Let \(T\) be a bounded linear operator on a complex Banach space \(X\). The operator \(T\) is called power bounded if there is a constant \(C\) such that \(\| T^{n} \| \leq C\) for all \(n = 1,2,\dots\). The main results of the paper are the following. Assume that the operator \(T\) satisfies the Tauberian condition \[ \sup_{n \geq 1} (n + 1) \| (I - T) T^{n} \| \leq M 0\). Then \(T\) is power bounded with the estimates \[ \| T^{n}\|\leq 2 + C\| T\| + 2 M \text{ and } \lim\sup_{n \longrightarrow \infty}\| T^{n}\| \leq 2 + C \| T \| + (1 + 1/\varepsilon) M. \] Under the mentioned Tauberian condition, the following statements are equivalent: {\parindent9mm \begin{itemize}\item[(i)] \(T\) is power bounded. \item[(ii)] There exists \(0 0\), \(| \theta | 1\) and \(k \in \mathbb{N}.\) \item[(iv)] For some \(k \in \mathbb{N}\), there exists \(0 1\) and \(k \in \mathbb{N}\) is satisfied. \item[(vi)] \(A = T - I\) generates an uniformly bounded, norm continuous, analytic semigroup \(t \mapsto e^{A t}\) of linear operators. \item[(vii)] The operators \(M_{n} : = \frac{1}{n + 1} \sum_{j = 0}^{n} T^{j}\) are uniformly bounded. \item[(viii)] There exists \(C_{U A} 1\) and \(n\in\mathbb{N}\). \end{itemize}}