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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.

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The following paper seems to fully answer the question: http://arxiv.org/pdf/1205.0846.pdf.

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Up to http://www.math.tamu.edu/~terdelyi/papers-online/erdos.pdf , "For example, it has been observed by Bernstein [B58] that Markov’s inequality for monotone polynomials is not essentially better than for arbitrary polynomials ".

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  • $\begingroup$ Thanks for your answer! As I said I am especially interested at bounding $P'$ near $0$. There is still hope for improvements at that aspect. $\endgroup$
    – NoamL
    Commented Jun 11, 2014 at 16:36
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    $\begingroup$ The hope is there only as long as you restrict yourself to the formulations and never look into the proofs. However, if you take any explicit convolution type polynomial approximation operator (like replanting the function to the circle and taking the Fejer sum, say), you will realize that the width of its kernel support is $1/d$ for all practical purposes, so an attempt to approximate the function that is $0$ for $x<0$ and $1$ for $x>0$ will result in a non-decreasing polynomial of degree $d$ that jumps from nearly $0$ to nearly $1$ within the distance of order $1/d$. $\endgroup$
    – fedja
    Commented Jun 12, 2014 at 3:05

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