×

Linear problems and convexity techniques in geometric function theory. (English) Zbl 0581.30001

Monographs and Studies in Mathematics, 22. Boston - London - Melbourne: Pitman Adavanced Publishing Program. XVII, 182 p. £26.50 (1984).
The class S consists of all functions f analytic and univalent in the unit disk \(\Delta\), normalized by \(f(0)=0\) and \(f'(0)=1\). This book deals mainly with extremal problems for convex functionals defined over special suclasses of S: the convex, starlike, close-to-convex, typically real functions, etc. The dominant theme is the use of the Krein-Milman theorem to solve these problems. This idea would have seemed preposterous before about 1970, when L. Brickman, D. R. Wilken, and T. H. MacGregor initiated a systematic program to study these classes by the methods of linear functional analysis. With D. J. Hallenbeck and others, they found the extreme points of special subclasses and applied this information to solve various linear and convex extremal problems. Their approach led also to new insights in the full class S, which is more difficult to handle because no integral representation of Herglotz type is available.
The book gives a unified presentation of the theory that has evolved, including material on subordination and majorization. It begins with introductory chapters on the classical distortion theorem and related topics, special families, subordination, the Herglotz formula, the Krein- Milman theorem (stated without proof), and the structure of continuous linear functionals over the space \({\mathcal A}\) of all functions analytic in \(\Delta\). A key theorem asserts that if \({\mathcal F}\) is a compact subset of \({\mathcal A}\) and if J is a continuous convex functional on its closed convex hull H\({\mathcal F}\), then the maximum of J over H\({\mathcal F}\) is attained in \({\mathcal F}\) and is attained at an extreme point of H\({\mathcal F}\). In later chapters the extreme points of the various families are identified and used to solve extremal problems for linear functionals (description of support points) and for certain convex functionals such as integral means. The same approach is then adapted to the study of extremal problems over \(H^ p\) spaces. A final chapter uses extreme- point theory to describe the regions of variability \[ V_ n({\mathcal F},\zeta)=\{(f(\zeta),f'(\zeta),...,f^{(n-1)}(\zeta)):\quad f\in {\mathcal F}\} \] for fixed \(\zeta\in \Delta\), for some families \({\mathcal F}\) where \(V_ n({\mathcal F},\zeta)\) is a convex set in \({\mathbb{C}}^ n.\)
The book is well organized and clearly written. Although the topic is rather specialized, the authors have made it accessible and attractive to outsiders while producing a valuable reference for research workers in the field. It is an interesting and unusual book.
Reviewer: P.L.Duren

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
46A55 Convex sets in topological linear spaces; Choquet theory
30D55 \(H^p\)-classes (MSC2000)