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On the finite-increment theorem for complex polynomials. (English. Russian original) Zbl 1246.30003
Math. Notes 88, No. 5, 647-654 (2010); translation from Mat. Zametki 88, No. 5, 673-682 (2010).
The main results of the article under review are two reverse inequalities of the finite increment formula for polynomials with complex coefficients. Let \(P(z)\in \mathbb{C}[z] \) be a complex polynomial of degree at most \( n\) and let \(z_1, z_2 \in \mathbb{C}\).
1. There exists a point \(\zeta \in \mathbb{C}\) such that \(P(\zeta) = P(z_2)\) and \[ |P(z_1) - P(z_2)| \geq \frac{4^{\frac{1}{n}-1}}{n} |P'(z_1)||z_1-\zeta|. \] 2. Let \(\gamma\) be the connected component of the pre-image of the interval \([P(z_1), P(z_2)]\) under the mapping \(P\) containing the point \(z_1\). Then, for any point \(\zeta \in \gamma\), the following inequality holds: \[ |P(z_1) - P(z_2)| \geq \frac{1}{n^2}|P'(z_1)||z_1-\zeta |. \tag{1} \] The equality in (1) is attained for the Chebyshev polynomial \(T_n(z)\), the points \(z_1 = 1, \zeta= -1\), and for an arbitrary point \(z_2\) satisfying the condition \(T_n(z_2) = -1\).

MSC:
30-XX Functions of a complex variable
39B32 Functional equations for complex functions
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