Dimension gap under conformal mappings

We give an estimate for the Hausdorff gauge dimension of the boundary of a simply connected planar domain under p-integrability of the hyperbolic metric, p>1. This estimate does not degenerate when p tends to one; for p=1 the boundary can even have positive area. The same phenomenon is extended to general planar domains in terms of the quasihyperbolic metric. We also give an example which shows that our estimates are essentially sharp.