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Integral mean estimates for $(α,β)$-harmonic functions

Zhi-Gang Wang, Brindha Valson E, R. Vijayakumar

Abstract

We establish sharp $L^p$ integral mean estimates for $(α,β)$-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 $α$-harmonic functions to the $(α,β)$-harmonic setting.

Integral mean estimates for $(α,β)$-harmonic functions

Abstract

We establish sharp integral mean estimates for -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 -harmonic functions to the -harmonic setting.
Paper Structure (8 sections, 16 theorems, 102 equations)

This paper contains 8 sections, 16 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jensen's inequality
  4. Hardy space
  5. Lebesgue measurable space and subharmonic functions
  6. Gauss hypergeometric functions
  7. Integral means of $(\alpha,\beta)$-harmonic functions
  8. Properties of $(\alpha,\beta)$-harmonic functions

Key Result

Lemma 1

(See KlintbergOlofsson) Let $\alpha,\beta \in \mathbb{C} \setminus \mathbb{Z}^-$ such that $\hbox{\rm Re}(\alpha+\beta)>-1.$ Let $f\in \mathcal{C}(\mathbb{T})$. Then a function $w$ in ${\mathbb D}$ satisfies Dirichlet_problem if and only if it has the form of a Poisson type integral where is the $(\alpha,\beta)$-harmonic Poisson kernel in ${\mathbb D}$, and

Theorems & Definitions (25)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • ...and 15 more