Suppose $f:\mathbb{C}^2 \to \mathbb{R}$ is plurisubharmonic, and let $\Omega \subset \mathbb{C}^2$ be a connected, open set. I've got a function $u:\Omega \to \Omega$ that is nonholomorphic, but real-analytic as a function of the real and imaginary parts of its argument; e.g. something like $u(\mathbf{z})= \overline{z_1\!} \, \mathbf{z}$, where $\mathbf{z} = (z_1,z_2)$.
Is there any literature that specifies conditions on such $u$ so that $f\circ u$ is plurisubharmonic? Currently I can only seem to find results when $u$ is assumed to be holomorphic; e.g., Corollary 2.9.5 in Klimek's Pluripotential theory.
What I'm really trying to do is show that the complex Hessian of some specific composition $f\circ u$ is positive-definite, but dealing with this matrix directly is prohibitively complicated. So I am trying to work with an equivalent characterization of this property. Directly checking if my function $f\circ u$ is PLSH also seems intractable via the "subharmonic on complex lines" characterization, so I'm running out of ideas.
Even results akin to Klimek's Theorem 2.9.19 and its corollaries could be useful to me - basically, I'm interested in general ways of building PLSH functions from "parts" that could be nonholomorphic.
Of course, if there is some strong limit on the extent that what I am asking for is achievable, I would be interested to know that, too.