Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $a,b,c >0$

Prove the inequality $\displaystyle{\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}+\frac{1}{a+b+c+1} \geq 1}$

I dont even know where to begin.

Only interested in hints (not solution)

share|improve this question

2 Answers 2

up vote 11 down vote accepted

Hint :

$$\frac{a}{a+1} > \frac{a}{a+b+c+1}$$

$$\frac{b}{b+1} > \frac{b}{a+b+c+1}$$

$$\frac{c}{c+1} > \frac{c}{a+b+c+1}$$

share|improve this answer
    
Duh! I get it now –  Kirthi Raman Mar 15 '12 at 13:06
1  
The whole expression is $> \frac{a+b+c+1}{a+b+c+1} = 1$ –  Kirthi Raman Mar 15 '12 at 13:08

We have $a+1\leq a+b+c+1$, $b+1\leq a+b+c+1$ and $c+1\leq a+b+c+1$ so $\frac 1{x+1}\geq \frac 1{1+a+b+c}$ where $x=a,b$ or $c$. Now multiply by $x$ to get $\frac x{x+1}\geq \frac x{1+a+b+c}$ and sum.

share|improve this answer
    
If $a+1 \geq a+b+c+1$ doesn't that mean $b+c \leq 0$? –  Kirthi Raman Mar 15 '12 at 13:05
    
Sorry I reversed the inequalities, now I think it's correct. –  Davide Giraudo Mar 15 '12 at 13:07

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.