Let $n \in \Bbb N$ be a positive integer and let $a_1,...,a_n \in \Bbb C$. Let $D \subset \Bbb C$ be a compact subset of $\Bbb C$ and let $M \in \Bbb R_+$
I would like to know if there exist $z_1,...,z_n \in D$ such that $$\left|\sum_{j=1}^n z_ja_j\right|>M.$$ Thanks
No, that is false.
As $D$ is compact, it is also limited, let's say a bound for $|z|$ for any $z\in D$ is $K$. Consider also $L:=\max_i(|a_i|)$. So we have that for any $z_1,z_2,\dots ,z_n\in D$:
$$|\sum_{i=1}^{n}z_ia_i|\leq\sum_{i=1}^{n}|z_ia_i|\leq\sum_{i=1}^{n}KL=nKL$$
So we have that such statement is not valid for any $M>nKL$.
Take your compact subspace to be the unit disc. Then $$\left|\sum a_iz_i\right|\leq \sum \left|a_i\right|\leq n\max_i{\left|a_i\right|}$$ by Cauchy Schwartz. So if $a_k$ is the largest element in the sequence by absolute value, then setting. $M=n|a_k|+1$ gives a counter example.