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study the quantity $$ \\tau_{n,k}:=\\frac{|T_n^{(k)}(\\omega_{n,k})|}{T_n^{(k)}(1)}\\,, $$ where $T_n$ is the Chebyshev polynomial of degree $n$, and $\\omega_{n,k}$ is the rightmost zero of $T_n^{(k+1)}$.\n  Since the absolute values of the local maxima of $T_n^{(k)}$ increase monotonically towards the end-points of $[-1,1]$, the value $\\tau_{n,k}$ shows how small is the largest critical value of $\\,T_n^{(k)}\\,$ relative to its global maximum $\\,T_n^{(k)}(1)$.\n  In this paper, we improve and extend earlier estimates by Erd\\H{o}s--Szeg\\H{o}, Eriksson and Nikolov in several directions.\n  Firstly, ","authors_text":"Alexei Shadrin, Geno Nikolov, Nikola Naidenov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-17T06:43:32Z","title":"On the largest critical value of 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