If it holds that , then find the value of
From the Power-Mean Inequality we have that
So, multiplying by we have that
Taking now in hand the left hand side, we know that
So, we know that . Doing the manipulations on both sides we get that
, hence , Q.E.D.
2nd solution (An idea by Vo Quoc Ba Can):
The inequality is symmetric on . So, we only need to find the maximum of those two constants for the values of which
, that is . S
o, plugging on the above inequality the value
we get the desired maximum result, Q.E.D.