Let $\bar D$ denote the closed unit disc in the complex plane. Consider the function $f:\bar D\longrightarrow \mathbb{C}$, defined as $f(z)=z$ for all $z\in \bar D$.

Let $n\in \mathbb{N}$. For $1\leq i\leq n$, let $p_i:\bar D\longrightarrow \mathbb{C}$ be monic complex polynomials on $\bar{D}$, such that $p_i$’s have no roots in $\bar{D}$ and $c_i\in\mathbb{C}$ such that $$f=\sum_{i=1}^{n}c_ip_i.$$ Can we say that for any such representation of $f$, the value $$K=|c_1|\|p_1\|_\infty+|c_2|\|p_2\|_\infty+\cdot+|c_n|\|p_n\|_\infty\geq 2?$$

$\|.\|_\infty$ denotes the uniform norm, defined as $\|p\|_\infty=sup\{|p(t)|: t\in \bar D\}$ for any complex polynomial $p$ on $\bar{D}$.