I wonder that whether there exists a complex polynomial of the form $$ P(z)= z+\sum_{i\not=1} a_iz^i, a_i,z\in \mathbb{C},$$ (i.e. its first order term has coefficient 1) s.t. its modulus is less than 1 on $|z|\leq 1$, i.e.

$$ |P(z)|<1,\forall |z|\leq 1. $$

I know by modulus maximum principle, we only need to find $$ |P(z)| <1, \forall |z|=1.$$

Does there exist such polynomial? I have tried the chebyshev polynomial but didn't get through. Any ideas?

Any help would be appreciated!