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تسجيل مستخدم جديدJulia theory for slice regular functions
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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
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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
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 expone
ntial 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.
Given a quaternionic slice regular function $f$, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself
. Afterwards, we compare the coefficients of $f$ with those of its slice derivative $partial_{c}f$ obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.
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 space
s, 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 $uin partialmathbb{B}$.
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.
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