# Inequality

• Jan 1st 2008, 07:49 AM
james_bond
Inequality
1. Prove that if $\displaystyle a;b\in\mathbb{R}^+$ and $\displaystyle a+b=1$, then
$\displaystyle \left(a+\frac 1a\right)^2+\left(b+\frac 1b\right)^2\ge \frac {25}2$.
2. Similarly to the previous problem:
Prove that if $\displaystyle a;b;c\in\mathbb{R}^+$ and $\displaystyle a+b+c=1$, then
$\displaystyle \left(a+\frac 1a\right)^2+\left(b+\frac 1b\right)^2+\left(c+\frac 1c\right)^2\ge \frac {100}3$.
3. Let's generalize the problems above:
Prove that if $\displaystyle a_1;a_2;\ldots ;a_n\in\mathbb{R}^+$, $\displaystyle n\ge 2$, $\displaystyle n\in\mathbb{N}$, and $\displaystyle a_1+a_2+\ldots +a_n=1$ then
$\displaystyle \left(a_1+\frac 1{a_1}\right)^2+\left(a_2+\frac 1{a_2}\right)^2+\ldots +\left(a_n+\frac 1{a_n}\right)^2\ge K$.
Find the maximum possible value of $\displaystyle K$.
• Jan 1st 2008, 08:48 AM
Isomorphism
Quote:

Originally Posted by james_bond
1. Prove that if $\displaystyle a;b\in\mathbb{R}^+$ and $\displaystyle a+b=1$, then
$\displaystyle \left(a+\frac 1a\right)^2+\left(b+\frac 1b\right)^2\ge \frac {25}2$.

Equality for $\displaystyle a=b=\frac12$
By Cauchy-Schwarz we have $\displaystyle (a+\frac 1a)^2+(b+\frac 1b)^2 \geq \frac{(a+\frac 1a + b+\frac 1b)^2}{1 + 1} = \frac{(1+\frac{1}{ab})^2}{2}$

Since$\displaystyle (a+b)^2 = 1 \geq 4ab \Rightarrow \frac{(1 + \frac1{ab})^2}{2} \geq \frac{25}{2}$.

Thus we have our solution :D
• Jan 1st 2008, 09:07 AM
Isomorphism
I post the remaining separately because a small trick there, wont work any longer ;)

I will directly attack the generalization,

Claim:$\displaystyle K = \frac{(n^2 + 1)^2}{n}$
Proof:
Denote LHS of your inequality by I,
AM-HM says:
$\displaystyle (\sum a_i)(\sum \frac1{a_i}) \geq n^2 \Rightarrow (1 + \sum \frac1{a_i})^2 \geq (n^2 + 1)^2$-------------(*)
Applying CS:
$\displaystyle I \geq \frac{{(a_1+\frac 1{a_1} + a_2+\frac 1{a_2} + \ldots a_n+\frac 1{a_n}})^2}{1+1+1 \ldots +1}$
But $\displaystyle ({a_1+\frac 1{a_1} + a_2+\frac 1{a_2} + \ldots a_n+\frac 1{a_n}} = 1 + \sum \frac1{a_i}$
So $\displaystyle I \geq \frac{{(1 + \sum \frac1{a_i})}^2}{n}$.
But by (*), $\displaystyle \frac{{(1 + \sum \frac1{a_i})}^2}{n}\geq \frac{(n^2 + 1)^2}{n}$
Thus $\displaystyle I \geq \frac{(n^2 + 1)^2}{n}$
Equality for all $\displaystyle a_i$s equal.
K cant get any better than that because we showed equality achieving ;)