Answering all questions---concerning proper holomorphic mappings between generalized Hartogs triangles---posed by Jarnicki and Plfug (First steps in several complex variables: Reinhardt domains, 2008) we characterize the existence of proper holomorphic mappings between generalized Hartogs triangles and give their explicit form. In particular, we completely describe the group of holomorphic automorphisms of such domains and establish rigidity of proper holomorphic self-mappings on them.Comment: 15 page

In the paper we study the geometric properties of a large family of domains, called the generalized tetrablocks, related to the μ-synthesis, containing both the family of the symmetrized polydiscs and the family of the μ 1,n -quotients E n , n ≥ 2, introduced recently by G. Bharali. It is proved that the generalized tetrablock cannot be exhausted by domains biholomorphic to convex ones. Moreover, it is shown that the Carathéodory distance and the Lempert function are not equal on a large subfamily of the generalized tetrablocks, containing i.a. E n , n ≥ 4. We also derive a number of geometric properties of the generalized tetrablocks as well as the μ 1,n -quotients. As a by-product, we get that the pentablock, another domain related to the μ-synthesis problem introduced recently by J. Agler, Z.A. Lykova, and N.J. Young, cannot be exhausted by domains biholomorphic to convex ones.

Abstract.We give an example of a Zalcman-type domain in C which is complete with respect to the integrated form of the (k +1)st Reiffen pseudometric, but not complete with respect to the kth one.

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