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We continue the research on the structure of complex geodesics in tube domains over (bounded) convex bases. In some special cases a more explicit form of the geodesics than the existing ones are provided. As one of the consequences of our study an effective formula for the Kobayashi—Royden metric in the tube domain T B n at the origin is given. The results on the Kobayashi—Royden metric in a natural way provide versions of the Schwarz Lemma for harmonic mappings. We also present a result on harmonic mappings defined on the disc that may be seen as a generalisation of the Radó—Kneser—Choquet Theorem for a class of harmonic bivalent mappings that lets understand better the geometry of complex geodesics in tube domains. [ABSTRACT FROM AUTHOR]

In this paper the complex geodesics of a convex domain in ℂn are studied. One of the main results provides a certain necessary condition for a holomorphic map to be a complex geodesic for a convex domain in ℂn. The established condition is of geometric nature and it allows to find a formula for every complex geodesic. The ℂ-convexity of semitube domains is also discussed. [ABSTRACT FROM AUTHOR]

We give a description of complex geodesics and we study the structure of stationary discs in some non-convex domains for which complex geodesics are not unique. [ABSTRACT FROM AUTHOR]

One of the long-standing problems in the quasiconformal theory is finding sharp distortion bounds for k-quasiconformal maps for arbitrary $$k <1$$ . We provide a general distortion theorem for univalent functions in arbitrary quasiconformal disks with k-quasiconformal extensions to $$\mathbb {C}$$ giving a universal power bound. Generically, this power cannot be strengthened. [ABSTRACT FROM AUTHOR]

Complex geodesics are fundamental constructs for complex analysis and as such constitute one of the most vital research objects within this discipline. In this paper, we formulate a rigorous description, expressed in terms of geometric properties of a domain, of all complex geodesics in a convex tube domain in $${\mathbb {C}}^n$$ containing no complex affine lines. Next, we illustrate the obtained result by establishing a set of formulas stipulating a necessary condition for extremal mappings with respect to the Lempert function and the Kobayashi-Royden metric in a large class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in $${\mathbb {C}}^2$$ and for those in $${\mathbb {C}}^n$$ whose logarithmic image is strictly convex in the geometric sense. [ABSTRACT FROM AUTHOR]

In the paper we discuss the problem of uniqueness of left inverses (solutions of two-point Nevanlinna-Pick problem) in bounded convex domains, strongly linearly convex domains, the symmetrized bidisc and the tetrablock. [ABSTRACT FROM AUTHOR]

We investigate the invariant metrics and complex geodesics in the universal Teichmüller space and the Teichmüller space of the punctured disk using Milin's coefficient inequalities. This technique allows us to establish that all non-expanding invariant metrics in either of these spaces coincide with its intrinsic Teichmüller metric. Other applications concern the variational theory for univalent functions with quasiconformal extension. It turns out that geometric features caused by the equality of metrics and connection with complex geodesics provide deep distortion results for various classes of such functions and create new phenomena which do not appear in the classical geometric function theory. [ABSTRACT FROM AUTHOR]

In this paper we show that the Lempert property (i.e., the equality between the Lempert function and the Carathéodory distance) holds in the tetrablock, a bounded hyperconvex domain which is not biholomorphic to a convex domain. The question whether such an equality holds was posed by Abouhajar et al. in J. Geom. Anal. 17(4), 717-750 (). [ABSTRACT FROM AUTHOR]

Using the Kobayashi distance, we provide sufficient conditions for the intersection of a family of holomorphic retracts in a Banach space to also be a holomorphic retract. [ABSTRACT FROM PUBLISHER]

LetD be the class of domains forn≥2,a≥0 and p=(p1,...,n) ∈ (ℝ+)n such thatDa,p is convex. The classD is a class of convex bounded Reinhardt domains of ℂn which are a generalization of complex ellipsoids. In this paper we compare Carathéodory balls and norm balls of the domainsD∈D. We prove that in this case a Carathéodory ball inD∈D is a norm ball if, and only if,D is a complex ellipsoid such thatpk=1 for exactly onek∈{1,…,n},pj=1/2 for allj≠k and the centre lies on thezk-axis. [ABSTRACT FROM AUTHOR]

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifold M, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies of M. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair ( p; v) ∈ T M, with v ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing through p tangent to v iff the Finsler metric is Kähler, has constant holomorphic sectional curvature −4, and its curvature tensor satisfies a specific simmetry condition-which are the differential geometric conditions we were after. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature −4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric. [ABSTRACT FROM AUTHOR]