As we know Bernstein's inequality for polynomials states that, if $P(z)$ is a polynomial of degree $n$ then
$$\max\_{|z|=1}|P'(z)|\leq n \max\_{|z|=1}|P(z)|. $$
There are results related to the reverse of the above inequality for some restricted class of polynomials. Are there any literature about the lower bound of $\frac{\max\_{|z|=1}|P'(z)|}{\max\_{|z|=1}|P(z)|} $ for any unrestricted non-constant polynomial $P(z)$? $



Lower bound for polynomials
For unrestricted polynomials of a given degree $n$, there is no lower bound. Indeed, consider
$$
P(z)=cz^n+1,
$$
with $|c|$ small. Then
$$
\frac{\|P'\|\_\infty}{\|P\|\_\infty}=\frac{|c|n}{1+|c|},
$$
which is arbitrarily small for $|c|$ small enough.


By the way, if the roots of $P$ lie in the closed disk, then there is the lower bound $n/2$ by a result of Turan:


P. Turan, Uber die Ableitung von Polynomen. Comp. Math. 7, 1939, 89–95.


