On Polynomials with Curved Majorants

Abstract

A well-known result of Chebyshev is that if pn ∊ Pn, (Pn is the set of polynomials of degree at most n) and(1)then an(pn), the leading coefficient of pn, satisfies(2)with equality holding only for pn = ±Tn, where Tn is the Chebyshev polynomial of degree n. (See [6, p. 57].) This is an example of an extremal problem in which the norm of a given linear operator on Pn is sought. Another example is A. A. Markov's result that (1) implies that(3)There are also results for the linear functionals pn(k)(x0), x0 real, k = 1, … n – 1 ([8]).Suppose φ(x) ≧ 0 on [–1, 1] and (1) is generalized toas suggested by Rahman [4] (polynomials with curved majorants), what can then be said about the analogue of (3) or similar extremal problems?