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On Polynomials with Curved Majorants

  • D. J. Newman (a1) and T. J. Rivlin (a2)

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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?

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References

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1. Dzjadyk, V. K., On a problem of Chebyshev and Markov, Analysis Mathematica 3 (1977), 171175.
2. Lachance, M., E. B., Saft” and Varga, R. S., Inequalities for polynomials with a prescribed zero, Math. Z. 168 (1979), 105116.
3. Pierre, R. and Rahman, Q. I., On a problem of Turân about polynomials, Proc. Amer. Math. Soc. 56 (1976), 231238.
4. Rahman, Q. I., On a problem of Turân about polynomials with curved majorants, Trans. Amer. Math. Soc. 163 (1972), 447455.
5. Rahman, Q. I., Addendum to “On a problem of Turân about polynomials with curved majorants'', Trans. Amer. Math. Soc. 168 (1972), 517518.
6. Rivlin, T. J., The Chebyshev polynomials (John Wiley & Sons, New York, 1974).
7. Schur, I., Ûber das Maximum des absoluten Betrages eines Polynoms in einem gegebenen Interval, Math. Z. 4 (1919), 271287.
8. Voronovskaja, E. V., The functional method and its applications Translations of Mathematical Monographs 28 (Amer. Math. Soc, Providence, 1970).
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On Polynomials with Curved Majorants

  • D. J. Newman (a1) and T. J. Rivlin (a2)

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