Dual mean value problem for complex polynomials. (English) Zbl 1208.30005

Smale proved that, for a critical point \(\zeta\) of a polynomial \(P\) of degree \(d\geq 2\) over \(\mathbb{C}\) \((P(z)= 0)\), there exists a noncritical point \(z\in\mathbb{C}\) such that \(\left|{P(\zeta)- P(z)\over \zeta-z}\right|\leq 4|P'(g)|\). His conjecture that the factor 4 can be replaced by \(1\) or even by \(1-1|d\) has not been solved. The authors prove a dual problem: For a noncritical point \(z\) of a polynomial \(P\) of degree \(d\geq 2\), there exists a critical point \(\zeta\) of \(P\) such that \({|P(z)|\over d4^d}\leq \left|{P(\zeta)- P(z)\over \zeta-z}\right|\).
Reviewer: Yu Jiarong (Wuhan)


30C10 Polynomials and rational functions of one complex variable
30C55 General theory of univalent and multivalent functions of one complex variable
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