This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains).

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Iteration of meromorphic functions

Studies in Advanced Mathematics

The space of all Bloch functions will be denoted by «^. There are several distinct characterisations of Bloch functions: (i) f € @l if and only if the f amily of functions f,(*) = ΐ(φ(*ϊ) f(?(0)); φ(ζ) = -ff^·, | a |< l, is finitely normal in D [28]. (ii) Let df(z) denote the largest schlicht disc around the point f(z) on the Riemann image surf ace by /. Then f € SS if and only if b = sup df(z) < oo. z€D This is equivalent to a well-known theorem of Bloch. In quantitative terms

On Bloch functions and normal functions.

Modular elliptic curves and fermat's last theorem

A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski inequality for convex sets is proved as a corollary.

A mass transportation approach to quantitative isoperimetric inequalities

Research Problems in Function Theory

Suppose that 1.1 is a transcendental integral function. In this article we develop the theory initiated by Wiman [22, 23] and deepened by other writers including Valiron [18, 19, 20], Saxer [15], Clunie [4, 5] and Kovari [10, 11], which describes the local behaviour of f(z), near a point where | f(z) | is large, in terms of the power seriesf of f(z).

Walter K. Hayman

Papers

Research Problems in Function Theory

The Local Growth of Power Series: A Survey of the Wiman-Valiron Method

On Analytic Functions of Bounded Mean Oscillation

Research Problems in Complex Analysis

On the Second Hankel Determinant of Mean Univalent Functions