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

TL;DR

This paper derives sharp integral mean estimates for a generalized class of harmonic functions called $(eta,eta)$-harmonic functions, providing explicit bounds and extending classical inequalities to this broader setting.

Contribution

It introduces new sharp $L^p$ integral mean estimates for $(eta,eta)$-harmonic functions, including explicit bounds and applications to coefficient and Hardy space estimates.

Findings

Established sharp $L^p$ integral mean estimates.

Derived explicit bounds for functions and derivatives.

Extended classical inequalities to $(eta,eta)$-harmonic functions.

Abstract

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.