Some Results On Normal Families Of Meromorphic Functions

In 1907, P. Montel ([82]) introduced the concept of normal families. A family T of meromorphic (holomorphic) functions in a domain D C C is said to be normal in D, if each sequence {f_n}(?) F has a subsequence {f_nj} which converges spherically locally uniformly in D, to a meromorphic (holomorphic) function or oo. See Hayman ([62]), Schiff ([98]) and Yang ([111]). The theory of normal families plays an important role in complex dynamics.A heuristic principle attributed to A. Bloch relates normal families in a domain and meromorphic (holomorphic) functions on the plane. It says that the family of all meromorphic (holomorphic) functions in a given domain with a certain property is likely to be normal under the hypothesis that meromorphic (entire) functions on the plane with this property are constant, or under a weaker hypothesis, suggested by W. Bergweiler and L. Zalcman., cf. [17], that the set of all meromorphic (entire) functions on the plane with this property forms a normal family in C. For an excellent discussion of Bloch's principle we refer to the books by W. K. Hayman [62], J. Schiff [98], L. Yang [111] and the papers by W. Bergweiler [17] and L. Zalcman [116, 117]. Although Bloch's principle is not true in general (interesting counterexamples can be found in [96] and in §2.6 and §3.6 of the present thesis), it has stimulated a lot of research in the theory of normal families. Indeed, all normality criteria in this thesis are closely related to Bloch's principle.The main aim of this thesis is to study normal families with conditions on fixed points of iterated functions and their applications. Normal families with conditions on fixed points of composite functions and shared values, as well as their applications are also studied. This thesis is divided into six chapters.In Chapter I, we state the concept of normal families and some basic results such as the theorems of Nevanlinna and Ahlfors and the Zalcman...