proof of arithmetic-geometric-harmonic means inequality
For the Arithmetic Geometric Inequality, I claim it is enough to prove that if with then . The arithmetic geometric inequality for will follow by taking . The geometric harmonic inequality follows from the arithmetic geometric by taking .
So, we show that if with then by induction on .
Clear for .
Induction Step: By reordering indices we may assume the are increasing, so . Assuming the statement is true for , we have . Then,
by adding to both sides and subtracting . And so,
The last line follows since .
|Title||proof of arithmetic-geometric-harmonic means inequality|
|Date of creation||2013-03-22 15:09:37|
|Last modified on||2013-03-22 15:09:37|
|Last modified by||Mathprof (13753)|