I am working on the following problem:
Let $f$ be a holomorphic function on a Riemann surface $X$. Show that for $\alpha>0$, the functions $|f|^{\alpha}$ and $\log (1+|f|^{\alpha})$ are subharmonic on $X$.
My progress:
When $\alpha>1$, by Jensen's inequality, we have $$\begin{aligned}|f|^{\alpha}&=\left|\frac{1}{2\pi i}\int_{0}^{2\pi}f(z_0+re^{it})\, dt\right|^{\alpha}\\&\leq\left(\frac{1}{2\pi }\int_{0}^{2\pi}\left|f(z_0+re^{it})\right|\, dt\right)^{\alpha}\\&\leq \frac{1}{2\pi }\int_{0}^{2\pi}\left|f(z_0+re^{it})\right|^{\alpha}\, dt\end{aligned}$$
I have no idea about the rest: when $0<\alpha<1$; $\log (1+|f|^{\alpha})$...
The only possible relevant thing I came up is when $\alpha>1$, we have $$\log(1+|f|^{\alpha})\leq \alpha |f|.$$
Any hint and answers are welcomed!