The content of this thesis can be divided into two broad topics. The first half investigates the deficient values and deficient functions of certain classes of meromorphic functions. Here a value is called deficient if a function takes that value less often than it takes most other values. It is shown that the derivative of a periodic meromorphic function has no finite non-zero deficient values, provided that the function satisfies a necessary growth condition. The classes B and S consist of those meromorphic functions for which the finite critical and asymptotic values form a bounded or finite set. A number of results are obtained about the conditions under which members of the classes B and S and their derivatives may admit rational, or slowly-growing transcendental, deficient functions. The second major topic is a study of real functions -- those functions which are real on the real axis. Some generalisations are given of a theorem due to Hinkkanen and Rossi that characterizes a class of real meromorphic functions having only real zeroes, poles and critical points. In particular, the assumption that the zeroes are real is discarded, although this condition reappears as a conclusion in one result. Real entire functions are the subject of the final chapter, which builds upon the recent resolution of a long-standing conjecture attributed to Wiman. In this direction, several conditions are established under which a real entire function must belong to the classical Laguerre-Polya class LP. These conditions typically involve the non-real zeroes of the function and its derivatives.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:523573 |
| Date | January 2010 |
| Creators | Nicks, Daniel A. |
| Publisher | University of Nottingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://eprints.nottingham.ac.uk/11327/ |
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