Question P. Can a polynomial $\sum_{n=0}^ma_nx^n$ with coefficients $a_n\in\{-1,1\}$ have a multiple root in the interval $(\tfrac12,1)$?

Also I am interested in a similar question for analytic functions.

Question A. 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 $(0,1)$?

Question P. Can a polynomial $P(x)=\sum_{n=0}^ma_nx^n$ with coefficients $a_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?

Also I am interested in a similar question for analytic functions.

Question A. 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 $(0,1)$?

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 $(0,1)$?

Question P. Can a polynomial $\sum_{n=0}^ma_nx^n$ with coefficients $a_n\in\{-1,1\}$ have a multiple root in the interval $(\tfrac12,1)$?

Also I am interested in a similar question for analytic functions.

Question A. 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 $(0,1)$?