On Local Behavior of Analytic Functions

Abstract

We prove local inequalities for analytic functions defined on a convex body in R n which generalize well-known classical inequalities for polynomials. Academic Press

Local dimension-free estimates for volumes of sublevel sets of analytic functions

“…The main feature of the result is its dimensionless character. Dimension-dependent versions of the theorem were obtained by N. Garofalo and P. Garrett (see [GG]), and A. Brudnyi (see [Br1], [Br2]). …”

“…For other developments, see A. Brudnyi [Theorem 1.11,Br2]), S. Bobkov [Bo], and Carbery and Wright [CW].…”

Bernstein Type Inequalities for Quasipolynomials

“…1] it was proved that the number of zeros of a univariate quasipolynomial q of degree at most m in B c (z, R)/C does not exceed 4(m&1)+3R=(q). Using this estimate and an estimate of the Chebyshev degree of an analytic function by its local valency given in [Br1,Prop. 1.7] we obtain that : c(4m+(3Â2) =(q) diam(V)).…”

“…Remark 1.6. The best constant l for which the inequality of Theorem 1.5 is valid with c 1 =4 is called according to [Br1,Def. 1.5] the Chebyshev degree of q in B c (z, r).…”

Graphs of zeros of analytic families

“…Such problems arise in several diverse areas of Modern Analysis, including Dynamical Systems (started by the classical works of Poincaré, Dulac and Bautin related to the so-called second part of Hilbert's 16 th problem [I]), Transcendental Number Theory (going back to the classical works of Gelfond, Mahler and Siegel related to Hilbert's 7 th problem [B]) and Approximation Theory (Bernstein-Markov-Remez' type inequalities for analytic functions see e.g. [Br1], [RY]). In [Br2] we studied this problem in the case of analytic families appearing as displacement maps of certain planar polynomial vector fields.…”