Hi,

firstly, we can notice that is concave on (by 2 derivations). So, if a,b, and c are in , we can apply the inegality of convexity, and the result is obvious.

Now, if at least one of the 3 numbers is . Then, the 2 others are . Let's say that and and , for example.

Then and and (because has a maxima in and is decreasing on .

finally,

is there a mistake ?