Man, Chiang Yik; (1991) Schwarzian Derivatives and Second Order Differential Equations. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis is divided into two parts. The first part consisting of the first four chapters. We study mainly the properties of complex function f when some conditions are imposed on the Schwarzian derivative of f. In Chapter 1, we define the notions of quasiconformal mappings and investigate conditions that allow f to have a quasiconformal extension outside the unit disc A to the extended complex plane. We used the method of Ahlfors to obtain and extend the criteria, involving Schwarzian derivatives, obtained earlier by Ahlfors, Krzyz and Lewandowski etc. In Chapter 2 we shall look at the domain constant Ω(A) introduced by Lehto with the norm of the Schwarzian and logarithmic derivatives. In Chapter 3, we consider the Schwarzian S(f, z) alone and show that if it is sufficiently small and the second coefficient is also small (depending on S(f, z)), then f is a a-strongly starlike function for one such constant, and convex for a smaller constant. Other properties of f when S(f, z) is small are also investigated. The method used depends heavily on the second order differential equations. Chapter 4 considers the same problems as in Chapter 3, but solved by the use of the Clunie-Jack principle. The advantage of this principle is that it enables us to consider a more restricted class of functions. The results obtained complement that of Chapter 3. With the Clunie-Jack principle, we give alternative proofs of results, in one case with an extension, obtained previously by Miller and Mocanu. Chapter 5 is our second part. Here we consider the distribution of the zero sequences of the solutions of a second order differential equation, with the given coefficient being an entire transcendental function of finite order. This has been considered by Bank, Laine, Langley and Rossi etc.

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