A sharp inequality of Markov type for polynomials associated with Laguerre weight (Q5957489)

In this interesting and elegant paper, the authors obtain the sharp constant in a Markov inequality with the Laguerre weight. Let \(w(x)= e^{-x}\), \(x\in [0,\infty)\) and for \(n\geq 1\), let \(T_n(x;w)=x^n+\dots\) denote the monic weighted Chebyshev polynomial, so that NEWLINE\[NEWLINE\bigl\|T_n(x,w)w(x) \bigr \|_{L_\infty [0,\infty)}= \min_{P(x)= x^n+\dots}\bigl\|P(x)w(x) \bigr \|_{L_\infty [0,\infty)}.NEWLINE\]NEWLINE The authors show that for all polynomials \(P\) of degree \(\leq n\) NEWLINE\[NEWLINE\bigl\|(Pw)'\bigr \|_{L_\infty [0,\infty)} \leq{\biggl \|\bigl(T_n (\cdot,w)w\bigr)' \biggr\|_{L_\infty [0,\infty)} \over\bigl \|T_n(x,w)w(x) \bigr\|_{L_\infty [0,\infty)}} \bigl\|P(x)w(x)\bigr \|_{L_\infty [0,\infty)}.NEWLINE\]NEWLINE Of course, there is equality if \(P(x)+T_n(x,w)\). The authors also show that \(\|(T_n (\cdot,w)w)' \|_{L_\infty [0,\infty)}\) is attained at 0. The paper is essential reading for anyone interested in weighted polynomials, sharp constants and Markov inequalities.