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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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##### References:
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