Growth behaviour and zero distribution of rational approximants

We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree $\leq n$ and denominator degree $\leq m_n$ for meromorphic functions $f$ on a compact set $E$ of $\CC$ where $m_n = o(n/\log n)$ as $n \rightarrow \infty$. We obtain a Jentzsch-Szeg\H{o} type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain $E_{\rho(f)}$ of meromorphy of $f$ if $f$ has a singularity of multivalued character on the boundary of $E_{\rho(f)}$. The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator. As applications, Padé approximation and real rational best approximants are considered