Sharp Estimates for the Maximum over Minimum Modulus of Rational Functions

Let m, n ≥ 0, λ > 1, and R be a rational function with numerator, denominator of degree ≤ m,n, respectively. In several applications, one needs to know the size of the set S ⊂ [0, 1] such that for r ∈ S, max |z|=r |R (z)| / min |z|=r |R (z)| ≤ λ. In an earlier paper, we showed that meas (S) ≥ 1 4 exp ( − 13 log λ ) , where meas denotes linear Lebesgue measure. Here we obtain, for each λ, the sharp version of this inequality in terms of condenser capacity. In particular, we show that as λ→ 1+, meas (S) ≥ 4 exp ( − π 2 2 log λ ) ( 1 + o(1) ) .