Lemma 5.24.5E6AC64216A47

Proof. First let $f$ be a smooth subharmonic function on $\mathbb C$ with $f\leq M$. Write $\sigma =(\Delta f)\, \mathrm{d}A$ for its nonnegative Laplacian measure, where $\Delta $ and $\mathrm{d}A$ are Euclidean. Green's formula for circular means gives, for $R>0$,

For fixed $r_0>0$ and $R>r_0$, the right-hand side is at least $\sigma (B(0,r_0))\log (R/r_0)/(2\pi )$, while the left-hand side is at most $M-f(0)$. Letting $R\to \infty $ gives $\sigma (B(0,r_0))=0$. Thus $f$ is harmonic, and the Liouville theorem for the positive harmonic function $M+1-f$ shows that $f$ is constant.

Now let $p$ be plurisubharmonic on $\mathbb C^ m$ and bounded above. Its convolutions with a smooth nonnegative radial probability kernel are smooth plurisubharmonic functions with the same upper bound. By the planar case, their restrictions to every affine complex line are constant, so each convolution is constant on $\mathbb C^ m$. Since the convolutions converge to $p$ in $L^1_{\mathrm{loc}}$, the function $p$ is almost everywhere equal to a constant, and hence constant by lemma 5.24.3. $\square$