Computational Methods and Function Theory 4 (2004), No. 1, 183--188
Copyright Heldermann Verlag 2004
Richard Fournier
fournier@crm.umontreal.ca , Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal, Qc H3C 3J7, Canada.
Let $\mathbb{D}$ denote the unit disc of the complex plane and $\mathcal{P}_n$ the class of polynomials of degree at most $n$ with complex coefficients. We give a new proof of $$ \left| p(z) - \frac{zp'(z)}{n}\right| + \left| \frac{zp'(z)}{n}\right| \leq |p|_{\mathbb{D}}, \qquad z \in \overline{\mathbb{D}},\, p \in \mathcal{P}_n, $$ together with a complete discussion of all cases of equality. We also discuss an extension, due to Ruscheweyh, of the above inequality.
Keywords: Complex polynomials, Bernstein inequality, generalizations of Bernstein inequality.
MSC 2000: Primary: 41A17.
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