Let two convex combinations $\lambda = \sum_{i = 1}^N \lambda_i \theta_i$ and $\mu = \sum_{i = 1}^N \mu_i \theta_i$ of real constants $\lambda_i$ and $\mu_i$, and variables $\theta_i$ such that $\sum_{i = 1}^N \theta_i = 1$.
The paper below claims that the product $\lambda \mu$ is a harmonic function in the plane $\lambda-\mu$ (to prove (b) of Theorem 1). I'm trying to demonstrate it using the definition of harmonic function (https://en.wikipedia.org/wiki/Harmonic_function), but I can't get 0 on the left side of the equation. By the symmetric between $\lambda$ and $\mu$, we need to have $\frac{\partial^2(\lambda \mu)}{\partial\lambda^2} = \frac{\partial^2(\lambda \mu)}{\partial\mu^2} = 0$ to satisfy the equation in general. If we set $\lambda_i = \mu_i, \forall i$, $x := \lambda = \mu$ and then $\frac{\partial^2(\lambda \mu)}{\partial\lambda^2} + \frac{\partial^2(\lambda \mu)}{\partial\mu^2} = 2 \frac{\partial^2 x^2}{\partial x^2} = 4 \neq 0$. What am I doing wrong?
Furthermore, if $\lambda_i$ $\mu_i$ are complex constants instead real and we consider the function $| \lambda \mu|$, the mentioned item (b) will be true?, i.e., the maximum principle would be applied to such function?
Paper: https://www.sciencedirect.com/science/article/pii/S0024379587903090