\frac{ab+bc+ca}{a+b+c}$

Let $a$ , $b$ and $c$ be non-negative numbers such that $a+b+c\neq0$ . Prove that: $$\sqrt{8a(b+c)+(b-c)^{2}}+\sqrt{8b(c+a)+(c-a)^{2}}+\sqrt{8c(a+b)+(a-b)^{2}}\ge\frac{12(ab+bc+ca)}{a+b+c}.$$ This question was deleted here . I solved this problem by…
Michael Rozenberg
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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 hold?…
30 Anh Ti 711
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