It is (relatively) easy to show that a polynomial $f(x)=\sum_{n=0}^na_nx^n$ with coefficients $a_n\in\{-1,1\}$ cannot have a multiple root $x_0$ in the interval $(0,1):=\{x\in\mathbb R:0<x<1\}$. I am interested in the same fact for analytic functions.

**Question.** *Let $f(x)=\sum_{n=0}^\infty a_nx^n$ a series with coefficients $a_n\in\{-1,1\}$ such that $\sup_{m\in\mathbb N}|\sum_{n=0}^ma_i|<\infty$. Can the analytic function $f$ have a multiple root in the interval $(0,1)$?*