W. K. Hayman

The Strength of Cartan's Version of Nevanlinna Theory

In 1933 Henri Cartan proved a fundamental theorem in Nevanlinna theory, namely a generalization of Nevanlinna’s second fundamental theorem. Cartan’s theorem works very well for certain kinds of problems. Unfortunately, it seems that Cartan’s theorem, its proof, and its usefulness, are not as widely known as they deserve to be. To help give wider exposure to Cartan’s theorem, the simple and general forms of the theorem are stated here. A proof of the general form is given, as well as several applications of the theorem.

On the computation of modules of long quadrilaterals

AbstractWe study quadrilateralsQ which are given by two intervals on {ζ:Im ζ = 0} and {ζ:Im ζ = 1}, and two Jordan arcsγ1,γ2, in {ζ:0 ⩽ Im ζ ⩽ 1} connecting these two intervals. Many practical problems require the determination of the modulem(Q) ofQ, but ifQ is “long,” i.e., if

Characteristic, maximum modulus and value distribution

h: = \min \{ \operatorname{Re} \zeta :\zeta \in \gamma _2 \} - \max \{ \operatorname{Re} \zeta :\zeta \in \gamma _1 \}

Some inequalities in the theory of functions

is large, the conformal mapping ofQ onto a rectangle becomes difficult because of the crowding effect. However, it turns out thatm(Q)−h approaches a limit very quickly, ash→∞, and we can therefore estimatem(Q)−[m(Q1)+m(Q2)] whenQ is decomposed into two smaller quadrilateralsQ1,Q2. Several numerical examples are presented. This domain decomposition method goes back to Papamichael and Stylianopoulos [9].

On the coefficients of certain automorphic functions

Suppose that f is bounded and continuous in a domain D in Rk. Then there exists a best harmonic approximant h to f in the uniform norm. If D is a Jordan domain, f is continuous in D, and h is continuous in D, then h is unique and can be characterised in terms of the sets in D where h — f assumes the extreme values +m. Examples are given to show that if these hypotheses are relaxed in various ways the conclusion may fail. For instance h need not be continuous in D, even if f is continuous in D, and if f is only bounded and continuous in D, h need not be unique.

Conformal Mapping of Parabola-Shaped Domains

Let/be an entire function such that log M(r, f) T(r, f) on a set E of positive upper density. Then / has no finite deficient values. In fact, if we assume that E has density one and / has nonzero order, then the roots of all equations f(z) = a are equidistributed in angles. In view of a recent result of Murai [6] the conclusions hold in particular for entire functions with Fejér gaps.

Tauberian theorems for multivalent functions

Let be regular in U = {| z | f ( z ) all lie in a domain D in the w -plane. If certain geometrical restrictions are made on D we can deduce growth conditions on the maximum modulus the means and the coefficients α n . Good bounds for M ( r , f ) have been obtained under various conditions on D .

Univalent functions of fast growth with gap power series

In this paper we investigate the following problem. We suppose given a sequence of complex values w n , defined for n = 0, 1, 2, …, and for n = ∞, and such that while at least one w n differs from zero and ∞. We consider functions f ( z ), which are regular in | z | w n , and we investigate the effect of this restriction on the rate of growth of the function, as given by the maximum modulus