$h:\mathbb{C}^n\rightarrow [0,\infty)$ is a homogeneous plurisubharmonic function. Assume $g:\mathbb{C^n}\rightarrow [0,\infty)$ is another homogeneous function. If we have to prove that $h(z)=g(z)$ for all $z\in \mathbb{C}^n$, why it suffices to prove that $h(a)=g(a)$ for some $a\in \mathbb{C}^n$ such that $h(a)=1$?
Since $h,g$ are homogeneous $h(\lambda z)=|\lambda|h(z)$ and $g(\lambda z)=|\lambda|g(z)$ for any $\lambda \in \mathbb{C}$ and any $z\in \mathbb{C}^n$. So we are done for $z\in \mathbb{C}^n$ which are of the form $\lambda a$ for some $\lambda \in \mathbb{C}$. But how about other elements? I came across this problem while reading a proof in a SCV book. Any help is appreciated.