Plurisubharmonicity of Mobius distance function

Question

Let $\mathbb{D}$ denotes the unit disc in $\mathbb{C}$. Consider the function $\mathscr{M}:\mathbb{D}\times\mathbb{D}\longrightarrow[0,\,1)$ defined by $$\mathscr{M}((z,\,w))\,=\,\frac{|z-w|}{|1-\bar{w}z|} \quad \forall(z,\,w) \in \mathbb{D}\times\mathbb{D}.$$

We recall this is the Möbius distance function. I have the following question:

Is the function $\mathscr{M}$ plurisubharmonic in $\mathbb{D}\times\mathbb{D}$ ?

Answer 1

It would be plurisubharmonic if the complex hessian, that is the matrix $[ \frac{\partial^2 f}{\partial z_j \partial \bar{z}_k} ]$ was positive semidefinite. This is a two by two matrix in this case, kind of ugly, but doable easily with the computer. Then compute the determinant. There are points where this determinant is negative, meaning the eigenvalues are of opposite signs, meaning it cannot possibly be positive semidefinite. I get for example that the determinant is negative at $z=0.9$ and $w=0$.

Here's what I did in maxima:

rho:sqrt((z-w)*(zb-wb)/((1-wb*z)*(1-w*zb)));

To set up the function. Note that I use zb for $\bar{z}$, as $z$ and $\bar{z}$ are formally different and so we can just treat them as different variables for simplicity. While that does make a bit of a difference because of the square root, it should all be OK as what we are plugging in makes everything under the square roots nonnegative.

detH:determinant(matrix(
 [diff(diff(rho,z),zb),diff(diff(rho,z),wb)], 
 [diff(diff(rho,w),zb),diff(diff(rho,w),wb)]
));

Compute the determinant of the complex hessian. This will be a mess.

ratsimp(subst([w=0,wb=0,z=9/10,zb=9/10],detH));

Substitute $z=0.9$ and $w=0$ into it and simplify. I get $-31/200$