A well-known result of Chebyshev is that if pn ∊ Pn, (Pn is the set of polynomials of degree at most n) and
(1)![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00033046/resource/name/S0008414X00033046_eqn1.gif?pub-status=live)
then an(pn), the leading coefficient of pn, satisfies
(2)![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00033046/resource/name/S0008414X00033046_eqn2.gif?pub-status=live)
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)![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00033046/resource/name/S0008414X00033046_eqn3.gif?pub-status=live)
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 to
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00033046/resource/name/S0008414X00033046_eqn4.gif?pub-status=live)
as suggested by Rahman [4] (polynomials with curved majorants), what can then be said about the analogue of (3) or similar extremal problems?