Nicks, Daniel A. (2010) Value distribution of meromorphic functions and their derivatives. PhD thesis, University of Nottingham.
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.
|Item Type:||Thesis (PhD)|
|Uncontrolled Keywords:||Meromorphic function, entire function, Nevanlinna theory, value distribution, deficient value|
|Faculties/Schools:||UK Campuses > Faculty of Science > School of Mathematical Sciences|
|Deposited By:||Mr Daniel Nicks|
|Deposited On:||17 Dec 2010 11:26|
|Last Modified:||17 Dec 2010 11:26|
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