Sharp estimates for the maximum over minimum modulus of rational functions (Q2750845)

Let \(F(b)\) be the condenser capacity of the pair \([b,1]\), \(i \mathbb{R}\), \(1>b>0\). Let \(\lambda>1\) and \(m,n\in\mathbb{N}_+\). The author presents the following sharp estimate of meas\((S)\) where \(S\subset[0,1]\) is, for given \(R(z)= {p_m(z) \over q_n(z)}\), \(z\in\mathbb{C}\), the set of \(1>r>0\) for which NEWLINE\[NEWLINE\max_{|z|=r} \bigl|R(z)\bigr |/ \min_{|z|=r} \bigl|R(z) \bigr |\leq a^{m+n}.NEWLINE\]NEWLINE Here \(p_m\) and \(q_n\) are polynomials of degree \(m\) and \(n\), respectively. Theorem (a) meas\((S)\geq F^{[-1]}(1/ \log\lambda)\) (b) Given \(\varepsilon>0\), there exists for large enough \(m\), a polynomial \(R\) of degree \(m\), such that (with \(n=0)\) \(\text{meas}(S)\leq F^{[-1]}({1\over\log \lambda})+ \varepsilon\).