A.M.>G.M. of four numbers

1,500

Solution 1

Hint: If you can use the AM-GM on 2 numbers, then you should use that.

First prove with this inequality on $(a,b)$ $$\frac{a+b}{2} \geq \sqrt{ab}$$

Then prove with this inequality on $(c,d)$ $$\frac{c+d}{2} \geq \sqrt{cd}$$

Then use the AM-GM on 2 numbers again on the terms above.

Solution 2

The inequality $\text{AM}>\text{GM}$ is equivalent to $\log\text{AM}>\log\text{GM}$, because the logarithm is a monotone increasing function.

Expanding $\log\text{AM}$ and $\log\text{GM}$,

\begin{align} \log\text{AM} & = \log \frac{a+b+c+d}{4} \\ \log\text{GM} & = \log(abcd)^{1/4} = \frac{\log a+\log b+\log c+\log d}{4} \end{align}

The result then follows because the logarithm is a concave function (see Jensen's inequality)

Solution 3

we have by AM-GM: $$\frac{\frac{a+b}{2}+\frac{c+d}{2}}{2}\geq \sqrt{\frac{a+b}{2}\frac{c+d}{2}}\geq \sqrt{\sqrt{ab}\sqrt{cd}}$$

1,500

Wisha

Updated on February 07, 2020

Comments

Wisha over 3 years

Prove that arithmetic mean of $4$ numbers is greater than geometric mean of the same $4$ numbers, i.e. prove that $$\dfrac{a+b+c+d}{4} > (abcd)^{\frac1{4}}$$
- 5xum almost 8 years
  
  Hi and welcome to the site! Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it?
- Ruben almost 8 years
  
  For correctness, I think you should add that a, b, c, d should all be different and non-negative.
Thomas Ahle almost 8 years

I don't think your expansion is correct. You need to divide everything by $4^4$.
NeilRoy almost 8 years

Sorry ... my bad!
wythagoras almost 8 years

This is true by Jensen's inequality.
Luis Mendo almost 8 years

@wythagoras Thanks. I've included that
Ruben almost 8 years

$ 4^4 $ > 24, making the proof incorrect. Would have been a nice proof though!
wythagoras almost 8 years

Well you could make all kinds of horrible estimates (e.g $a^4+b^4+c^4+d^4>4abcd$ by the rearrangement inequality) but I wouldn't recommend that.
Wisha almost 8 years

hi your method is interesing.
Wisha almost 8 years

can u pls tell me what is concave function?
Luis Mendo almost 8 years

It's a function that lies above any of its chords. It's the opposite of a convex function. Did you follow the link in my question?
NeilRoy almost 8 years

@MonKeePoo yeh thats right... but if we shift $4^4=256$ to rhs...the lhs will still be huge!
Jolien almost 8 years

Why is this clearly bigger than $abcd$? It could be the case that for example $4c^3d<0$.