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

First, we establish some Schwarz type inequalities for mappings with bounded Laplacian, then we obtain boundary versions of the Schwarz lemma.

Let α , β ∈ ( - 1 , ∞ ) such that α + β > - 1 . Given two continuous functions g ∈ C ( D ¯ ) and f ∈ C ( T ) , we provide various Schwarz type lemmas for mappings u satisfying the inhomogeneous ( α , β ) -harmonic equation L α , β u = g in D and u = f in T , where D is the unit disc of the complex plane C and T = ∂ D is the unit circle. The obtained results provide a significant improvement over previous research on the subject.

In this paper, we provide some algebraic structures of convex subrings of ∗C{}^*\\mathbb {C}, a nonstandard extension of the field of complex numbers C\\mathbb {C}. In particular, a detailed description of the prime spectrum of any convex subring of ∗C{}^*\\mathbb {C} is given. To achieve our goal, first we investigate prime ideals and we characterize two consecutive elements in the spectrum of a divided domain. We also show that the prime spectrum of the ring of bounded hypercomplex numbers has two peculiar properties: there are no three consecutive elements in the spectrum; moreover, nonzero elements are a disjoint union of three subsets where one of them is strongly dense and the other two are dense in the spectrum.

In this paper, we construct new nonstandard hulls of topological vector spaces using convex subrings of ∗R{}^*\\mathbb {R} (or ∗C{}^*\\mathbb {C}) and we show that such spaces are complete. Some examples of locally convex spaces are provided to illustrate our construction. Namely, we show that the new nonstandard hull of the space of polynomials is the algebra of Colombeau’s entire holomorphic generalized functions. The proof is based on the existence of global representatives of entire generalized functions.

Using convex subrings of ∗ℂ, a nonstandard extension of ℂ, we define several kinds of complex bounded polynomials and we provide their associated analytic functions obtained by taking the quasistandard part.

Using convex subrings of *$\\mathbb{C}$, a nonstandard extension of$\\mathbb{C}$, we define several kinds of complex bounded polynomials and we provide their associated analytic functions obtained by taking the quasistandard part.

In this paper we investigate the solutions of the so-called α¯-Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and (p,q)-harmonic functions.

In this paper we investigate the solutions of the so-called α¯ -Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and (p,q) -harmonic functions.

Suppose \\(\\alpha,\\beta \\in \\mathbb{R}\\backslash \\mathbb{Z}^-\\) such that \\(\\alpha+\\beta>-1\\) and \\(1\\leq p \\leq \\infty\\). Let \\(u=P_{\\alpha,\\beta}[f]\\) be an \\((\\alpha,\\beta)\\)-harmonic mapping on \\(\\mathbb{D}\\), the unit disc of \\(\\mathbb{C}\\), with the boundary \\(f\\) being absolutely continuous and \\(\\dot{f}\\in L^p(0,2\\pi)\\), where \\(\\dot{f}(e^{i\\theta}):=\\frac{d}{d\\theta}f(e^{i\\theta})\\). In this paper, we investigate the membership of the partial derivatives \\(\\partial_z u\\) and \\(\\partial_{\\overline{z}}u\\) in the space \\(H_G^{p}(\\mathbb{D})\\), the generalized Hardy space. We prove, if \\(\\alpha+\\beta>0\\), then both \\(\\partial_z u\\) and \\(\\partial_{\\overline{z}}u\\) are in \\(H_G^{p}(\\mathbb{D})\\). For \\(\\alpha+\\beta<0\\), we show if \\(\\partial_z u\\) or \\(\\partial_{\\overline{z}}u \\in H_G^1(\\mathbb{D})\\) then \\(u=0\\) or \\(u\\) is a polyharmonic function.