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)
(x
0), x
0 real, k = 1, … n – 1 ([8]).
Suppose φ(x) ≧ 0 on [–1, 1] and (1) is generalized to
as suggested by Rahman [4] (polynomials with curved majorants), what can then be said about the analogue of (3) or similar extremal problems?