Proof for Inequality

  • Thread starter Undirrast
  • Start date
  • #1
15
0

Homework Statement


If a, b and c are distinct positive numbers, show that
[itex]
2 (a^3 + b^3 + c^3) > a^2b + a^2c + b^2c + b^2a + c^2a + c^2b
[/itex]

Homework Equations





The Attempt at a Solution


I have tried to expand from [tex](a+b+c)^3 > 0[/tex], also tried [tex](a+b)^3 + (b+c)^3 + (c+a)^3 > 0[/tex], and then [tex]\frac{a+b+c}{3} > \sqrt[3]{abc}[/tex]. But with no avail. I guess I'm heading in the wrong direction?
 
Last edited:

Answers and Replies

  • #2
hunt_mat
Homework Helper
1,741
25
Try playing with:
[tex]
a(a-b)^{2}+b(b-c)^{2}+c(c-a)^{2}>0\quad a(a-c)^{2}+b(b-a)^{2}+c(c-b)^{2}>0
[/tex]
along with other things like this to get the answer.
 
  • #3
15
0
Hmm, thanks for your tip off, but I still can't seem to make it...

The 2 is always appearing on the right side of the inequality, but as of the question it's on the right side, if I expand [tex](a-b)^2[/tex] the 2 sticks together with ab instead of [tex]a^3[/tex]...
 
  • #4
disregardthat
Science Advisor
1,854
33
Have you heard of the arithmetic mean and geometric mean and their relationship? Try using the AM-GM inequality (it is extremely useful in situations like this, look it up!). Rearrange the left hand side as such: [tex]\frac{a^3+a^3+b^3}{3} +...[/tex] Apply the AM-GM inequality at each term and note exactly when you have equality. An alternative way is to use the rearrangement inequality. Assuming without loss of generality a >= b >= c, the inequality follows immediately, and strict inequality follows from a>c.

Symmetric inequalities as these can very often be solved using the inequalities QM-AM-GM-HM (quadratic, arithmetic, geometric and harmonic means), Rearrangement, Chebychev, Cauchy-Schwarz and Jensens; or a combination of these. Note also that this particular inequality is homogenous, so you can assume e.g. a+b+c=1, or a^3+b^3+c^3=1. That might also be useful in some situations.
 
Last edited:
  • #5
15
0
Have you heard of the arithmetic mean and geometric mean and their relationship? Try using the AM-GM inequality (it is extremely useful in situations like this, look it up!). Rearrange the left hand side as such: [tex]\frac{a^3+a^3+b^3}{3} +...[/tex] Apply the AM-GM inequality at each term and note exactly when you have equality.
I think I got it, by using [tex]\frac{a+b}{2}>\sqrt{ab}[/tex], etc and by multiplying a and b respectively then adding up all together made it. Thanks for all the tips!
 

Related Threads on Proof for Inequality

  • Last Post
Replies
12
Views
776
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
936
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
2
Views
2K
Replies
22
Views
2K
Replies
6
Views
1K
Top