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Register a new userOn two Bloch type theorems for quaternionic slice regular functions
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In this paper we prove two Bloch type theorems for quaternionic slice regular functions. We first discuss the injective and covering properties of some classes of slice regular functions from slice regular Bloch spaces and slice regular Bergman spaces, respectively. And then we show that there exits a universal ball contained in the image of the open unit ball $mathbb{B}$ in quaternions $mathbb{H}$ through the slice regular rotation $widetilde{f}_{u}$ of each slice regular function $f:overline{mathbb{B}}rightarrow mathbb{H}$ with $f(0)=1$ for some $uin partialmathbb{B}$.
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The purpose of this paper is twofold. One is to enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau-Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow to prove the minimum principle and one version of the open mapping theorem. Another is to strengthen a version of boundary Schwarz lemma first proved in cite{WR} for quaternionic slice regular functions, with a completely new approach. Our quaternionic boundary Schwarz lemma with optimal estimate improves considerably a well-known Osserman type estimate and provides additionally all the extremal functions.
In this paper, we introduce the quaternionic slice polyanalytic functions and we prove some of their properties. Then, we apply the obtained results to begin the study of the quaternionic Fock and Bergman spaces in this new setting. In particular, we give explicit expressions of their reproducing kernels.
In this paper, we study the (possible) solutions of the equation $exp_{*}(f)=g$, where $g$ is a slice regular never vanishing function on a circular domain of the quaternions $mathbb{H}$ and $exp_{*}$ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function $f$ which satisfies $exp_{*}(f)=g$ is called a $*$-logarithm of $g$. We provide necessary and sufficient conditions, expressed in terms of the zero set of the ``vector part $g_{v}$ of $g$, for the existence of a $*$-logarithm of $g$, under a natural topological condition on the domain $Omega$. By the way, we prove an existence result if $g_{v}$ has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of $g_{v}$ are finite, the domain is either the unit ball, or $mathbb{H}$, or $mathbb{D}$ and a further condition on the ``real part $g_{0}$ of $g$ is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of $g_{v}$, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.
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 quaternion
The theory of slice regular functions is nowadays widely studied and has found its elegant applications to a functional calculus for quaternionic linear operators and Schur analysis. However, much less is known about their boundary behaviors. In this paper, we initiate the study of the boundary Julia theory for quaternions. More precisely, we establish the quaternion
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