This follows straightforwardly from proposition 3.9:

$$T_{\omega}(f^{j}(K))\geq T_{(f^{j})^{*}\omega}(K)=\left[T_{\lambda^{-j}(f^{j})^{*}\omega}(K)\right]^{\lambda^{j}}\geq\left[\alpha T_{\omega}(K)\right]^{\lambda^{j}},$$

where the first two inequalities follow from 3.9.4 and 3.9.2 and last one follows from 3.9.3 and the fact that $\lambda^{-j}(f^{j})^{*}\omega=\omega+dd^{c}g_{j}$, where $g_{j}$ is uniformly bounded. ∎