Thesis (Ph.D.)--University of Melbourne, Dept. of Mathematics and Statistics, 2005. / Typescript (photocopy). Includes bibliographical references and index (leaves 143-151).

I use novel splittings of conjugate exponents in Holder’s inequality and other techniques to obtain new inequalities of Hilbert, Hilbert-Pachpatte and Hardy type for series and integrals. The Thesis gives far reaching generalisations of the work of Dragomir-Kim (2003), Pachpatte (1987, 1990, 1992),Handley-Koliha-Pecaric (2000), Hwang-Yang (1990), Hwang(1996), Love-Pecaric (1995) and Mohapatra-Russell (1985) and inequalities for fractional derivatives of integrable functions. (For complete abstract open document)

This thesis concerns the classes of analytic functions on bounded, n-connected domains known as the Smirnov classes Ep, where p > 0. Functions in these classes satisfy a certain growth condition and have a relationship to the more well known classes of functions known as the Hardy classes Hp. In this thesis I will show how the geometry of a given domain will determine the existence of non-constant analytic functions in Smirnov classes that possess real boundary values. This is a phenomenon that does not occur among functions in the Hardy classes. The preliminary and background information is given in Chapters 1 and 3 while the main results of this thesis are presented in Chapters 2 and 4. In Chapter 2, I will consider the case of the simply connected domain and the boundary characteristics that allow non-constant analytic functions with real boundary values in certain Smirnov classes. Chapter 4 explores the case of an n-connected domain and the sufficient conditions for which the aforementioned functions exist. In Chapter 5, I will discuss how my results for simply connected domains extend Neuwirth-Newman's Theorem and finish with an open problem for n-connected domains.