The paper presents sharp inequalities for the moduli of rational functions under certain constraints on the modulus of the independent variable. These inequalities supplement some results of Govil, Mohapatra, and Dubinin. Bibliography: 6 titles.
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N. C. Ankeny and T. J. Rivlin, “On a theorem of S. Bernstein, Pacific J. Math., 5, 849–852 (1955).
N. K. Govil and R. N. Mohapatra, “Inequalities form maximum modulus of rational funtions with prescribed poles,” in: Approximation Theory: in Memory of A. K. Varma (Monogr. Textbooks Pure Appl. Math., 212), Marcel Dekker, Inc., New York (1998), pp. 255–263.
V. N. Dubinin, “Shwarz’s lemma and estimates of coefficients for regular functions with free domain of definition,” Mat. Sbor., 196, No.11, 53–74 (2005).
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Dedicated to the 80th anniversary of Igor’ Petrovich Mityuk’s birthday
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 371, 2009, pp. 109–116.
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Kalmykov, S.I. An estimate for the modulus of a rational function. J Math Sci 166, 186–190 (2010). https://doi.org/10.1007/s10958-010-9858-4
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DOI: https://doi.org/10.1007/s10958-010-9858-4