Subharmonic function and holomorphically parametrized integrals

Question

Let $f_\lambda$ be a family of $L^1$ functions (say on $\mathbb{C}$) such that for all $z$ the map $\lambda \mapsto f_\lambda(z)$ is holomorphic.

Consider the map $N(\lambda)=\log \int |f_\lambda(z)| dm(z)$. Is $N$ subhmarmonic ?