On the largest critical value of $T_n^{(k)}$

Abstract

We 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 $T_n^{(k+1)}$. 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 $T_n^{(k)}(1)$. In this paper, we improve and extend earlier estimates by Erdős--Szegő, Eriksson and Nikolov in several directions. Firstly, we show that the sequence $\{{\tau }_{n,k}{\}}_{n=k+2}^{\infty }$ is monotonically decreasing in $n$, hence derive several sharp estimates, in particular $${\tau }_{n,k}\le \{\begin{array}{llc}{\tau }_{k+4,k}=\frac{1}{2k+1}\frac{3}{k+3},& n\ge k+4{\tau }_{k+6,k}=\frac{1}{2k+1}(\frac{5}{k+5}{)}^2{\beta }_k,& n\ge k+6,\end{array}$$ where ${\beta }_k<\frac{2+\sqrt{10}}{5}\approx 1.032$. We also obtain an upper bound which is uniform in $n$ and $k$, and that implies in particular $${\tau }_{n,k}\approx (\frac{2}{e}{)}^k,n\ge k^{3/2};{\tau }_{n,n-m}\approx (\frac{em}{2}{)}^{m/2}{n}^{-m/2};{\tau }_{n,n/2}\approx (\frac{4}{\sqrt{27}}{)}^{n/2}.$$ Finally, we derive the exact asymptotic formulae for the quantities $$ τ_k^{*} := \lim_{n\to\infty}τ_{n,k} \quad \mbox{ and }\quad τ_m^{**} := \lim_{n\to\infty} n^{m/2} τ_{n,n-m}\,, $$ which show that our upper bounds for ${\tau }_{n,k}$ and ${\tau }_{n,n-m}$ are asymptotically correct with respect to the exponential terms given above.

19 pages, 2 figures