Markov-type inequalities on certain irrational arcs and domains

Let denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set KR^d set Then the Markov factors on K are defined by (Here, as usual, S^d-1 stands for the Euclidean unit sphere in R^d.) Furthermore, given a smooth curve ΓR^d, we denote by D_TP the tangential derivative of P along Γ (T is the unit tangent to Γ). Correspondingly, consider the tangential Markov factor of Γ given by Let . We prove that for every irrational number α>0 there are constants A,B>1 depending only on α such that for every sufficiently large n.Our second result presents some new bounds for M_n(Ω_α), where (d=2,α>1). We show that for every α>1 there exists a constant c>0 depending only on α such that M_n(Ω_α)n^clogn.