Let \(U\) be a bounded open set, and let \(F\) be a closed subset of \(\overline U\). If for each function \(f\), analytic and bounded in \(U\), and each \(\varepsilon>0\) there exists a sequence of polynomials \(p_ m\) such that \(p_ m(z)\to f(z)\) for \(z\in U\), \[ \sup_{z\in U} | p_ m(z)| \leq \sup_{z\in U}| f(z)|, \qquad \sup_{z\in F} | p_ m(z)|\leq \sup_{z\in F\cap U} | f(z)|+ \varepsilon, \] then the authors call \((U,F)\) a Farrell pair. They characterize Farrell pairs and extend a result of A. Stray [Pac. J. Math. 51, 301-305 (1974; Zbl 0288.30035)].