Markov inequality on the graph of holomorphic function




Markov inequality, graph of holomorphic function, pluripolar sets


The purpose of this paper is to show that the Markov inequality does not hold on the graph of holomorphic function.


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R.P. Boas, Inequalities for the derivatives of polynomials, Mathematics Magazine, 1969, 42, 165-174.   Google Scholar

A.A. Markov, On a problem of D.I. Mendeleev (Russian), Zapishi Imp. Akad. Nauk, 1889, 62, 1-24.   Google Scholar

M. Baran, Bernstein Type Theorems for Compact Sets in Rn Revisited, J. Approx. Theory, 1994, 79 (2), 190-198.   Google Scholar

M. Baran, Markov inequality on sets with polynomial parametrization, Ann. Polon. Math., 1994, 60 (1), 69-79.   Google Scholar

M. Baran and W. Pleśniak, Markov's exponent of compact sets in Cn, Proc.Amer. Math. Soc., 1995, 123 (9), 2785-2791.   Google Scholar

M. Baran and W. Pleśniak, Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves, Studia Math., 1997, 125, 83-96.   Google Scholar

M. Baran and W. Pleśniak, Polynomial Inequalities on Algebraic Sets, Studia Math., 2000, 41 (3), 209-219.   Google Scholar

L. Białas-Cież, Markov Sets in C are not Polar. Bull. Pol. Acad. Sci., Math., 1998, 46(1), 83-89.   Google Scholar

L. Białas-Cież, Equivalence of Markov's property and Hölder continuity of the Green function for Cantor-type sets, East Journal on Approximations, 1995, 1(2), 249-253.   Google Scholar

L. Białas-Cież and A. Volberg, Markov's property of the Cantor ternary set, Studia Math., 1993, 104, 259-268.   Google Scholar

L. Białas-Cież and R. Eggink, L-regularity of Markov sets and of m-perfect sets in the complex plane. Constr. Approx., 2008, 27, 237-252.   Google Scholar

L. Bos, N. Levenberg, P. Milman and B.A. Taylor, Tangential Markov Inequalities Characterize Algebraic Submanifolds of RN, Indiana Univ. Math. Journal, 1995, 44 (1), 115-138.   Google Scholar

L. Bos and P. Milman, On Markov and Sobolev type inequalities on compact subsets in Rn, In "Topics in Polynomials in One and Several Variables and Their Applications" (Th. Rassias et al. eds.), World Scientific, Singapore (1992), 81-100.   Google Scholar

L. Bos and P. Milman, Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains, Geometric and Functional Analysis, 1995, 5 (6), 853-923.   Google Scholar

W. Gautschi, "The Incomplete Gamma Functions Since Tricomi". In Tricomi's ideas and contemporary applied mathematics, Atti Convegni Lincei, Rome, 1998, pages 203-237.   Google Scholar

P. Goetgheluck, Inégalité de Markov dans les ensembles efillés, J. Approx. Theory, 1980, 30, 149-154.   Google Scholar

P. Goetgheluck, Polynomial Inequalities on General Subsets of RN , Colloq. Math., 1989, 57 (1), 127-136.   Google Scholar

P. Goetgheluck and W. Ple±niak, Counter-examples to Markov and Bernstein Inequalities, J. Approx. Theory, 1992, 69, 318-325.   Google Scholar

A. Goncharov, A compact set without Markov's property but with an extension operator for C1 functions, Studia Math., 1996, 119, 27-35.   Google Scholar

L.A. Harris, A Bernstein-Markov theorem for normed spaces, J. Math. Anal. Appl., 1997, 208, 476-486.   Google Scholar

A. Jonsson, Markov's inequality on compact sets, In: "Orthogonal Polynomials and Their Applications" (C. Brezinski, L. Gori and A. Ronveaux, eds.), 1991, 309-313.   Google Scholar

A. Jonsson, Markov's Inequality and Zeros of Orthogonal Polynomials on Fractal Sets, J. Approx. Theory, 1994, 78, 87-97.   Google Scholar

M. Klimek, Pluripotential Theory, Oxford Univ. Press, London, 1991.   Google Scholar

W. Pawłucki and W. Pleśniak, Markov's inequality and C1 functions on sets with polynomial cusps, Math. Ann., 1986, 275, 467-480.   Google Scholar

W. Pawłucki and W. Pleśniak, Extension of C1 functions from sets with polynomial cusps, Studia Math., 1988, 88, 279-287.   Google Scholar

R. Pierzchała, UPC condition in polynomially bounded o-minimal structures, J. Approx. Theory, 2005, 132, 25-33.   Google Scholar

W. Pleśniak, Compact subsets of Cn preserving Markov's inequality, Mat. Vesnik, 1988, 40, 295-300.   Google Scholar

W. Pleśniak, A Cantor regular set which does not have Markov's property, Ann. Polon. Math., 1900, 51, 269-274.   Google Scholar

W. Pleśniak, Markov's inequality and the existence of an extension operator for C1 functions, J. Approx. Theory, 1990, 61, 106-117.   Google Scholar

T. Ransford, Potential Theory in the Complex Plane. In: Lond. Math. Soc. Stud. Texts, vol. 28. Cambridge, 1995.   Google Scholar

J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc., 1962, 105, 322-357.   Google Scholar

J. Siciak, Highly noncontinuable functions on polynomially convex sets, Univ. Jagello. Acta Math., 1985, 25, 95-107.   Google Scholar

V. Totik, Markoff constants for Cantor sets, Acta Sci. Math. (Szeged), 1995, 60, 715-734.   Google Scholar

A. Volberg, An estimate from below for the Markov constant of a Cantor repeller, In: "Topics in Complex Analysis", eds. P. Jakóbczak and W.Pleśniak, Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences 31, 393-390.   Google Scholar

A. Zeriahi, Inégalités de Markov et développement en série de polynômes orthogonaux des fonctions C1 et A1, in: "Proceedings of the Special Year of Complex Analysis of the Mittag-Letter Institute 1987-88" (ed. J.F. Fornaess), Princeton Univ. Press, Princeton New Jersey, 1993, 693-701.   Google Scholar

M. Zerner, Développement en séries de polynômes orthonormaux des fonctions indéfiniment différentiables, C. R. Acad. Sci. Paris Sér. I, 1969, 268, 218-220.   Google Scholar




How to Cite

Beberok, T. (2017). Markov inequality on the graph of holomorphic function. Science, Technology and Innovation, 1(1), 57–63.



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