Convergence rates results for Tikhonov regularization of nonlinear ill-posed
operator equations in abstract function spaces require the handling of both
smoothness conditions imposed on the solution and structural conditions
expressing the character of nonlinearity. Recently, the distinguished role of
variational inequalities holding on some level sets was outlined for obtaining
convergence rates results. When lower rates are expected such inequalities
combine the smoothness properties of solution and forward operator in a
sophisticated manner. In this paper, using a Banach space setting we are going
to extend the variational inequality approach from H\"older rates to more
general rates including the case of logarithmic convergence rates.