Showing papers in "Complex Variables and Elliptic Equations in 1999"
TL;DR: The authors generalize the definition of a certain derivative of regular functions of variables from the four-dimensional real associative algebra of quaternions to monogenic functions of hypercomplex variables in Rn+1.
Abstract: We generalize the definition of a certain derivative of regular functions of variables from the four-dimensional real associative algebra of quaternions to monogenic functions of hypercomplex variables in Rn+1. Using this concept of derivation we look for primitives of monogenic functions in the set of monogenic functions. The results will be applied for proving a final result about the invertibility of a hypercomplex II-operator.
138 citations
TL;DR: In this paper, a definition of Qp-spaces for quaternion-valued functions of three real variables is proposed and some basic properties of its basic properties are studied.
Abstract: We consider a definition of Qp-spaces for quaternion-valued functions of three real variables and study some of its basic properties.
37 citations
TL;DR: For a finitely generated rational semigroup G, this article established the existence of a probability measure μ=μ G on the Julia set J(G) which has a certain invariance property with respect to the semigroup.
Abstract: For a finitely generated rational semigroup G; we establish the existence of a probability measure μ=μ G on the Julia set J(G)which has a certain invariance property with respect to the semigroup.
33 citations
TL;DR: In this paper, it was shown that there exists a URSM-IM with 17 elements, which is the largest known URSSM-IM for nonconstant non-constant meromorphic functions ignoring multiplicities.
Abstract: Two meromorphic functions are said to share a set ignoring multiplicities (IM) if S has the same preimage under both functions If any two nonconstant meromorphic functions sharing a set IM must be identical, it is called a unique range set for meromorphic functions ignoring multiplicities (URSM-IM) In this paper, we show that there exists a URSM-IM with 17 elements
24 citations
TL;DR: In this paper, it was shown that the Cayley transform does not give the maximum or minimum distortion among all mappings of the unit ball of n ≥ 2, onto convex domains.
Abstract: We continue the study, begun by the first author, of linear-invariant families in . We obtain the unexpected result that the Cayley transform (the analogue of the half-plane mapping in the complex plane) does not give the maximum or minimum distortion among all mappings of the unit ball of n≥ 2, onto convex domains. In addition a result analogous to that of Pommerenke that a linear invariant family has order 1 (the smallest possible order) if and only if it consists of convex mappings of the disk does not hold for the ball in n≥ 2. Finally, we extend these ideas to the polydisk in . The theory for this case bears some similarity to the theory for the ball but there are some striking differences as well. For example it is true that the family of convex holomorphic mappings of the polydisk in has minimum order (which is nin this case) but it is not true that for n>1 every linear-invariant family of minimum order consists of convex mappings.
23 citations
TL;DR: In this paper, it was proved that the analogous result holds for the unique extremality, that is, μ∈M(R) is uniquely infmitesimally extremal if and only if μ/(1−|μ|2)∈L∞(R).
Abstract: Let M(R) denote the unit ball of the space L∞(R) of all essentially bounded Beltrami differentials on a Riemann surface R. It is well known that μ∈M(R) is infinitesimally extremal if and only if μ/(1−|μ|2)∈L∞(R) is infinitesimally extremal. In this paper, it is proved that the analogous result holds for the unique extremality, that is, μ∈M(R) is uniquely infmitesimally extremal if and only if μ/(1−|μ|2)∈L∞(R) is uniquely infinitesimally extremal. An application is also given.
20 citations
TL;DR: In this paper, a function theory connected with principal series representation of the discrete series was explored in contrast to standard complex analysis connected with discrete series, and the Hardy space, Cauchy-Riemann equation and Taylor expansion were constructed.
Abstract: We explore a function theory connected with the principal series representation of ) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy space, the Cauchy-Riemann equation and the Taylor expansion.
19 citations
TL;DR: In this paper, the author proves Parseval-Goldstein-type theorems involving a Laplace-type integral tranform, the Widder transform and the K-transform.
Abstract: In the present paper the author proves Parseval-Goldstein-type theorems involving a Laplace-type integral tranform, the Widder transform and the K-transform. The theorem is then shown to yield a number of new identities involving several well-known integral transforms. Using the theorems and its corollaries, a number of interesting infinite integrals of elementary and special functions are presented. Some illustrative examples are also given.
18 citations
TL;DR: In this paper, the generalized Jacobi polynomials were used to find explicit formulas for the coefficients of these differential equations, which is a consequence of the Jacobi inversion formula which is proved in this paper.
Abstract: We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [−1,1] with respect to the weight function where α>−1, β>−1M≥0 and N≥0. In order to find explicit formulas for the coefficients of these differential equations we have to solve systems of equations of the form where the coefficients are independent of n. This system of equations has a unique solution given by This is a consequence of the Jacobi inversion formula which is proved in this paper.
15 citations
TL;DR: Univalence criteria related to Ruscheweyh and Salagean derivatives are obtained in this article, and useful corollaries are also given in this paper. But they do not consider the relation between Salageans and Salcheyh derivatives.
Abstract: Univalence criteria related to Ruscheweyh and Salagean derivatives are obtained. Some useful corollaries are also given.
15 citations
TL;DR: In this article, a spatial modified Miura transform is used to map solutions of the modified Korteweg-de Vries equation into solutions of a non-linear first-order system of partial differential equations.
Abstract: We investigate a spatial modified Miura transform. To describe this transform we have to solve a non-linear first-order system of partial differential equations. This investigation will be done by the help of quaternionic analysis. The main goal is to find a hypercomplex factorization of the Schrodinger equation. In one dimension Miura's transformation is needed to map solutions of the modified Korteweg-de Vries equation into solutions of Korteweg-de Vries equation.
TL;DR: In this paper, it was shown that there exists a set S with 11 elements such that any two nonconstant meromorphic functions f and g satisfying E3((S,f) = E3)(S, g) must be identical.
Abstract: This paper studies the problem of uniqueness of meromorphic functions and shows that there exists a set S with 11 elements such that any two nonconstant meromorphic functions f and g satisfying E3)(S,f) = E3)(S, g) must be identical, which improves a result of G. Frank and M. Reinders.
TL;DR: In this article, the Taylor and Laurent series for spherical monogenics with point singularities has been given explicit formulae for this class of functions, leading to an explicity residue theory.
Abstract: Spherical monogenics of complex degree correspond to local eigenfunctions of the (Atiyah-Singer) Dirac operator on the unit sphere Sm-1 of R m. In this paper we will given explicit formulae for the Taylor and Laurent series for this class of functions. This leads to an explicity residue theory for spherical monogenics having point singularities.
TL;DR: In this article, the authors introduce two new tools to study holomorphic self-maps, namely the inner space A (f) and the generalized inner space AG(f), which is defined as the span of the eigenvectors not belonging to the complex tangent space of ∂ B n at the Wolff point τ and contained in A(f).
Abstract: We introduce two new tools to study a holomorphic self-map f of B n (the unit ball of C n, n>1): the inner space A (f) and the generalized inner space AG(f) After having defined the differential at the boundary for f, k-dfτ, in its Wolff pont τ ∊ ∂B n, we prove that the boundary dilatation coefficient α(f) is an eigenvalue for k-dfτ and we define AG(f) to be the generalized eigenspace associated to α(f); the inner space A(f) will be the span of the eigenvectors not belonging to the complex tangent space of ∂ B n at the Wolff point τ and contained in AG(f). Among other things it turns out that A(f) is the space of all the direction of complex geodesics that are mapped into themselves by f, and that the generalized inner space AG(f) is a direct addend of a boundary Cartan-type decomposition for C n. Using A(f) and AG(f) we obtain several new results on the geometry of holomorphic self-maps of B n, including some necessary conditions for commutation under composition.
TL;DR: In this article, it was shown that log |h(eit)| and log |g| are BMO when some Bloch conditions are satisfied, where h and g are analytic functions on the unit disk.
Abstract: Let f = h + g be a harmonic, univalent and orientation-preserving function on the unit disk, where h and g are analytic. We show that log|h(eit)| and log|g(eit)| are BMO when some Bloch conditions are satisfied.
TL;DR: In this article, it was proved that Julia sets of parabolic rational maps are porous, which implies that the box counting dimension of these Julia sets is always strictly less than two, which was shown before by Denker and Urbanski using conformal measures.
Abstract: It is proved that Julia sets of parabolic rational maps are porous This implies that the box counting dimension of these Julia sets is always strictly less than two, which was shown before by Aaronson Denker and Urbanski using conformal measures Our proof uses only elementary techniques from complex analysis and iteration of rational maps.
TL;DR: In this article, the authors give a full description of the class of monic orthogonal polynomials with respect to a positive Borel measure on the unit circle with infinite support.
Abstract: Let μ be a positive Borel measure on the unit circle with infinite support and φn(z)=φn(z,μ) be monic orthogonal polynomials with respect to μ. We give a full description of the class of measures possessing the following property: there exists a sequence of complex numbers (αn) such that monic polynomials ψn=φn − αnφn−1 are also orthogonal with respect to a certain measure v.
TL;DR: In this paper, the authors investigated some criteria of p-vaience for conformal mappings and showed that for any z > 0 and any n∊N there exists n-valent analytic function f(z) such that.
Abstract: We investigate some criteria of p-vaience for conformal mappings. In particular, we show that for any z>0 and any n∊N there exists n-valent analytic function f(z) such that . This result is connected with theorem by J.Becker and Ch. Pommerenke. A generalization of Goodman's theorem is also provided.
TL;DR: In this article, the authors present necessary and sufficient conditions for a real analytic function to have a Cauchy-Kovalevska extension monogenic of higher spin and show how these conditions are similar to the hypothesis in a generalized version of Poincare's lemma.
Abstract: In this paper we will present necessary and sufficient conditions for a real-analytic function to have a Cauchy-Kovalevska extension monogenic of higher spin and show how these conditions are similar to the hypothesis in a generalized version of Poincare's lemma. This leads to a local characterization of monogenic functions of higher spin.
TL;DR: In this paper, the authors obtained some distortion results for convex mappings defined on the unit ball of C n in terms of Caratheodory distance, and some interesting consequences are presented.
Abstract: In this paper we obtain some distortion results for convex mappings defined on the unit ball of C n in terms of Caratheodory distance. Also, some interesting consequences are presented.
TL;DR: In this article, the authors studied extremal functions of a finite type with respect to an arbitrary proximate order and deduced well-known Bernstein's inequality and some of its generalizations for entire functions.
Abstract: The paper mainly concerns with functions f, analytic in S|Imz|<1 and bounded by a given constant. We state sharp estimates for supR|f′| under the additional condition SupR|f|≤1. Using these estimates we deduce well-known Bernstein's inequality and some of its generalizations for entire functions of a finite type with respect to an arbitrary proximate order. Parallelly we investigate also the next extremal problem, related to the mentioned class of functions: if for some ζ∊S, what is the minimal value for sup R|f|? Also we present the description of extremal functions for these problems.
TL;DR: In this article, the authors examined properties of analytic self-maps ϕ of the unit disc which induce isometries on non-Hilbertian Bergman or Hardy spaces.
Abstract: In this note we examine properties of analytic self-maps ϕ of the unit disc which induce isometries on non-Hilbertian Bergman or Hardy spaces. All isometries of these spaces are given by weighted composition operators Tf=w.foϕ, and it is our purpose to isolate properties of the inducing map ϕ. In both settings, maps inducing isometries are either essentially m-to-1 or infinitely many-to-one. We examine the relationship between ϕ and the multiplier w in the Bergman space setting.
TL;DR: In this paper, the third Painleve equation is transformed into an equivalent one the solutions of which are meromorphic on the whole complex plane, and value distribution properties of transcendental solutions of it are examined.
Abstract: The third Painleve equation is transformed into an equivalent one the solutions of which are meromorphic on the whole complex plane. We examine value distribution properties of transcendental solutions of it.
TL;DR: In this article, the Neumann problem for the Laplace equation in a connected plane region bounded by y closed and open curves is studied, and the existence of classical solution is proved by potential theory.
Abstract: The Neumann problem for the Laplace equation in a connected plane region bounded y closed and open curves is studied. The existence of classical solution is proved by potential theory. The problem is reduced to a Fredholm equation of the second kind, which is uniquely solvable.
TL;DR: In this paper, the authors investigated the properties of the integral transform where λ is a non-negative real valued function normalized by ∫ 1 0 λ(t)dt=1.
Abstract: LetAbe the class of functions analytic in the unit disk and normalized by f(0)=f′(0)+1=0 Let S, S∗(γ), and K(γ)be respectively the classes of normalized univalent functions, starlike functions of order γ and convex functions of order γ. In this paper we investigate the properties of the integral transform where λ is a non-negative real valued function normalized by ∫1 0 λ(t)dt=1. From our main results we get conditions on λ and the classFso that V λ (f) mapsF into various subclasses of the class of univalent functions. As a corollary to our results, we give an affirmative answer in support of a conjecture of Kim: f is a member ofSor S∗(γ) or K(γ), then the function φ(3,3+ αz) ∗ f(z) belongs to the same class for α >1, where φ(b,c;z) ∗ f(z) stands for the convolution of incomplete beta function with f∊ A
TL;DR: In this paper, it was shown that the uniform Carleson condition is sufficient for an analytic variety of a strict finite type convex domain in C n to be difined by an holomorphic function in some Hardy space Hp.
Abstract: We prove that the uniform Carleson condition is sufficient for an analytic variety of a strict finite type convex domain in C n to be difined by an holomorphic function in some Hardy space Hp . This extends a result of Varopoulos for the case of strictly pseudoconvex domains.
TL;DR: In this article, a fractional differintegral operator Ωλ z(−∞ <λ <1) was proposed, which unifies the concepts of fractional derivative and fractional integral.
Abstract: By making use of a fractional differintegral operator Ωλ z(−∞<λ<1), which unifies the concepts of fractional derivative and fractional integral, we introduce and study a nested class Tλ(α) (0≤α<1) of analytic functions with negative coefficients. Various growth and distortion theorems, and some general results pertaining to the Strohacker-Marx problem, are obtained for the class Tλ(α).
TL;DR: In this paper, it was shown that the equicontinuity domain and the Montel domain of a complex manifold satisfy the strong disk property in the complex manifold M ⊂O(M).
Abstract: Let M be a complex manifold. Then we prove that for any family F⊂O(M) the equicontinuity domain E(F) and the Montel domain B(F) of F satisfy the strong disk property in M.