Convergence rates for the Tikhonov regularization of nonlinear ill-posed operator equations in abstract function spaces using general convex penalty functionals require the handling of both smoothness conditions imposed on the solution and structural conditions expressing the character of nonlinearity. Recently, in the distinguished role of variational inequalities holding on some level sets was outlined for this purpose. When not too high rates are expected such inequalities combine the smoothness properties of solution and forward operator in a sophisticated manner. In this talk, we will present that theory and moreover some extensions of the variational inequality approach from Holder rates to more general cases including logarithmic convergence rates. To formulate the results in a Banach space setting we use Bregman distances for measuring the regularization error. In this context, we also exploit the method of approximate source conditions.