  We prove a H\"older-type inequality for Hamiltonian diffeomorphisms relating
the $C^0$ norm, the $C^0$ norm of the derivative, and the Hofer/spectral norm.
We obtain as a consequence that sufficiently fast convergence in Hofer/spectral
metric forces $C^0$ convergence. The second theme of our paper is the study of
pseudo-rotations that arise from the Anosov-Katok method. As an application of
our H\"older-type inequality, we prove a $C^0$ rigidity result for such
pseudo-rotations.
