A Bloch-Landau Theorem for Slice Regular Functions

Rocchetta, Chiara Della; Gentili, Graziano; Sarfatti, Giulia

doi:10.1007/978-88-470-2445-8_4

Cited by 19 publications

(28 citation statements)

References 19 publications

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“…In Section 6, as an application, we obtain a quaternionic Bloch-Landautype result in the spirit of [12]. Although finding a full-fledged quaternionic generalization of Theorem 1.4 is still an open problem, the new approach used here opens a new path towards such a generalization, which will be the subject of future research.…”

Section: Introductionmentioning

confidence: 93%

“…This section presents an application of Theorem 5.2 to a new version of the quaternionic Bloch-Landau-type result proven in [12]. We will first prove a lemma and give a technical definition.…”

Section: A New Approach To the Quaternionic Bloch-landau Theoremmentioning

confidence: 99%

“…As we already mentioned, in the present work we establish perfect analogs of Theorems 1.2 and 1.3 for slice regular functions. Other original results are proven along with them and a new version of Theorem 6 of [12] is obtained as an application. The paper is structured as follows.…”

Section: Introductionmentioning

confidence: 97%

“…We define f p as the (unique) regular function on the ball B(0, 1 − |p|) that coincides with z → f (p + z) on the disk B I (0, 1 − |p|). When p is in the real interval (−1, 1) then f p (q) = f (p + q) for all q ∈ B(0, 1 − |p|).We now proceed with the announced new version of Theorem 6 of[12].Theorem 6.4. Let Ω ⊆ H be a symmetric slice domain that contains the closure of the unit ball B and let f : Ω → H be a regular function with |∂ c f (0)| = 1.…”

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confidence: 99%

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Landau’s theorem for slice regular functions on the quaternionic unit ball

Bisi

Stoppato

2017

Int. J. Math.

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Along with the development of the theory of slice regular functions over the real algebra of quaternions H during the last decade, some natural questions arose about slice regular functions on the open unit ball B in H. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of B fixing the origin, it establishes two variants of the quaternionic Schwarz-Pick lemma, specialized to maps B → B that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps f of the complex unit disk with f (0) = 0. Landau had computed, in terms of a := |f ′ (0)|, a radius ρ such that f is injective at least in the disk ∆(0, ρ) and such that the inclusion f (∆(0, ρ)) ⊇ ∆(0, ρ 2 ) holds. The analogous result proven here for slice regular functions B → B allows a new approach to the study of Bloch-Landau-type properties of slice regular functions B → H.

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Section: Introductionmentioning

confidence: 93%

Section: A New Approach To the Quaternionic Bloch-landau Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

mentioning

confidence: 99%

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Landau’s theorem for slice regular functions on the quaternionic unit ball

Bisi

Stoppato

2017

Int. J. Math.

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“…In the literature, some other geometric properties of this class of functions have been already considered and some results have been proved: the Bloch-Landau theorem, [14], the Bohr theorem, [18], Landau-Toeplitz theorem, [15], (some of these results are collected in [19]) Schwarz-Pick lemma, see [6] and [2] for an alternative, shorter proof. Recently also the BorelCarathéodory theorem has been proved, see [27] and also [3] for a weaker version.…”

Section: Introductionmentioning

confidence: 99%

On some geometric properties of slice regular functions of a quaternion variable

Gal

González-Cervantes²,

Sabadini

2015

Complex Variables and Elliptic Equations

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Abstract. The goal of this paper is to introduce and study some geometric properties of slice regular functions of quaternion variable like univalence, subordination, starlikeness, convexity and spirallikeness in the unit ball. We prove a number of results, among which an Area-type Theorem, Rogosinski inequality, and a Bieberbach-de Branges Theorem for a subclass of slice regular functions. We also discuss some geometric and algebraic interpretations of our results in terms of maps from R 4 to itself. As a tool for subordination we define a suitable notion of composition of slice regular functions which is of independent interest.

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