Let
\[ \Delta_{\alpha, \beta}= (1-|z|^2)\bigg( \sum_{i,j=1}^n (\delta_{i j}- z_i\bar{z}_j)D_i\bar{D}_j+\alpha R +\beta \bar{R}+\alpha\beta \bigg) , \]
where \(R= \sum_{j=1}^n z_j D_j\) and \(D_j= \frac{\partial}{\partial z_j}\).
The solutions of the equation \( \Delta_{\alpha, \beta} f = 0\) are called \((\alpha, \beta)\)-harmonic functions. Such functions have a special mean-value property.
Let \(n+\alpha >0 \), \(n+\beta >0 \), \(n+\alpha + \beta >0\). Let \(B_n\) be the unit ball in \(\mathbb{C}^n\). The authors prove that the functions \(f\in L_1(B_n)\) which have the mean-value property are \((\alpha, \beta)\)-harmonic functions if \(n+\alpha + \beta \leq \rho_0\).
The definition of \(\rho_0\) is given. It is known that \(\rho_0 \in (11{.}025; 11{.}069)\).
The article is motivated by the work of [P. Ahern, M. Flores and W. Rudin, J. Funct. Anal. 111, No. 2, 380–397 (1993; Zbl 0771.32006)], where the case \(\alpha = \beta =0\) was considered.