estimates for Abel's theorem

Question

Suppose $a_1,a_2,\dots$ is a sequence of real numbers with $\displaystyle\sum_{n=1}^\infty a_n =s<\infty$. For $0<z<1$, define $f(z):=\displaystyle\sum_{n=1}^\infty z^n a_n$. By Abel's theorem, we know that $\displaystyle\lim_{z\uparrow 1}f(z)=s$. Is it possible to bound $f(z)$ in terms of $z$ and $s$ without knowing anything about $f$?

Answer 1

Looks doubtful. Consider $f_n(z) = nz - z^{n+1} - z^{n+2} -\cdots- z^{2n}.$ Then $s=0$ for each $f_n.$ But at the fixed point $z=1/2,$ we have $f_n(1/2) \ge n/2 - n/2^{n+1} \to \infty.$