Well, in particular I'm looking for an upper bound that's tighter than 1/(1-a).

Interesting. Are there any characterizations about how slowly the partial sums grow with respect to the growth of the partial sums of the geometric series?

I follow everything, except for the base step to establish $a_n\in\mathbb{R}$. I presume that it's because $f(z)/z^n$ has an accumulation of real values on values from the $[0,1]$ interval, and since the image of $[0,1]$ contains the accumulation point, $f(0)/0^n = a_n$ must be real valued as well.