Let $f(x,s)=\sum_{n=0}^\infty a_n(s)x^n$ where $|a_n(s)|\le1$ is a bounded function theory. Suppose for every $|x|<1$, $f(x,s)$ is Holder-$\alpha$ for $s$-variable, i.e. $|f(x,s_1)-f(x,s_2)|\le C|s_1-s_2|^\alpha$. Then can we have $a_n(s)$ is Holder-$\alpha$ for every $n$?

Equivalently, can we have, for every $z\in\mathbb C$, $|z|<1$, $f(z,s)$ is of Holder-$\alpha$ for $s$?