Estimates of Green potentials. Applications

Let D be a simply-connected domain in the complex plane ℂ, and let f = u + iv be a univalent, orientation-preserving, harmonic mapping from D into ℂ. Then f can be written in the form f = h + \(\bar g\) where h and g belong to the linear space H(D) of analytic functions on D. In addition, f can be viewed as a solution of the elliptic partial differential equation $$ {\bar f_{\bar z}} = a{f_z} $$ (1.1) where the function a = g′/h′ belongs to H(D) and satisfies |a(z)| < 1 for all z ∈ D. Hence the mapping f is locally quasiconformal. Conversely, any univalent solution of (1.1) with a analytic and |a| < 1 is an orientation-preserving harmonic mapping of D (see [4]). Observe that if φ is a conformal mapping from D 1 onto D, then f 1 = f o φ is a harmonic mapping defined on D 1 with D 1 with a 1 = a o φ.

The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting.

Let Ω be a bounded, simply connected domain in the complex plane, with C ∞ smooth boundary and take u ∈ C∞(∂Ω) real-valued, then there exists a unique real-valued harmonic function for which the restriction to the boundary ∂Ω is precisely u. Let be the conjugate harmonic to U in the sense that F = U+iV is holomorphic in Ω, for which moreover V(a)=0, a ∈ Ω, then the restriction v of V to the boundary ∂Ω is called the Hilbert transform of u. In this article, we extend the principles of this construction to m-dimensional space , more specifically in a Clifford analysis setting, in order to define a Hilbert-like integral transform on the unit sphere S m−1, based upon a specific notion of conjugate harmonic functions using spherical co-ordinates. ‡Dedicated to Richard Delanghe on the occasion of his 65th birthday.

We investigate the uniqueness of transcendental analytic fun- ctions that share three values DM in one angular domain instead of the whole complex plane. 1. Introduction and main results In this paper, a transcendental meromorphic (analytic) function is mero- morphic (analytic) in the whole complex plane C and not rational. We as- sume that the reader is familiar with the Nevanlinna's theory of meromorphic functions and the standard notations such as m(r, f ), T (r, f ). For references, see (2). We say that two meromorphic functions f and g share the value a (a ∈ C = C ∪ {∞}) in X ⊆ C provided that in X, we have f (z) = a if and only if g(z) = a. We will state whether a shared value is by DM (differential mul- tiplicities), or by IM (ignoring multiplicities). R. Nevanlinna (see (4)) proved that if two meromorphic functions f and g have five distinct IM shared values in X = C, then f (z) ≡ g(z). After his very work, the uniqueness of meromor- phic functions with shared values in the whole complex plane attracted many investigations (for references, see (7)). E. Mues consider DM shared values and proved the following theorem.

In this paper we consider the meromorphic function G(z) with a pole of order 1 at -a and analytic function F(z) with a zero -a of order 2 in <TEX>$\mathbb{D}=\{z :{\mid}z{\mid}</TEX><TEX><</TEX><TEX>1\}$</TEX>, where -1 < a < 1. From these functions we obtain the regular simply-connected minimal surface <TEX>$S=\{(u(z),\;{\nu}(z),\;H(z)):z{\in}\mathbb{D}\}$</TEX> in <TEX>$E^3$</TEX> and the harmonic function <TEX>$f=u+i{\nu}$</TEX> defined on <TEX>$\mathbb{D}$</TEX>, and then we investigate properties of the minimal surface S and the harmonic function f.

Given a harmonic function U in a domain Ω in Euclidean space, the problem of finding a harmonic conjugate V, generalizing the well-known case of the complex plane, was considered in [4] in the framework of Clifford analysis. By the nature of the given construction, which is genuinely cartesian, this approach lead to geometric constraints on the domain Ω. In this paper we consider the problem in a larger class of domains, by a spherical approach. Starting from a real-valued function u, and singling out the radial direction, we explicitly construct a harmonic function of the form w = erv, with v ∈ span(eθ1,…, eθm−1), such that u+w is monogenic, i.e. a null solution of the Dirac operator. As an illustration, the construction is applied to important classes of homogeneous monogenic polynomials and functions. Finally, it is investigated to which extent the approach also applies to the complex plane case.

Let Ω be a domain in the complex plane C whose complement E=\(\overline C \backslash \Omega \) is a subset of the real line (i.e., Ω a Denjoy domain). If each point of E is regular for the Dirichlet problem in Ω, we provide a geometric description of the structure of E near infinity such that the Martin boundary of Ω has one or two “infinite” points.

Preface

Let D denote the unit disc in the complex plane. If p and q are two complex numbers, let x ( p , q ) denote the chordal distance between p and q on the Riemann sphere. In particular, we have the formula and The following problem was posed by Paul Gauthier: if f (z) and g (z) are meromorphic functions in D such that Clunie [4] has answered this problem in the negative by constructing different meromorphic functions f (z) and g (z) with the desired property. However, the functions constructed by Clunie both have an infinity of poles in D . It is the purpose of this note to give an example of two analytic functions―which thus have no poles―which also give a negative answer to the problem of Gauthier.

Let C[a,b] denote the set of all real-valued continuous function x defined on [a,b] such that x(a) = 0. In this thesis, we define the Wiener and Feynman integrals with complex parameters for functionals defined on C[a,b] , and we have a sufficient condition for the existence or the complex sequential Wiener integral for analytic and harmonic functionals. Moreover we have that this integral equals the ordinary Wiener integral, Finally, we define the limiting sequential Wiener and Feynman integrals, and we also establish an existence theorem for the limiting Feynman integral for analytic and harmonic functionals in such a way that it equals the ordinary Wiener integral.

1. In our research on approximation by polynomial and rational interpolation in the complex plane we were led to some problems on balayage (sweeping out) which we want to describe. Let D be a bounded simply connected domain in the complex plane C. Let C = C ∪∞ be the extended complex plane and let A be defined by A = C\D, where D = D ∪ ∂D and ∂D is the boundary of D. In the problems below we consider unit (probability) measures α and β such that α(D) = 1 or α(D) = 1, and β(A) = 1 or β(A) = 1. In all cases α′ and β ′ denote the balayage onto ∂D of α and β, respectively. We denote by δ∞ the Dirac measure at infinity, and by δ′ ∞ its balayage onto ∂D. Then δ ′ ∞ is the equilibrium measure for logarithmic potentials of ∂D. We refer to [6] for this and other facts on balayage. In the problems below we are interested in the equality α′ = β ′. In Problem 1 we consider the case β = δ∞.

Let Ω be a domain in the complex plane \(\mathbb{C}\) containing the origin and let \({\cal P}_{n}^{0}(\Omega)\) denote the set of all polynomials P of degree ≤ n satisfying the conditions \(P(0)=0\) and \(P(\mathbb{D})\subset \Omega\), where \(\mathbb{D}=\left\{z\,:\,\,|z|<1\right\}\) is the unit disk in the complex plane. The maximal range, denoted by Ω n , is then defined as the union of all sets \(P(\mathbb{D}), P \in {\cal P}_{n}^{0}\) .We shall construct extremal polynomials and maximal polynomial range for a domain of intersection of two circular disks.

Motivation: In designing linear feedback systems, tracking quality specifications are typically mapped into admissible domains in the complex plane and then the root locus technique is used to synthesize a controller that places closed loop poles in the desired locations. This methodology motivates the first goal of this chapter: to generalize the notion of admissible domains and the root locus technique to LPNI systems. Accomplishing this goal leads to a quasilinear method for PID controller design.

Near-isometric duality of Hardy norms with applications to harmonic mappings

If f is an entire function in the complex plane such thatwhere 0 ≤ α &lt; 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).