The Schwarz-Pick lemma for slice regular functions

Bisi, Cinzia; Stoppato, Caterina

doi:10.1512/iumj.2012.61.5076

Cited by 36 publications

(37 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for all q ∈ B. The work [2] proved the quaternionic Schwarz-Pick Lemma and, in particular, the inequality log |f (q)| ≤ log |M q0 (q)| valid for all regular f : B → B with f (q 0 ) = 0. It is therefore natural to ask ourselves whether an equality may hold in (12).…”

Section: A Significant Examplementioning

confidence: 92%

Some Notions of Subharmonicity over the Quaternions

Stoppato

2019

Trends in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

Section: A Significant Examplementioning

confidence: 92%

Some Notions of Subharmonicity over the Quaternions

Stoppato

2019

Trends in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The proof was based on the following lemmas, proven in [8] and in [7], respectively. Finally, the following lemma, proven in [7] as a further tool for the proof of Theorem 2.15, will also be useful later in this paper.…”

Section: Preliminary Materialsmentioning

confidence: 99%

“…For slice regular functions on the quaternionic unit ball B, the Schwarz lemma and its boundary version were proven in [20,22]. Slice regular analogs of the Möbius transformations of B have been introduced and studied in [8,23,31], leading to the generalization of the Schwarz-Pick lemma in [7]. Other results concerning slice regular functions on B have been published in [2,3,11,12,14,15,17].…”

Section: Introductionmentioning

confidence: 99%

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

Bisi

Stoppato

2017

Int. J. Math.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Such a notion has its source in the work by Fueter, further developed by Ghiloni and Perotti. It represents a counterpart in several variables of the notion of slice-regular function in one quaternionic variable studied in [GS06,GS07], which appeared to share with holomorphic functions a rich theory from the analytic point of view, [BS12], [BS17]; see also [CSS12]. We refer to [GSS13,GP12] for precise definitions and for results.…”

Section: Introductionmentioning

confidence: 99%

Slice-Quaternionic Hopf Surfaces

Angella

Bisi

2018

J Geom Anal

Self Cite

View full text Add to dashboard Cite

We investigate slice-quaternionic Hopf surfaces. In particular, we construct new structures of slice-quaternionic manifold on S 1 × S 7 , we study their group of automorphisms and their deformations.In the following cases, the quotient of H 2 \ {(0, 0)} by the subgroup generated by f yields a structure of slice-quaternionic Hopf surface: Case A.: Case A.1.: when λ = 0, α = β ∈ H with 0 < |α| < 1; Case A.2.: when λ = 0, α, β ∈ H with 0 < |α| ≤ |β| < 1 and α = β; Case A.3.: when λ ∈ H with λ = 0, p = 1, α = β ∈ H with 0 < |α| < 1; Case B.: when λ ∈ H with λ = 0, p ∈ N with p > 1, β ∈ R with 0 < |β| < 1, and α = β p .Remark. We wonder whether other slice-quaternionic structures on S 1 × S 7 may be constructed; see Remark 2.3.The automorphism groups of the above slice-quaternionic structures are investigated in Theorem 3.1. Notice that, in general, a slice-quaternionic structure does not induce a holomorphic structure; compare also Remark 2.2. (Note indeed that (where I, J are orthogonal complex structures on R 4 .) Therefore the slice-quaternionic Hopf surfaces in case (B) do not underlie a complex Calabi-Eckmann structure.We prove the following result. Theorem 3.1. Let X = H 2 \ {(0, 0)} f be a slice-quaternionic Hopf surface, where f is as in equation (0.1). The dimension of the group of automorphisms of X is as follows: Case A.1.: dim R Aut(X) ∈ {8, 16}; Case A.2.: dim R Aut(X) ∈ {4, 6, 8}; Case A.3.: dim R Aut(X) ∈ {4, 8}; Case B.: dim R Aut(X) = 5. Finally, we provide families of slice-quaternionic Hopf surfaces, connecting cases (A.1) and (A.3), respectively cases (A.2.1) and (B), see Section §4.Remark 0.1. As suggested by the anonymous Referee, we observe that analogous constructions can be performed to obtain slice-quaternionic Hopf manifolds of higher dimension, and that similar techniques might be possibly developed for the study of other slice-quaternionic manifolds, for Suppose now α ∈ R, and that λ ∈ R. By imposing the slice regularity of the left-hand side, we get that: a h,k = 0 and b h,k = 0 for (h, k) ∈ {(1, 0), (0, k) : k ≥ 1}. Therefore we are reduced to:We equal: firstly, the coefficients in z; secondly, the coefficients in µ. We are reduced to: α · a 1,0 = a 1,0 · α + b 1,0 · λ , λ · a 1,0 + α · a 0,1 = a 0,1 · α + b 0,1 · λ , α k · a 0,k = a 0,k · α + b 0,k · λ , for k > 1 , α · b 1,0 = b 1,0 · α , λ · b 1,0 + α · b 0,1 = b 0,1 · α ,

show abstract

The Schwarz-Pick lemma for slice regular functions

Cited by 36 publications

References 22 publications

Some Notions of Subharmonicity over the Quaternions

Some Notions of Subharmonicity over the Quaternions

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

Slice-Quaternionic Hopf Surfaces

Contact Info

Product

Resources

About