Abstract
The presented article is devoted to differential inequalities for polynomials. The theme goes back to the problem posed by the famous chemist D. I. Mendeleev. This problem was repeatedly modificated and extended by many mathematicians. In these studies, it was usually assumed that all the zeros of a majorizing polynomial belong to the closed unit disk. We remove this requirement, replacing it with a weaker one and obtain a generalization of the Smirnov type inequality for polynomials having one zero in the exterior of the unit disk. This allow us to obtain a refinement of the Bernstein inequality, proving it not only outside the unit disk, but also in a part of the this disk.
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References
D. I. Mendeleev, Investigation of Aqueous Solutions by Specific Gravity (Tip. V. Demakova, St. Petersburg, 1887) [in Russian].
A. A. Markov, “On a problem posed by D. I. Mendeleev,” Izv. Akad. Nauk. St. Petersburg 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 Deviating Least from Zero in the Given Interval (Izdat. Imp. Akad. Nauk, St. Petersburg, 1892) [in Russian].
V. I. Smirnov and N. A. Lebedev, Constructive Theory of Functions of a Complex Variable (Nauka, Moscow, Leningrad, 1964) [in Russian].
V. I. Smirnoff, “Sur quelques polynomes aux propriétés extrémales,” Transactions of the Kharkov Mathematical Society 4 (2), 67–72 (1928).
S. Bernstein, “Sur la limitation des dérivées des polynomes,” C. R. Acad. Sci. Paris, 338–341 (1930).
S. N. Bernstein, Constructive Functions Theory (1905-1930), in Collected Works (Publishing house of Academy of Science of USSR, Moscow, 1952), Vol. I [in Russian].
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials (Oxford University Press, New York, 2002).
V. N. Dubinin, “Methods of geometric function theory in classical and modern problems for polynomials,” Uspekhi Mat. Nauk 67 (4 (406)), 3–88 (2012).
E. G. Ganenkova and V. V. Starkov, “The Möbius transformation and Smirnov’s Inequality for Polynomials,” Mat. Zametki 105 (2), 228–239 (2019).
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 (2), 696–714 (2019).
E. G. Kompaneets and V. V. Starkov, “Generalization of the Smirnov operator and differential inequalities,” Lobachevskii J. Math. 40 (12), 2043–2051 (2019).
S. L. Wali, W. M. Shah, and A. Liman, “Inequalities concerning \(B\)-operators,” Probl. Anal. Issues Anal. 5 (23) (1), 55–72 (2016).
A. V. Olesov, “Differential inequalities for algebraic polynomials,” Sibirsk. Mat. Zh. 51 (4), 883–889 (2010).
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The authors thank the referee for remarks that improved the exposition of the results of this paper.
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Kompaneets, E.G., Starkov, V.V. On the Smirnov-Type Inequality for Polynomials. Math Notes 111, 388–397 (2022). https://doi.org/10.1134/S0001434622030063
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DOI: https://doi.org/10.1134/S0001434622030063