Abstract
The question raised in this article goes back to the problem posed by the famous chemist D. I. Mendeleev in 1887 (solved by A. A. Markov in 1889). In the next 100 years, the Mendeleev problem was repeatedly modificated and solved. Its essence is in the description of conditions under which the inequality ∣f(z)∣ ≤ ∣F(z)∣ for polynomials f and F and for z from a fixed set implies the inequality ∣L[f](z)∣ ≤ ∣L[F](z)∣ for some differential operator L. In the presented paper, we consider a differential operator of special type and arbitrary order. In particular, we obtain a sharp upper estimate for higher order derivatives of arbitrary polynomial in terms of the polynomial values.
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References
A. I. Aptekarev, A. Draux, and D. N. Tulyakov, “On asymptotics of the sharp constants of the Markov—Bernshtein inequalities for the Sobolev spaces,” Lobachevskii J. Math. 39, 609–622 (2018). https://doi.org/10.1134/S1995080218050025
S. N. Bernstein, Collected Works, Vol. 1: Constructive Functions Theory (1905–1930) (Akad. Nauk SSSR, Moscow, 1952) [in Russian].
S. N. Bernstein, Collected works, Vol. 2: Constructive Functions Theory (1931–1953) (Akad. Nauk SSSR, Moscow, 1954) [in Russian].
S. N. Bernstein, Extremal Properties of Polynomials and Best Approximation of Continuous Functions of One Complex Variable (ONTI NKTP SSSR, Leningrad, 1937; Chelsea, reprint, 1970) Vol. 1.
S. Bernstein, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable reelle (Gauthier-Villars, Paris, 1926).
S. N. Bernstein, “On a theorem by V. A. Markov,” Tr. Leningr. Idu. Inst., Fiz.-Mat. Nauki 5, 8–13 (1938) [in Russian].
S. Bernstein, “Sur la limitation des dérivées des polynomes,” C. R. Acad. Sci. 190, 338–341 (1930).
Abdullah Mir, K. K. Dewan, and Imtiaz Hussain, “On an inequality of Paul Turan concerning polynomials,” Lobachevskii J. Math. 37, 155–159 (2016). https://doi.org/10.1134/S1995080216020104
V. N. Dubinin, “Methods of geometric function theory in classical and modern problems for polynomials,” Russ. Math. Surv. 67, 599–684 (2012). https://doi.org/10.1070/RM2012v067n04ABEH004803
E. G. Ganenkova and V. V. Starkov, “The Möbius transformation and Smirnov’s inequality for polynomials,” Math. Notes 105(2), 58–68 (2019). https://doi.org/10.1134/S0001434619010243
E. G. Ganenkova and V. V. Starkov, “Variations on a theme of the Marden and Smirnov operators, differential inequalities for polynomials,” J. Math. Anal. Appl. 476, 696–714 (2019). https://doi.org/10.1016/j.jmaa.2019.04.006
M. Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, No. 3 of Mathematical Surveys (Am. Math. Soc., New York, 1949).
A. A. Markov, “On a problem posed by D.I. Mendeleev,” Izv. Akad. Nauk 62, 1–24 (1889).
A. A. Markov, Selected Works on Theory of Continued Fractions and Theory of Functions Deviating Least from Zero (OGIZ, Moscow, Leningrad, 1948) [in Russian].
V. A. Markov, On Functions that Deviates Least from Zero at the Given Interval (Tip. Imperator. Akad. Nauk, St. Petersburg, 1892) [in Russian].
D. I. Mendeleev, Investigation of Aquenous Liquids by Specific Gravity (Tip. V. Demakova, St. Petersburg, 1887) [in Russian].
A. V. Olesov, “Differential inequalities for algebraic polynomials,” Siber. Math. J. 51, 706–711 (2010). https://doi.org/10.1007/s11202-010-0071-y
Q. I. Rahman, “Functions of exponential type,” Trans. Am. Math. Soc. 135, 295–309 (1969). https://doi.org/10.1090/S0002-9947-1969-0232938-X
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials (Oxford Univ. Press, New York, 2002).
V. I. Smirnoff, “Sur quelques polynomes aux propriétés extremales,” Trans. Kharkov Math. Soc. 4(2), 67–72 (1928).
V. I. Smirnov and N. A. Lebedev, Functions of a Complex Variable: Constructive Theory (Nauka, Moscow, Leningrad, 1964; Massachusetts Inst. Technol., Cambridge, MA, 1968).
S. L. Wali, W. M. Shah, and A. Liman, “Inequalities concerning B-operators,” Probl. Anal. Issues Anal. 5(23), 55–72 (2016). https://doi.org/10.15393/j3.art.2016.3250
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This work is supported by the Russian Science Foundation under grant 17-11-01229.
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Submitted by A. M. Elizarov
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Kompaneets, E., Starkov, V. Generalization of the Smirnov Operator and Differential Inequalities for Polynomials. Lobachevskii J Math 40, 2043–2051 (2019). https://doi.org/10.1134/S1995080219120047
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DOI: https://doi.org/10.1134/S1995080219120047