This is on page 32 of Rudin's Real and Complex Analysis, 3rd Edition:
Suppose $\mu$ is a positive measure on $X$, $f: X \rightarrow [0, \infty]$ is measurable, $\int_X f d\mu = c$, where $0
, and $\alpha$ is a constant. Prove that$\lim_{n \rightarrow \infty} \int_X n \log[1+(f/n)^{\alpha}]d \mu = \begin{cases} \infty & \text{ if } 0 < \alpha <1, \\ c & \text{ if } \alpha=1, \\ 0 & \text{ if } 1 < \alpha < \infty. \end{cases}$
The hint says "if $\alpha \geq 1$, the integrands are dominated by $\alpha f$". But why?
Thanks a lot.