Let $P$ be a polynomial with real coefficients, and $\deg P=d$. There is Markov-Berenstein inequality: $P′(x)\leq\frac{d\|P\|}{\sqrt{1-x^2}}$,where $\|P\|=\max\_{|x|\le1} |P(x)|$ and $|x|\leq1$. Are there any improvements when $P$ is increasing in the interval $[-1,1]$. I am particularly interested in bounding $|P'(x)|$ around $0$. Any reference would be very much appreciated.



Markov-Bernstein like inequalities for monotone polynomials
The following paper seems to fully answer the question:
<http://arxiv.org/pdf/1205.0846.pdf>.


