# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros (World Sci. Publ., Singapore, 1994). · Zbl 0848.26001  P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, in Grad. Texts in Math. (Springer-Verlag, New York, NY, 1995), Vol. 161. · Zbl 0840.26002  Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, in London Math. Soc. Monogr. (N. S.) (Oxford Univ. Press, Oxford, 2002), Vol. 26. · Zbl 1072.30006  S. Smale, ”The fundamental theorem of algebra and complexity theory,” Bull. Amer.Math. Soc. (N. S.) 4(1), 1–36 (1981). · Zbl 0456.12012 · doi:10.1090/S0273-0979-1981-14858-8  V. Dubinin and T. Sugawa, ”Dual mean value problem for complex polynomials,” Proc. Japan Acad. Ser. A Math. Sci. 85(9), 135–137 (2009). · Zbl 1208.30005 · doi:10.3792/pjaa.85.135  W. K. Hayman, Multivalent Functions, in Cambridge Tracts in Math. (Cambridge Univ. Press, Cambridge, 1994), Vol. 110.  V. N. Dubinin, ”Symmetrization in the geometric theory of functions of a complex variable,” Uspekhi Mat. Nauk 49(1), 3–76 (1994) [Russian Math. Surveys 49 (1), 1–79 (1994)]. · Zbl 0830.30014  A. F. Beardon, D. Minda, and T. W. Ng, ”Smale’s mean value conjecture and the hyperbolic metric,” Math. Ann. 322(4), 623–632 (2002). · Zbl 1001.30007 · doi:10.1007/s002080000184  V. N. Dubinin, ”Covering of vertical segments by conformal mappings,” Mat. Zametki 28(1), 25–32 (1980).  A. Eremenko, ”A Markov-type inequality for arbitrary plane continua,” Proc. Amer. Math. Soc. 135(5), 1505–1510 (2007). · Zbl 1112.41008 · doi:10.1090/S0002-9939-06-08640-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.