Sharp norm-inequalities, valid for functional Hilbert spaces of holomorphic functions on the polydisk, unit ball and C" are established by using the notion of reproducing kernels. These inequalities extend earlier results of Saitoh and ours.

BY WILHELM STOLL During the last fifty years value distribution in one complex variable has been established as one of the most beautiful branches of complex analysis. In several variables, value distribution was slow to grow up. Only a few people were concerned and many obstacles had to be overcome. However, recently, the theory has gained wide recognition. The outlook for the future is bright...

These notes are the outgrowth of a series of lectures given at MSRI in January 1995 at the beginning of the special semester in complex dynamics and hyperbolic geometry. In these notes, the primary aim is to motivate the study of complex dynamics in two variables and to introduce the major ideas in the field. Hence the treatment of the subject is mostly expository.

We deal with kernel convergence of domains in Cn which are biholomorphically equivalent to the unit ball B. We also prove that there is an equivalence between the convergence on compact sets of biholomorphic mappings on B, which satisfy a growth theorem, and the kernel convergence. Moreover, we obtain certain consequences of this equivalence in the study of Loewner chains and of starlike and co...