Well, why not just calculate its first derivative and equate it to zero.

if there are no stationary points inside (1/2,1) then f_n is strictly decreasing (btw, you meant ).

Yes this maybe also tough to calculate, you can let Maple,Matlab,Mathematica to calculate this. (I prefer Maple, it seems more quick than Mathematica).

Have you tried equating f' to zero, and see if there are no solutions?

Even if there is a solution, you need to check that at these stationary points the following cases are met:

1. we don't have one stationary point which its f_n value is negative and another two stationary points which f_n value is positive cause in this case we have two zeros(actually 3).

and other options, which I can't account for now cause I haven't calculated its first derivative, but the principle in this way to look for stationary and excluding any possibility for more than one zero.