Generalized harmonic functions and Schwarz lemma for biharmonic mappings

Suppose α, β ∈ R\Z − such that α + β > −1 and 1 ≤ p ≤ ∞. Let u = P α,β [f ] be an (α, β)-harmonic mapping on D, the unit disc of C, with the boundary f being absolutely continuous and ḟ ∈ L p (0, 2π), where ḟ (e iθ ) := d dθ f (e iθ ). In this paper, we investigate the membership of the partial derivatives ∂ z u and ∂ z u in the space H p G (D), the generalized Hardy space. We prove, if α + β > 0, then both ∂ z u and ∂ z u are in H p G (D). For α + β < 0, we show if ∂ z u or ∂ z u ∈ H 1 G (D) then u = 0 or u is a polyharmonic function.

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Generalized harmonic functions and Schwarz lemma for biharmonic mappings

Cited by 6 publications

References 28 publications

Some inequalities for self-mappings of unit ball satisfying the invariant Laplacians

Some inequalities for self-mappings of unit ball satisfying the invariant Laplacians

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