Sharp estimates for the maximum over minimum modulus of rational functions

Let m, n greater than or equal to 0, lambda > 1, and R be a rational function with numerator, denominator of degree less than or equal to m; n, respectively. In several applications, one needs to know the size of the set S subset of [0, 1] such that for r is an element of S, [GRAPHICS] In an earlier paper, we showed that meas ( S) greater than or equal to 1/4exp (-13/log lambda), where meas denotes linear Lebesgue measure. Here we obtain, for each lambda, the sharp version of this inequality in terms of condenser capacity. In particular, we show that as lambda --> 1+, meas ( S) greater than or equal to 4 exp (-pi (2)/2 log lambda) (1 + o(1)).