Integral mean estimates for $(\alpha ,\beta )$-harmonic functions

We establish sharp $L^p$ integral mean estimates for $(\alpha ,\beta )$-harmonic functions on the unit disk. Explicit bounds for the functions and their partial derivatives are obtained in terms of boundary data, by means of the associated Poisson-type kernel and hypergeometric function representations. As applications, we derive coefficient estimates and Hardy space-type results, extending well-known inequalities for classical harmonic and $\alpha$-harmonic functions to the $(\alpha ,\beta )$-harmonic setting.