  In this paper, we apply a new kind of smoothness concept, i.e. H\"older
stability estimates for the determination of convergence rates of Tikhonov
regularization for linear and non-linear inverse problems in Hilbert spaces.
For linear inverse problems, we obtain the convergence rates without
incorporating the classical concept of spectral theory and for non-linear
inverse problems, we obtain the convergence rates without incorporating any
additional non-linearity estimate. Further, we employ the smoothness concept of
inhomogeneous variational inequalities to deduce the convergence rates for
non-linear inverse problems. In addition to Tikhonov regularization, we also
consider Lavrentiev's regularization method for non-linear inverse problems and
determine its convergence rates by incorporating the H\"older stability
estimates as well as inhomogeneous variational inequalities. And finally, we
discuss the co-action between the variational inequalities and the H\"older
stability estimates.
