A Bernstein-type inequality for rational functions in weighted Bergman spaces

Given n⩾1 and r∈(0,1), we consider the set Rn,r of rational functions of degree at most n with no poles in , where D is the unit disc of the complex plane. We give an asymptotically sharp Bernstein-type inequality for functions in Rn,r in weighted Bergman spaces with “sub-polynomially” decreasing weights. We also prove that this result cannot be extended to weighted Bergman spaces with “super-polynomially” decreasing weights.