Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

19,265 questions
32 views

44 views

27 views

What are Contra Harmonic Mean and Inverse Contra Harmonic Mean? [closed]

Are they related to Inequality? Like the $$AM \times HM = GM^2$$
38 views

Tough substitution inequality

Prove that if $x, y, z >0$ and $xyz=x+y+z+2$, then $$\sqrt{x}+\sqrt{y}+\sqrt{z} \leq \frac{3}{2}\sqrt{xyz}.$$ By the way, the first equation implies the existence of positive $a, b, c$ such ...
63 views

30 views

64 views

AM-GM Inequality Involving Squares and Proof

Prove: $$(a^2 + b^2 + c^2)/3 \geq ((a + b + c)/3)^2$$ OR $$(a^2 + b^2 + c^2)/3 \leq ((a + b + c)/3)^2$$ for all $a, b, c \geq 0.$ The problem wants me to find which inequality is correct and then ...
44 views

Prove $(2a + b + c)(a + 2b + c)(a + b + 2c) ≥ 64abc$ using the AM-GM method and establishing when inequality holds

Prove: $$(2a + b + c)(a + 2b + c)(a + b + 2c) ≥ 64abc$$ for all $a, b, c ≥ 0$. Also, calculate/prove when equality holds. To prove this, the first thing that came to mind was the Arithmetic Mean -...
64 views

Prove that $|\frac{a+b}{1+ab}| < 1$ given that $|a|<1$ and $|b|<1$

Can I please get a hint/solution for why, if true, $$\bigg{|}\frac{a+b}{1+ab}\bigg{|}<1$$ given that $|a|<1$ and $|b|<1$, where $a,b \in \mathbb{R}$. I've tried the usual things like ...
34 views

Proof of Khintchine's inequality on $\mathbb{C}$?

I'm trying to understand this proof of Khintchine's inequality for the complex case ($a_n\in\mathbb{C}$). The author claims that the complex case follows from the real case by taking absolute values ...
Taking assigning an unknown variable $\theta = 0.5$ to solve an inequality
In order to solve the following inequality for $n$, where $0 < \theta < 1$, they use a trick that I do not understand $n \geq (\frac{1.96}{0.03})^2 \theta(1 - \theta)$ Since $\theta$ is ...