Let f be a conformal map of the unit disk D into Ĉ and let $$Q_{f}(z,\zeta) ={(1-\mid z\mid^2)\mid f'(z)\mid (1-\mid \zeta \mid^2)\mid f'(\zeta) \mid \over \mid f(z) - f(\zeta)\mid^2}\lambda_{\rm D}(z,\zeta)^2$$ , where λD denotes the hyperbolic distance. We introduce the family ML of all conformal maps f for which Qf(z, ζ) remains bounded. It contains all maps f that have a quasi-conformal extension to Ĉ but also some functions for which f(D) has outward-pointing cusps. We show that f has a continuous extension to \(\bar {\rm D}\) and study multiple boundary points and the Schwarzian derivative.