proof of general means inequality
The definition is extended to the case by taking the limit ; this yields the weighted geometric mean
(see derivation of zeroth weighted power mean). We will prove the weighted power means inequality, which states that for any two real numbers , the weighted power means of orders and of positive real numbers , , …, satisfy the inequality
with equality if and only if all the are equal.
First, let us suppose that and are nonzero. We distinguish three cases for the signs of and : , , and . Let us consider the last case, i.e. assume and are both positive; the others are similar. We write and for ; this implies . Consider the function
with equality if and only if . By substituting and back into this inequality, we get
which is the inequality we had to prove. Equality holds if and only if all the are equal.
If , the inequality is still correct: is defined as , and since for all with , the same holds for the limit . The same argument shows that the inequality also holds for , i.e. that for all . We conclude that for all real numbers and such that ,
|Title||proof of general means inequality|
|Date of creation||2013-03-22 13:10:26|
|Last modified on||2013-03-22 13:10:26|
|Last modified by||pbruin (1001)|