Markov-type inequalities on certain irrational arcs and domains
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Let Pnd denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set K Rd set PK = supxK P(x) . Then the Markov factors on K are defined by Mn(K): = max { DPK: P Pnd, PK 1, Sd-1}. (Here, as usual, Sd-1 stands for the Euclidean unit sphere in Rd.) Furthermore, given a smooth curve Rd, we denote by DTP the tangential derivative of P along (T is the unit tangent to ). Correspondingly, consider the tangential Markov factor of given by MnT (): = max {DTP : P Pnd, P 1}. Let := {(x, x) : 0x1}. We prove that for every irrational number > 0 there are constants A, B > 1 depending only on such that An MnT() Bn for every sufficiently large n. Our second result presents some new bounds for Mn(), where :={(x, y) 2 : 0x1; 1/2xy2x} (d=2, > 1). We show that for every > 1 there exists a constant c > 0 depending only on such that Mn ()nclogn. 2004 Elsevier Inc. All rights reserved.