A classical result due to Blaschke states that for every analytic self‐map f of the open unit disc of the complex plane there exists a Blaschke product B such that the zero sets of f and B agree. Indeed, a sequence is the zero set of an analytic self‐map of the open unit disc if and only if it satisfies the simple geometric condition known as the Blaschke condition. In contrast, the critical sets of analytic self‐maps of the open unit disc have not been completely described yet. In this paper, we show that for every analytic self‐map f of the open unit disc there is even an indestructible Blaschke product B such that the critical sets of f and B coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighted Bergman space as well as to the Berger–Nirenberg problem from differential geometry. By solving the Berger–Nirenberg problem in a special case, we identify the critical sets of bounded analytic functions with the zero sets of the weighted Bergman space 𝒜12.

On the Poincaré disc and its higher-dimensional analogs one has a canonical formal star product of Wick type. We define a locally convex topology on a certain class of real-analytic functions on the disc for which the star product is continuous and converges as a series. The resulting Fréchet algebra is characterized explicitly in terms of the set of all holomorphic functions on an extended and doubled disc of twice the dimension endowed with the natural topology of locally uniform convergence. We discuss the holomorphic dependence on the deformation parameter and the positive functionals and their GNS representations of the resulting Fréchet algebra. *

A boundary version of Ahlfors' Lemma is established and used to show that the classical Schwarz-Carathéodory reflection principle for holomorphic functions has a purely conformal geometric formulation in terms of Riemannian metrics. This conformally invariant reflection principle generalizes naturally to analytic maps between Riemann surfaces and contains among other results a characterization of finite Blaschke products due to M. Heins.

Abstract. We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation ∆u = 4 e 2u and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence {z j } in the unit disk there is always a Blaschke product with {z j } as its set of critical points. Our work is closely related to the Berger-Nirenberg problem in differential geometry.

We consider the classical problem of maximizing the derivative at a fixed point over the set of all bounded analytic functions in the unit disk with prescribed critical points. We show that the extremal function is essentially unique and always an indestructible Blaschke product. This result extends the Nehari-Schwarz Lemma and leads to a new class of Blaschke products called maximal Blaschke products. We establish a number of properties of maximal Blaschke products, which indicate that maximal Blaschke products constitute an appropriate infinite generalization of the class of finite Blaschke products.