Abstract
In this paper we establish an inequality for derivatives of rational functions with a fixed denominator generalizing V. S. Videnskii's inequality to the case of two intervals. To prove its asymptotic exactness, we use a new representation of Akhiezer-Zolotarev fractions with the least deviation from 0 on two intervals.
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References
P. B. Borwein, “Markov's and Bernstein's inequalities on disjoint intervals”Canad. J. Math.,33, No. 1, 201–209 (1981).
N. I. Akhiezer and B. Ya. Levin, “Generalization of the S. N. Bernstein inequality for derivatives of entire functions”, in:Studies in Modern Problems in the Theory of Complex Analysis (Markushevich A. I, editor) [in Russian], Moscow (1960), pp. 111–165.
V. S. Videnskii, “On estimates of derivatives of rational fractions”,Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],26, No. 3, 415–426 (1962).
P. B. Borwein, T. Erdelyi and J. Zhang, “Chebyshev polynomials and Markov-Bernstein type inequalities for rational spaces”,J. London Math. Soc. (2),50, 501–519 (1994).
V. N. Rusak,Rational Functions as an Approximation Tool [in Russian], Izd. BGU, Minsk (1979).
A. A. Pekarskii, “Bounds for derivative of integral of Cauchy type with meromorphic density and its applications”Mat. Zametki [Math. Notes],31, No. 3, 389–402 (1982).
X. Li, R. N. Mohapatra and R. S. Rodriguez, “Bernstein-type inequalities for rational functions with prescribed poles”,J. London Math. Soc. (2),51, 523–531 (1995).
V. I. Danchenko, “On separation of singularities of meromorphic functions”Mat. Sb. [Math. USSR-Sb.], Mat. Sb. [Russian Acad. Sci. Sb. Math.],125, No. 2, 181–198 (1984).
G. G. Lorentz, M. V. Golitschek and Y. Makovoz,Constructive Approximation. Advanced Problems, Springer, Berlin-Heidelberg (1996).
Q. I. Rahman and G. Schmeisser,Les inégalités de Markoff et de Bernstein, Presses Univ. Montréal, Montréal (1983).
A. L. Lukashov, “On the Chebyshev-Markov problem on two intervals”, in:Dep. VINITI [in Russian], No. 2426615-89, VINITI, Moscow (1989).
N. I. Akhiezer, “Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen. I”,Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],9, 1163–1202 (1932).
R. Levy, “Generalized rational function in finite intervals using Zolotarev functions”IEEE Trans. Microwave Theory Tech.,18, No. 12, 1052–1064 (1970).
E. T. Whittaker and G. N. Watson,A Course of Modern Analysis, Cambridge Univ. Press, New York (1962).
F. Peherstorfer, “Elliptic orthogonal and extremal polynomials”,Proc. London Math. Soc. (3),70, 605–624 (1995).
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Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 508–514, October, 1999.
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Lukashov, A.L. A bernstein type inequality for derivatives of rational functions on two intervals. Math Notes 66, 415–420 (1999). https://doi.org/10.1007/BF02679090
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DOI: https://doi.org/10.1007/BF02679090