Suppose $h:\mathbb{T}\to\mathbb{C}$ is a complex function on the unit circle with a valid Laurent series about zero: $$ h(z) = \sum_{k=-\infty}^\infty c_k z^k\qquad\text{for }|z|=1 $$ The coefficients $c_k$ satisfy certain constraints, and I would like to translate them to equivalent constraints on the function $h(z)$.
First question: is it true that the following statements are equivalent?
The coefficients satisfy $\sum_{k=-\infty}^\infty |c_k| \le 1$
The function satisfies $|h(z)| \le 1$ for all $|z| = 1$.
It's clear that 1 implies 2 because $|h(z)| \le \sum_k|c_k||z^k| = \sum_k|c_k|$ by the triangle inequality. Is the converse even true? How to prove it?
Second question: In addition to item 1 above, further assume that the coefficients $c_k$ are all real and nonnegative. Is there a nice way to represent this more complicated constraint on the coefficients as a constraint on $h$ ?