Asymptotically sharp inequalities for polynomials involving mixed Laguerre norms

The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial and its derivative is taken in L2L2 on the half-line with the weight tαe−ttαe−t and tβe−ttβe−t, respectively. Under the assumption that β−αβ−α is larger than the order of the derivative, we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity, thus proving a conjecture raised by Böttcher and Dörfler in 2010 [4].