Prove that $\widehat G = \left \{(z',w)\ :\ |w| < \rho (z') \right \}$ is pseudoconvex if and only if $-\log \rho$ is plurisubharmonic

Question

Let $G \subseteq \mathbb C^n$ be a domain and $\rho : G \longrightarrow \mathbb R$ be lower semicontinuous positive function. Prove that $$\widehat G = \left \{(z',w)\ :\ |w| < \rho (z') \right \}$$ is pseudoconvex if and only if $-\log \rho$ is plurisubharmonic.

This is an exercise problem to Grauert and Fritzsche From Holomorphic Functions to Complex Manifolds where I am getting stuck for almost a day but could not able to do anything fruitful. Could anyone give me some suggestion?

Thanks for your time.