Sharp estimates for the maximum over minimum modulus of rational functions

Abstract:

Let $m, n\geq 0, \lambda >1$, and $R$ be a rational function with numerator, denominator of degree $\leq m,n$, respectively. In several applications, one needs to know the size of the set ${\mathcal {S}}\subset \left [ 0,1\right ]$ such that for $r\in {\mathcal {S}}$, \[ \max _{\left | z\right | =r}\left | R\left ( z\right ) \right | /\min _{\left | z\right | =r}\left | R\left ( z\right ) \right | \leq \lambda ^{m+n}.\] In an earlier paper, we showed that \[ meas\left ( {\mathcal {S}}\right ) \geq \frac {1}{4}\exp \left ( -\frac {13}{\log \lambda }\right ) ,\] 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 $\lambda \rightarrow 1+$, \[ meas\left ( {\mathcal {S}}\right ) \geq 4\exp \left ( -\frac {\pi ^{2}}{2\log \lambda }\right ) \big ( 1+o(1) \big ) .\]