Let $P$ be a polynomial with real coefficients, and degP=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 at bounding $|P'(x)|$ around $0$. Any reference would be very much appreciated.