Let $P(z)=\sum_{k=0}^na_kz^k $ be a polynomial of degree $n$ and $z_k (1\leq k\leq n)$'s be $n$th roots of $-1$. Then when $\theta=0$ the inequality $$|P'(e^{i\theta})|\leq \frac{4}{n}\left|\sum_{k=1}^nP(e^{i\theta}z_k)\frac{z_k}{(z_k-1)^2}\right|$$ does not hold. Suppose we modify it as $$|P'(e^{i\theta})|\leq \frac{4}{n}\sum_{k=1}^n\left|\frac{z_k}{(z_k-1)^2}\right| \max_{|z|=1}|P(z)|.$$ Is this modification true?
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$\begingroup$ What is the summands in RHS corresponding to $z_k=1$? $\endgroup$– Fedor PetrovSep 3, 2018 at 6:02
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3$\begingroup$ This is wrong. Choose $P(z)=z^n+1$ and $\theta=0$. Then the RHS vanishes by construction. The LHS does not vanish since $P'(z)=nz^{n-1}$. $\endgroup$– Philipp LampeSep 3, 2018 at 7:47
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4$\begingroup$ Numerical experiments suggest $$\sum_{k=1}^n\left\lvert\frac{z_k}{(z_k-1)^2}\right\rvert=\sum_{k=1}^n\left\lvert\frac{1}{(z_k-1)^2}\right\rvert=\frac{n^2}{4}.$$ This substitution transforms the statement into a Theorem of Bernstein. $\endgroup$– Philipp LampeSep 3, 2018 at 14:55
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3$\begingroup$ @PhilippLampe: I suggest that you turn your comment into an answer, so that this question can be closed. $\endgroup$– GH from MOSep 3, 2018 at 19:36
1 Answer
Terms that are very similar to the terms given here can be found in an article by Aziz and Mohammad, see [1].
Suppose that $a\neq -1$ is a complex number of modulus $1$. Let us denote the roots of the polynomial $z^n+a$ by $z_1,\ldots,z_n$. The authors show that \begin{align*} \left\lvert\frac{(1+a)^2}{na}\right\rvert\sum_{k=1}^n\left\lvert\frac{z_k}{(z_k-1)^2}\right\rvert=n. \end{align*} We put $a=1$. Then the right hand side in the considered inequality becomes $n\operatorname{max}_{\lvert z\rvert=1}\lvert P(z)\rvert$.
The desired inequality follows from a Theorem of Bernstein, see [2], which states $$\operatorname{max}_{\lvert z\rvert=1}\vert P'(z)\rvert \leq n\operatorname{max}_{\lvert z\rvert=1}\lvert P(z)\rvert.$$
- A. Aziz and Q. G. Mohammad: Simple Proof of a Theorem of Erdős and Lax, Proc. Amer. Math. Soc. 80, no. 1 (1980), 119--122
- S. Bernstein: Sur la limitation des dérivées des polynomes, C. R. Math. Acad. Sci. Paris 190 (1930), 338--340