Meromorphic functions of one complex variable. A survey

This is an appendix to the English translation of the book by A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphic functions, Moscow, Nauka, 1970. An English translation of this book is to be published soon by the AMS. In this appendix we survey the results obtained on the topics of the book after 1970. The literature on meromorphic functions is very large. There is a comprehensive survey [62] that contains everything that was reviewed on the topic in the Soviet “Referativnyi Zhurnal” in 1953–1970, and a later large survey [67]. More recent surveys [30], [80] and [48] are shorter and have narrower scope. Some books on specific topics in the theory of meromorphic functions published after 1970 are [18], [20], [79], [100], [137], [134], [138], [162]. A survey of the fast developing subject of iteration of meromorphic functions is [7]. Here we give a short survey of some results which are closely related to the problems considered in this book. Thus we do not include many important topics, like geometric theory of meromorphic functions, iteration, composition, differential and functional equations, normal families, Borel and Julia directions, uniqueness theorems, and most regrettably, holomorphic curves and quasiregular mappings. Chapter I. Mokhon’ko [121] proved the following generalization of (6.29), Ch. I. Let R(w, z) be a rational function of w whose coefficients are meromorphic functions h(z) satisfying T (r, h) = O(φ(r)), where φ is a fixed positive increasing If the domain is not specified explicitly, we mean meromorphic functions in C.