A refinement of an inequality of S. Bernstein. (English) Zbl 0691.30007

Aziz, Abdul

J. Math. Anal. Appl. 144, No. 1, 226-235 (1989).

Let p(z) be a polynomial of degree n. Recently, C. Frappier, Q. I. Rahman and S. Ruscheweyh [Trans. Am. Math. Soc. 288, 69-99 (1985; Zbl 0567.30006)] have shown that \[ (*)\quad_{| z| =1}| p'(z)| \leq n_{1\leq k\leq 2n}| p(e^{ik\pi /n})|. \] In the paper under review, the author shows that the bound in (*) can be considerably improved. In addition, the author obtains a sharp lower bound for the maximum of \(| p'(z)|\) on \(| z| =1\).

Reviewer: G.Csordas

MSC:

30C10

Polynomials and rational functions of one complex variable

Keywords:

Bernstein inequality

Citations:

Zbl 0567.30006

References:

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