SCHWARZ-PICK ESTIMATES FOR BOUNDED HOLOMORPHIC FUNCTIONS ON CLASSICAL DOMAINS

In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.

A Prop erty of Convex Mappings on the Classical Domains

In this paper, we give a property of normalized biholomorphic convex mappings on the first, second and third classical domains: for any Z0 belongs to the classical domains,f maps each neighbourhood with the center Z0, which is contained in the classical domains,to a convex domain.

Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains

In this paper,a class of holomorphic invariant metrics is introduced on the irreducible classical domains of typesⅠ-Ⅳ,which are strongly pseudoconvex complex Finsler metrics in the strict sense of Abate and Patrizio(1994).These metrics are of particular interest in several complex variables since they are holomorphic invariant complex Finsler metrics found in the literature which enjoy good regularity as well as strong pseudoconvexity and can be explicitly expressed to admit differential geometry studies.They are,however,not necessarily Hermitian quadratic as the Bergman metrics.These metrics are explicitly constructed via deformation of the corresponding Bergman metric on the irreducible classical domains of typesⅠ-Ⅳ,respectively,and they are all proved to be complete K?hler-Berwald metrics.They enjoy very similar curvature properties as those of the Bergman metric on the irreducible classical domains,i.e.,their holomorphic sectional curvatures are bounded between two negative constants and their holomorphic bisectional curvatures are always nonpositive and bounded below by negative constants,respectively.From the viewpoint of complex analysis,these metrics are analogs of Bergman metrics in complex Finsler geometry which do not necessarily have Hermitian quadratic restrictions in the sense of Chern(1996).

A boundary Schwarz lemma on the classical domain of type I

Let R_I(m,n) be the classical domain of type I in C^(m×n)with 1≤m≤n.We obtain the optimal estimates of the eigenvalues of the Fréchet derivative Df(Z) at a smooth boundary fixed point Z of R_I(m,n)for a holomorphic self-mapping f of R_L(m,n).We provide a necessary and sufficient condition such that the boundary points of R_I(m,n) are smooth,and give some properties of the smooth boundary points of R_L(m,n).Our results extend the classical Schwarz lemma at the boundary of the unit disk △ to R_I(m,n),which may be applied to get some optimal estimates in several complex variables.

Schwarz Lemma at the Boundary on the Classical Domain of Type Ⅲ

Let R_Ⅲ(n) be the classical domain of type Ⅲ with n≥2. This article is devoted to a deep study of the Schwarz lemma on R_Ⅲ(n) via not only exploring the smooth boundary points of R_Ⅲ(n) but also proving the Schwarz lemma at the smooth boundary point for holomorphic self-mappings of R_Ⅲ(n).

Existence of inner functions on classical domain

IN function of one complex variable, the inner function on the unit disc plays an important role in the theory of H^p functions. In the mid-1960s, W. Rudin and A. Vitushkin independently raised the question of whether there exist nonconstant inner functions on the unit ball B_n. Then quickly amassed considerable evidence indicated that such functions would be so pathological that they could not exist. Lately Rudin posed the conjecture on the nonexistence