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.

No abstract

No abstract

In this paper, we establish some Schwarz type lemmas for mappings Φ satisfying the inhomogeneous biharmonic Dirichlet problem ∆(∆(Φ)) = g in D, Φ = f on T and ∂ n Φ = h on T, where g is a continuous function on D, f, h are continuous functions on T, where D is the unit disc of the complex plane C and T = ∂D is the unit circle. To reach our aim, we start by investigating some properties of T 2 -harmonic functions. Finally, we prove a Landau-type theorem.