I am wondering if there is a closed form expression for
$\sum_{s=0}^N {N\choose s} q^s(1-q)^{N-s}\ln[x+s(y-x)/N]-\sum_{s=0}^{N-1} {N-1\choose s} q^s(1-q)^{N-1-s}\ln[x+s(y-x)/N]$
where y>x>0 are some constants. Any help is greatly appreciated.
More specifically, I want to show that this expression is decreasing in N (which I am able to verify numerically).