Exact Markov inequalities for the Hermite and Laguerre weights

Denote by πnπn the set of all real algebraic polynomials of degree at most nn and let Un≔{e-x2p(x):p∈πn}Un≔{e-x2p(x):p∈πn}, Vn≔{e-xp(x):p∈πn}Vn≔{e-xp(x):p∈πn}. We prove the following exact Markov inequalities:‖u(k)‖R⩽‖u*,n(k)‖R‖u‖R,∀u∈Un,∀k∈N,and‖v(k)‖R+⩽‖v*,n(k)‖R+‖v‖R+,∀u∈Vn,∀k∈N,where ‖·‖R‖·‖R (‖·‖R+‖·‖R+) is the supremum norm on RR (R+≔[0,∞)R+≔[0,∞)) and u*,nu*,n (v*,nv*,n) is the Chebyshev polynomial from UnUn (VnVn).