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On local behavior of analytic functions. (English) Zbl 0948.32006

The goal of this paper is to prove the generalizations of Chebyshev, Bernstein and Brudnyi-Ganzburg inequalities from the case of polynomials to the case of holomorphic functions. Let us formulate one of the results.
By \(B_C(0,r)\) denote the ball of radius \(r>1\) in \(\mathbb{C}^n\) centered at zero, by \(B(0,1)\) denote the unit ball in \(\mathbb{R}^n\).
Theorem. For any function \(f\) holomorphic in \(B_C(0,r)\) there exists a constant \(d_f(r)\) such that for any convex body \(V\subset B(0,1)\) and any measurable subset \(w\subset V\) the inequality \[ \sup_V|f|\leq \left({4n|V|\over|w |} \right)^{df(r)} \sup_w|f| \] holds.

MSC:

32A30 Other generalizations of function theory of one complex variable
30A10 Inequalities in the complex plane
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[1] J. Bourgain, On the distribution of polynomials on high dimensional convex sets, in Lecture Notes in Math., Vol. 1469, pp. 127-131, Springer-Verlag, Berlin/New York.; J. Bourgain, On the distribution of polynomials on high dimensional convex sets, in Lecture Notes in Math., Vol. 1469, pp. 127-131, Springer-Verlag, Berlin/New York. · Zbl 0773.46013
[2] Brudnyi, Yu.; Ganzburg, M., On an extremal problem for polynomials of \(n\)-variables, Math. USSR Izv., 37, 344-355 (1973)
[3] Bos, L.; Levenberg, N.; Milman, P.; Taylor, B. A., Tangential Markov inequalities characterize algebraic submanifolds of \(R^N\), Indiana Univ. Math. J., 44, 115-137 (1995) · Zbl 0824.41015
[4] Brudnyi, A., Bernstein-type inequality for algebraic functions, Indiana Univ. Math. J., 46, 93-115 (1997) · Zbl 0876.26015
[5] Brudnyi, A., Local inequalities for plurisubharmonic functions, Ann. Math., 149, 511-533 (1999) · Zbl 0927.31001
[6] Fefferman, C.; Narasimhan, R., On the polynomial-like behavior of certain algebraic functions, Ann. Inst. Fourier, 44, 1091-1179 (1994) · Zbl 0811.14046
[7] Fefferman, C.; Narasimhan, R., A local Bernstein inequality on real algebraic varieties, Math. Z., 223, 673-692 (1996) · Zbl 0911.32011
[8] Fefferman, C.; Narasimhan, R., Bernstein’s inequality and the resolution of spaces of analytic functions, Duke Math. J., 81, 77-98 (1995) · Zbl 0854.32006
[9] Güler, O., On the self-concordance of the universal barrier function, SIAM J. Optim., 7, 295-303 (1997) · Zbl 0872.90071
[10] Klimek, M., Pluripotential Theory (1991), Oxford Univ. Press: Oxford Univ. Press London · Zbl 0742.31001
[11] Khovanskii, A.; Yacovenko, S., Generalized Rolle theorem in \(R^n\) and \(C\), J. Dynam. Control Systems, 2, 103-123 (1996) · Zbl 0941.26009
[12] Levin, B. Ya., Distribution of Zeros of Entire Functions (1986), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0152.06703
[13] Remez, E., Sur une propriété extrémale des polynômes de Tchebychef, Zap. Nauk-Doslid. Inst. Math. Meh. Harkiv Mat. Tovar., 13, 93-95 (1936)
[14] Roytvarf, N.; Yomdin, Y., Bernstein classes, Ann. Inst. Fourier, 47, 825-858 (1997) · Zbl 0974.30524
[15] Sadullaev, A., An estimate for polynomials on analytical sets, Math. USSR Izv., 20, 493-502 (1983) · Zbl 0582.32023
[16] Stein, E. M., Beijing Lectures in Harmonic Analysis. Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., 112 (1986), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0595.00015
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