64 research outputs found
The Growth and Distortion Theorems for Slice Monogenic Functions
The sharp growth and distortion theorems are established for slice monogenic
extensions of univalent functions on the unit disc
in the setting of Clifford algebras, based on a new convex combination
identity. The analogous results are also valid in the quaternionic setting for
slice regular functions and we can even prove the Koebe type one-quarter
theorem in this case. Our growth and distortion theorems for slice regular
(slice monogenic) extensions to higher dimensions of univalent holomorphic
functions hold without extra geometric assumptions, in contrast to the setting
of several complex variables in which the growth and distortion theorems fail
in general and hold only for some subclasses with the starlike or convex
assumption.Comment: 24 pages; Accepted by Pacific Journal of Mathematics for publicatio
Growth theorems in slice analysis of several variables
In this paper, we define a class of slice mappings of several Clifford
variables, and the corresponding slice regular mappings. Furthermore, we
establish the growth theorem for slice regular starlike or convex mappings on
the unit ball of several slice Clifford variables, as well as on the bounded
slice domain which is slice starlike and slice circular
Gleason's problem in weighted Bergman space on egg domains
In the paper, we discuss on the egg domains: We show that Gleason's problem
can be solved in the weight Bergman space on theegg domains. The proof will
need the help of the recent work of the second named author on the weighted
Bergman projections on this kind of domain. As an application, we obtain a
multiplier theorem on the egg domains
Extremal functions of boundary Schwarz lemma
In this paper, we present an alternative and elementary proof of a sharp
version of the classical boundary Schwarz lemma by Frolova et al. with initial
proof via analytic semigroup approach and Julia-Carath\'eodory theorem for
univalent holomorphic self-mappings of the open unit disk . Our approach has its extra advantage to get the extremal functions
of the inequality in the boundary Schwarz lemma
Riemann slice-domains over quaternions II
We generalize the representation formula from slice-domains of regularity to
general Riemann slice-domains. This result allows us to extend the -product
of slice regular functions on axially symmetric domains to certain Riemann
slice-domains by introducing holomorphic stem systems and tensor holomorphic
functions. In particular, we construct a power series expansions of slice
regular functions on certain Riemann slice-domains, which implies a relation
between stem holomorphic, tensor holomorphic and slice regular functions.Comment: 49 page
Boundary Schwarz lemma for holomorphic self-mappings of strongly pseudoconvex domains
In this paper, we generalize a recent work of Liu et al. from the open unit
ball to more general bounded strongly pseudoconvex domains with
boundary. It turns out that part of the main result in this paper is in
some certain sense just a part of results in a work of Bracci and Zaitsev.
However, the proofs are significantly different: the argument in this paper
involves a simple growth estimate for the Carath\'eodory metric near the
boundary of domains and the well-known Graham's estimate on the boundary
behavior of the Carath\'eodory metric on strongly pseudoconvex domains, while
Bracci and Zaitsev use other arguments.Comment: Accepted by CAOT for publicatio
Julia theory for slice regular functions
Slice regular functions have been extensively studied over the past decade,
but much less is known about their boundary behavior. In this paper, we
initiate the study of Julia theory for slice regular functions. More
specifically, we establish the quaternionic versions of the Julia lemma, the
Julia-Carath\'{e}odory theorem, the boundary Schwarz lemma, and the
Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit
ball and of the right half-space . Our quaternionic
boundary Schwarz lemma involves a Lie bracket reflecting the non-commutativity
of quaternions. Together with some explicit examples, it shows that the slice
derivative of a slice regular self-mapping of at a boundary fixed
point is not necessarily a positive real number, in contrast to that in the
complex case, meaning that its commonly believed version turns out to be
totally wrong.Comment: To appear in Transactions of the American Mathematical Society. arXiv
admin note: substantial text overlap with arXiv:1412.420
Scaling limits of discrete holomorphic functions
One of the most natural and challenging issues in discrete complex analysis
is to prove the convergence of discrete holomorphic functions to their
continuous counterparts.
This article is to solve the open problem in the general setting. To this end
we introduce new concepts of discrete surface measure and discrete outer normal
vector and establish the discrete Cauchy-Pompeiu integral formula,
\begin{eqnarray*} f(\zeta)=\displaystyle{\int_{\partial B^h}}
\mathcal{K}^h(z,\zeta) f(z)dS^h(z)+\displaystyle{\int_{B^h}} E^h(\zeta-z)
\partial_{\bar z}^h f (z)dV^h(z),\end{eqnarray*} which results in the uniform
convergence of the scaling limits of discrete holomorphic functions up to
second order derivatives in the standard square lattices
Riemann slice-domains over quaternions I
We construct a counterexample to a well-known extension theorem for slice
regular functions, which motivates us to develop a theory of Riemann
slice-domains by introducing a new topology on quaternions. By some paths
describing axial symmetry in Riemann slice-domains, we rectify the classical
extension formula in the theory of slice regular functions and prove a
representation formula over slice-domains of regularity. This proof involves an
intertwining relation between imaginary units of quaternions and a fixed matrix
corresponding to a complex structure.Comment: 41 page
Complex Dunkl operators
The real theory of the Dunkl operators has been developed very extensively,
while there still lacks the corresponding complex theory. In this paper we
introduce the complex Dunkl operators for certain Coxeter groups. These complex
Dunkl operators have the commutative property, which makes it possible to
establish a corresponding complex analysis of Dunkl operators.Comment: 13 page
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