The theory of cluster sets / E. F. Collingwood and A. J. Lohwater

The present book is intended as an introduction to the theory of cluster sets. Chapter 1 is introductory and contains a beautiful historical note and summary. Chapter 2 gives a brief expository account of the classical theory of radial limit values with some additional material on Blaschke products; in particular, theorems of Fatou and F. and M. Riesz, and Lohwater-Piranian's theorem [the second author and G. Piranian, Ann. Acad. Sci. Fenn. Ser. A I No. 239 (1957); MR0091342 (19,950c)] are stated. In Chapter 3 the authors give some results from the theory of conformal mapping which will be needed in later chapters. In Chapter 4, intrinsic properties of cluster sets of functions defined in the unit disc, but not necessarily analytic, are introduced; existence theorems for global cluster sets, maximality theorem for a continuous function, maximality theorem for an arbitrary function and the Bagemihl ambiguous point theorem [F. Bagemihl, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 379–382; MR0069888 (16,1095d)] are stated. In Chapter 5, the authors study the boundary behavior of an important class of functions, Seidel's class U, together with the Gross-Iversen theorem on exceptional values, its generalizations and one-sided cluster-set theorems of J. L. Doob [Proc. London Math. Soc. (3) 13 (1963), 461–470; MR0166365 (29 #3642)]. In Chapter 6, the authors study the global cluster set of a function meromorphic in the unit disc and its relation with global exceptional and asymptotic sets (this is called boundary theory in the large). Chapter 7 is concerned with boundary theory in the small. Chapter 8 contains a number of recent results on the classification of the singularities of functions meromorphic in the unit disc and their distribution on the circumference, together with uniqueness and existence theorems under given boundary conditions. Chapter 9, which concludes the book, is a very important application of the theory of cluster sets to the theory of prime ends. The description of this final chapter is fresh and impressive.
It seems important to point out that the overlapping of the present book with the reviewer's book [Cluster sets, Springer, Berlin, 1960; MR0133464 (24 #A3295)] is very small, and the two books, to a considerable extent, complement one another. (MathSciNet)

Bibliogr. p. 190-205. Index