Q:

Inequality on the unit circle

Is $$\left|np(z)+(\alpha-1)zp'(z)\right|
\\\geq\left|np(z)+(\alpha-z)p'(z)\right|$$ on $|z|=1,$ and $|\alpha|\geq 1,$
where $p(z)$ is a polynomial of degree $n?$ 

A:

Counterexample: $n=1$, $p(z)=z+1$, $\alpha=1$, $z=-1$.  Then
$$ |np(z)+(\alpha-1)zp'(z)| = 0 < 2 = |np(z)+(\alpha-z)p'(z)| $$

