A.M.>G.M. of four numbers
Solution 1
Hint: If you can use the AMGM 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 AMGM 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 AMGM: $$\frac{\frac{a+b}{2}+\frac{c+d}{2}}{2}\geq \sqrt{\frac{a+b}{2}\frac{c+d}{2}}\geq \sqrt{\sqrt{ab}\sqrt{cd}}$$
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Wisha
Updated on February 07, 2020Comments

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 yearsHi 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 yearsFor correctness, I think you should add that a, b, c, d should all be different and nonnegative.


Thomas Ahle almost 8 yearsI don't think your expansion is correct. You need to divide everything by $4^4$.

NeilRoy almost 8 yearsSorry ... my bad!

wythagoras almost 8 yearsThis 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 yearsWell 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 yearshi your method is interesing.

Wisha almost 8 yearscan u pls tell me what is concave function?

Luis Mendo almost 8 yearsIt'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 yearsWhy is this clearly bigger than $abcd$? It could be the case that for example $4c^3d<0$.