For $p=\sum_{k=0}^{n}a_k X^k \in \mathbb{C}[X]$, we note $||p||_{\infty}= \max\{|a_k| \: / \: 0 \leq k \leq n \}$ and $||p||_{\mathbb{D}} = \max \{ |p(z)| \: / \: |z| \leq 1 \} = \max \{ |p(z)| \: / \: |z| = 1 \}$.
Considering the sequence $p_n(z) = 1 + z + ... + z^n$, it is easy to see that we cannot find a constant $C >0$ such that $||\cdot||_{\mathbb{D}} \leq C ||\cdot||_{\infty}$, as $||p_n||_{\infty}=1$ and $||p_n||_{\mathbb{D}} \geq |p_n(1)| = n+1 \xrightarrow[n \to \infty]{}+ \infty$.
But what about the reverse inequality ? Can we find a constant $C>0$ such that $||\cdot||_{\infty} \leq C ||\cdot||_{\mathbb{D}}$ ?
It follows from Cauchy's differentiation formula that $$ |a_k| \le \max \{ |p(z)| : |z| = 1 \} $$ for all coefficients, so that in your notation $\|\cdot\|_{\infty} \leq \|\cdot\|_{\mathbb{D}}$.