Proving a tricky inequality

Hi, this might be very easy but I am unable to prove it.

(From now on, the summations will be defined over the set {R,P,M})

Let ; where

and and

and

and also, define

Now the claim is:

1) If ; then

2) If ; then

Alpha's being in between 0 and 1 and summing to 1 seems to be crucial for this fact, since I tried around 50 examples in excel and it holds if thats the case.

Any help is appreciated.

p.s. forgot to add, we also have

edit: I got it, it's not tricky at all actually. If anyone's interested, just post a reply here and I'll write the solution.