arXiv:1003.5066v3 [math.FA] 25 Mar 2011 Bernstein-type inequalities for rational functions in weighted Bergman spaces Rachid Zarouf March 28, 2011 Abstract We prove Bernstein-type inequalities in weighted Bergman spaces of the unit disc D, for rational functions in D having at most n poles all outside of 1 r D, 0 <r< 1. The asymptotic sharpness of each of these inequalities is shown as n →∞ and r → 1. Our results extend a result of K. Dyakonov who studied Bernstein-type inequalities (for the same class of rational functions) in the standard Hardy spaces. 1. Introduction Estimates of the norms of derivatives for polynomials and rational functions (in different functional spaces) is a classical topic of complex analysis (see surveys given by A. A. Gonchar [Go], V. N. Rusak [Ru] and Chapter 7 of [BoEr]). Here, we present such inequalities for rational functions f of degree n with poles in {z : |z| > 1}, involving Hardy norms and weighted- Bergman norms. Some of these inequalities are applied in many domains of analysis: for example 1) in matrix analysis and in operator theory (see “Kreiss Matrix Theorem” [LeTr, Sp] or [Z1, Z5] for resolvent estimates of power bounded matrices), 2) to “inverse theorems of rational approximation” using the classical Bernstein decomposition (see [Da, Pel, Pek]), but also 3) to effective Nevanlinna-Pick interpolation problems (see [Z3, Z4]). Let P n be the complex space of polynomials of degree less or equal than n ≥ 1. Let D = {z ∈ C : |z| < 1} be the unit disc of the complex plane and D = 1