In the course of generalizing the Bernstein-Markov theorem to normed space, Harris came up with the following question.
Suppose that $p$ is a real polynomial satisfying $|p(x)| \leq (1+|x|)^d$. How large can $|p'(0)|$ be?
Harris obtained a bound of $O(d\log d)$, and Révész and Sarantopoulos gave a matching lower bound $\Omega(d\log d)$.
While attempting a different generalization of the theorem, we encountered the following question:
Suppose that $p$ is a real polynomial satisfying $|p(x)| \leq \max(1,|x|^d)$. How large can $|p'(1)|$ be?
Our conjecture is that $|p'(1)| = O(d)$. So far we have determined the optima for $d=0,1,2$, which are $0,1,1+\sqrt{2}$, the latter bound attained by $$ \left(\frac{1}{2} + \frac{1}{2\sqrt{2}}\right) (x^2 - 1) + \frac{1}{\sqrt{2}} x. $$
