Just off the top of my head, but since the change from to , and vice-versa, occurs at 0 and 1, I would guess that the only places we could have "=" would be at 0 or 1 or both.
I have seen a theorem:
If a_{1},a_{2},........,a_{n} are n positive numbers such that all are not equal, and m is any rational number not equal to 1 and 0,
then
{(a_{1}+a_{2}+........+a_{n})/n}^{m} ≤ or ≥ {(a_{1}^{m}+a_{2}^{m}+........+a_{n}^{m})/n} according as m doesn't or does lie between 0 and 1
I can't find out, when do they get equal?