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We study the extreme points of two classes of polynomials of degree at most n:
It turns out that f ∈ Ext if and only if Re f(eiθ) has exactly 2n zeros in [0, 2π). On the other hand, if f∈Hn and 1−|f(eiθ)|2 has 2n zeros in [0, 2π), then either f ∈ Ext Hn or else f(z) = α + βzn where |α|+|β| = l and αβ≠0; if 1−|f(eiθ)|2 has 2m zeros, 2n, then f may or may not belong to Ext Hn.
The split capital investment trust crisis brought into focus the need for more reliable risk assessment techniques for shares in the sector. We discuss the strengths and weaknesses of traditional pricing and risk description measures for split capital investment trusts (e.g. gross redemption yield, cover, hurdle rates) and ways of making these more useful. We then examine the application of traditional option pricing techniques and discuss the problems encountered in this approach. Finally, we propose the use of stochastic modelling to deal more effectively with the complexities involved in both pricing shares and understanding their risks.
An international conference on complex analysis was held in Canterbury in July 1973. Some of the world's most prominent complex analysts attended and some outstanding open problems had their first solutions announced there. These are reflected in this set of Proceedings. Almost all of the contributions are abstracts of talks given at the symposium. The final part of this volume is a section on research problems contributed by members of the conference and a report on a previous collection of problems edited by Professor W. K. Hayman after an earlier conference in 1964. This book is essential reading for research workers and graduate students interested in complex analysis.
In September 1964 a N. A. T. O. advanced study institute was held at Imperial College, London University. Arising out of this a book of problems with the above title was collected by me and published by the Athlone Press in 1967. It is my aim in this article to report on the progress that has been made on the problems of this book in so far as I am aware of it. I have given the references that I could easily trace. However, much of the work is as yet unpublished. Some of the solutions were first announced at the present conference in Canterbury. After discussing progress on the old problems I shall describe some new problems which were contributed by members of this conference.
In addition to the above problem collection the attention of the reader should also be drawn to the recent collection of Ch. Pommerenke.
I acknowledge with gratitude the very considerable help I have had from Professor Clunie, particularly in the tracing of references.
Progress on the previous problems
The problems are numbered as in the book referred to in the introduction [P], The notation and terminology of that book will be used throughout.
This problem has now been completely settled by Drasin in a result first announced at the Canterbury conference. An abstract appears elsewhere in these proceedings.
In this note we continue the study, initiated in , of the class S*(α) of functions
that are analytic and univalent in the unit disc U and satisfy the condition
S*(1) is the frequently studied class of univalent star-like functions. For each α, S*(α) is a subclass of the class K(α) of close-to-convex functions of order α introduced by Pommerenke . Properties of the class S*(α) proved useful in studying the coefficient behaviour of bounded univalent functions that are analytic and map U onto a convex domain . In this note we investigate the problem of determining
It has been generally held that the severe anaemia and other pathogenic effects of infestation by hookworm in man and dogs, and by Haemonchus contortus in sheep, are insufficiently explained by the amount of blood removed by the parasites. The action of haemotoxins has therefore been postulated.
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