Value distribution of meromorphic functions and their derivatives

Nicks, Daniel A. (2010) Value distribution of meromorphic functions and their derivatives. PhD thesis, University of Nottingham.

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Abstract

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)
Supervisors:Langley, J.K.
Uncontrolled Keywords:Meromorphic function, entire function, Nevanlinna theory, value distribution, deficient value
Faculties/Schools:UK Campuses > Faculty of Science > School of Mathematical Sciences
ID Code:1327
Deposited By:Mr Daniel Nicks
Deposited On:17 Dec 2010 11:26
Last Modified:17 Dec 2010 11:26

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