The sharp constant in Markov's inequality for the Laguerre weight

We prove that the polynomial of degree $n$ that deviates least from zero in the uniformly weighted metric with Laguerre weight is the extremal polynomial in Markov's inequality for the norm of the $k$th derivative. Moreover, the corresponding sharp constant does not exceed
$$ \frac{8^kn!\,k!}{(n-k)!\,(2k)!}. $$
For the derivative of a fixed order this bound is asymptotically sharp as $n\to\infty$.
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