Proving a tricky inequality

• Jan 14th 2012, 10:14 AM
ichoosetonotchoosetochoos
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 $\phi =\phi _{R}\alpha _{R}+\phi _{M}\alpha _{M}+\phi _{P}\alpha _{P}$ ; where

$\forall i\in \left \{ R,P,M \right \} \phi _{i}\in (0,\infty )$
and $\alpha _{i}\in (0,1 )$ and $\sum \alpha _{i}=1$
and $y_{i}\in (0,\infty )$

and also, define $y=\sum \alpha _{i}y_{i}$

Now the claim is:

1) If $\phi _{R}> \phi > \phi _{P}$; then $\frac{\sum \alpha _{i}\phi _{i}y_{i}}{\phi }> y$
2) If $\phi _{R}< \phi < \phi _{P}$; then $\frac{\sum \alpha _{i}\phi _{i}y_{i}}{\phi }< y$

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
$y_{R}> y_{M}>y_{P}$

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.