Let \(G\) be an open set in the complex plane, \(f\) analytic in \(G\) and continuous in \(\overline G\). Let \(\mu\) is a majorant in the sense that \(\mu(t)\) is a nonnegative, nondecreasing function defined for \(t\geq 0\) with \(\mu(2t)\leq 2\mu(t)\) for all \(t\geq 0\) and \[ |f(z_1)- f(z_2)|\leq \mu(|z_1- z_2|)\tag{1} \] for \(z_1\) and \(z_2\) in \(\partial G\). It is known that in this case \[ |f(z_1)- f(z_2)|\leq C\mu(|z_1- z_2|)\tag{2} \] for \(z_1\) and \(z_2\) in \(\partial G\) with an absolute constant \(C\) for all \(z_1,z_2\in\overline G\) if \(G\) is simply connected or doubly connected. In this paper the author shows that such a result is true if \(G\) is an open set with only bounded components. It is also shown that if (1) holds for a fixed \(z_1\in \partial G\) and for all \(z_2\in\partial G\) then (2) holds for this \(z_1\) and for all \(z_2\in\overline G\). A survey of results of this type is also given.