  In this paper we derive higher order convergence rates in terms of the
Bregman distance for Tikhonov like convex regularisation for linear operator
equations on Banach spaces. The approach is based on the idea of variational
inequalities, which are, however, not imposed on the original Tikhonov
functional, but rather on a dual functional. Because of that, the approach is
not limited to convergence rates of lower order, but yields the same range of
rates that is well known for quadratic regularisation on Hilbert spaces.
