Let $k\geq 2$, and let $P\_k$ be a sequence of polynomials, such that:


1. $P\_k=\sum\_{n=2}^{k+1}a\_{n,k}X^n \in \mathbb{Q}[X]$, $a\_{2,k}\neq 0$, $\deg P\_k \leq k+1$, and consider $P\_k :[0,1]\rightarrow \mathbb{R}$ as a real valued function.
2. $P\_k(1)=\frac{1}{k(k+1)}$ and $\mid a\_{n,k}\mid < \frac{2}{k}$, for all $n$ and $k$.
3. for any $a<1$, $\max\_{x\in[0,a]} \mid P\_k(x)\mid \rightarrow 0$, as $k\rightarrow +\infty$.


**Questions** Assume we are given a sequence of polynomials $(P\_k)\_{k\geq 2}$ satisfying conditions 1,2,3 from above. Do we have $\max\_{x\in[0,1]} \mid P\_k(x)\mid =O(\frac{1}{k^2})$, for $k$ large enough?


I am sorry if this question is trivial, however analysis is not my field. This question is related to a [question that I asked before](https://mathoverflow.net/questions/368725/m%C3%B6bius-function-and-polynomials).



Upper bound of a uniformly converging sequence of polynomials
Take
$$
 P\_k(x)=\frac{x^2}{k(k+1)}+\frac{x^2(1-x)}k.
$$


