Computational Methods and Function Theory 2 (2002), No. 2, 397--414
Copyright Heldermann Verlag 2002
Dimiter Dryanov
dryanovd@dms.umontreal.ca, University of Sofia, Department of Mathematics, James Boucher 5, 1164 Sofia, Bulgaria.
Richard Fournier
fournier@dms.umontreal.ca, Université de Montréal, Département de Mathématiques et de Statistique, Montréal H3C 3J7, Canada.
Let $\mathbb{D}$ be the unit disk in the complex plane $\mathbb{C}$ and $$ \Vert p\Vert:= \max_{z \in \partial \mathbb{D}}\vert p(z)\vert, $$ where $p(z)=\sum_{k=0}^{n}a_k(p)z^k$ is a polynomial of degree at most $n$ and $a_k(p) \in \mathbb{C}$. The following sharpening of Bernstein's inequality $$ \Vert p^{\prime}\Vert+\frac{2n}{n+2}\vert a_0(p)\vert\le n\Vert p\Vert $$ has been proved by Ruscheweyh. Our main contribution concerns the case of equality which has remained unsolved since 1982. We prove another inequality of Bernstein type that leads to an improvement of the upper bound for $\Vert p^{\prime}\Vert$ under some additional condition.
Keywords: Polynomials, bound-preserving operators, Bernstein type inequalities.
MSC 2000: 41A17, 30A10.
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