TITLE: Is it possible that $|z+\sum_{i\not=1} a_i z^i| <1$ for some $a_i \in \mathbb{C}$ and for all $|z|=1$?
QUESTION [7 upvotes]: 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!

REPLY [9 votes]: Hint: Cauchy's integral formula tells us that $\dfrac{1}{2\pi i}\displaystyle\oint_{|z| = 1}\dfrac{P(z)}{z^2}\,dz = P'(0) = 1$. 
Now suppose $|P(z)| < 1$ for all $|z| = 1$. Do you see the contradiction?