Letfbe a conformal mapping of a bounded Jordan domainDin the complex plane onto the unit disk. This paper examines the consequences for the local geometry ofDnear a boundary pointz0of the mappingf-or, to be more precise, of the homeomorphic extension of this mapping to the closure ofD—satisfying a Hölder condition atz0or, alternatively, of its inverse satisfying a Hölder condition at the pointf(z0). In particular, the compatibility of Hölder conditions with the presence of cusps in the boundary ofDis investigated.