I found a very useful inequality involving a polynomials of degree , having all its zeros in the unit disk from the following paper, doi:10.1016/j.jmaa.2009.07.049, published in J. Math. Anal. Appl.
Statement. If is a polynomial of degree , having all its zeros in the disk , then
Proof. Since all the zeros of lie in $latex|z| \leq 1$. Hence if are the zeros of , then for all . Clearly,
for every point , which is not a zero of . Note that
for every point , . Hence
for and this completes the proof.