×

# Need help on this 2

$\large \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{abc}}\geq\frac{4}{3}\bigg(\frac{a^2}{a^2+bc}+\frac{b^2}{b^2+ac}+\frac{c^2}{c^2+ab}\bigg)$ If $$a,b$$ and $$c$$ are positive reals, prove the inequality above.

I've proven $\sqrt[3]{\frac{(a+b)(b+c)(c+a)}{abc}}\geq 2$ So now I have to prove $\frac{a^2}{a^2+bc}+\frac{b^2}{b^2+ac}+\frac{c^2}{c^2+ab}\leq\frac{3}{2}$ I've tried to prove it but then realised that what I was trying to prove is wrong, can somebody help me?

Note by Gurīdo Cuong
1 year, 12 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

## Comments

Sort by:

Top Newest

Comment deleted Mar 06, 2016

Log in to reply

Oops, I believe you've made an error. This inequality is not trivial, because, you showed,
$$\dfrac{a^2}{a^2+bc}+\dfrac{b^2}{b^2+ac}+\dfrac{c^2}{c^2+ab} \ge \dfrac{(a+b+c)^2}{a^2+b^2+c^2+ab+bc+ca} \\$$ and $$\dfrac{3}{2} \ge \dfrac{(a+b+c)^2}{a^2+b^2+c^2+ab+bc+ca}$$
But this does not imply $$\dfrac{3}{2} \ge \dfrac{a^2}{a^2+bc}+\dfrac{b^2}{b^2+ac}+\dfrac{c^2}{c^2+ab}$$ which is not true for all (a, b, c).

- 1 year, 11 months ago

Log in to reply

Yes you are right. Do you know how to proceed with this problem?

- 1 year, 11 months ago

Log in to reply

I'm working on it, with some normalisation technique.

- 1 year, 11 months ago

Log in to reply

I proved it up to the lhs is greater than or equal to 427abc/( (a+b+c)^3+27abc)

- 1 year, 12 months ago

Log in to reply

can you show me how?

- 1 year, 12 months ago

Log in to reply

If you are on Slack, I can send you there.

- 1 year, 12 months ago

Log in to reply

yes I am, can you send it to me?

- 1 year, 12 months ago

Log in to reply

I am busy with my finals so I would send you it later make sure you check it

- 1 year, 12 months ago

Log in to reply

alright, good luck dude

- 1 year, 12 months ago

Log in to reply

Thanks, but please check it out later, but it may contain flaws and feel free to spot the mistakes

- 1 year, 12 months ago

Log in to reply

How did you proved the first part

- 1 year, 12 months ago

Log in to reply

First can be proved easily using AM-GM inequality.

- 1 year, 12 months ago

Log in to reply

Yes just saw that

- 1 year, 12 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...