I would like to know if the function $\frac{1}{(1+\vert z\vert^2)^2}$ is subharmonic on $\mathbf{C}=\mathbf{P}^1-\{\infty\}$.
Motivation. The Fubini-Study metric on the complex projective line $\mathbf{P}^1$ is given by $\frac{i}{2\pi} \frac{dz \wedge d\overline{z}}{(1+\vert z\vert^2)^2} ;$ this depends on the choice of the coordinate $z$.
I can think of two possiblities. Firstly, just an ugly computation with derivatives. Secondly, an argument using properties of the Fubini-study metric, i.e., it is a smooth positive real $(1,1)$-form.
Question. Is the above function subharmonic?
Question. If yes, then is it possible to prove this without calculating derivatives?