I am currently reading a paper in wich the author shows the subharmonicity of the Bergman kernel on the diagonal with respect to the weightfunction of the Bergman space. The statement I am struggling with is the following:
Let $f: \mathbb{C} \to \mathbb{R}_{\geq 0}$ be some smooth, subharmonic function, such that \begin{equation} e^{g(z)}f(z) \end{equation} is subharmonic for every smooth, subharmonic function $g: \mathbb{C} \to \mathbb{R}$. Then $\log f$ also is subharmonic.
So far I tried to calculate the Laplacian $\Delta \log f$ using the identity \begin{equation} \log f = \log (e^g f) - g, \end{equation} but there are always terms that I can not find good bounds of. Also I thougt one could find some approximation of $\log f$ by subharmonic functions, but I dont't think that actually works.
Any hints would be helpful. Thanks a lot.