The first and last papers of Harald Bohr deal with ordinary Dirichlet series and their order (or Lindelöf) function μ(σ) (= inf{α;f(σ + it) + 0(|t|α)}). The Lindelöf hypothesis is μ(σ) = inf(0, ½ − t) when an = (−1)n. Are there ordinary Dirichlet series with −l < μ′ (σ) < 0 for some σ? A negative answer would imply Lindelöf's hypothesis. This is the last problem of Harald Bohr. This paper gives (1) a review on Bohr's results on ordinary Dinchlet series; (2) a review on results of the author and of Queffelec on “almost sure” and “quasi sure” properties of series with the solution of a previous problem of Bohr; (3) the following answer to the last problem: μ′(σ) can approach − ½, and necessarily μ(σ + μ(σ) + ½) = 0. The characterization of the order functions of ordinary Dirichlet series remains an open question.

Let W+ denote the Banach algebra of all absolutely convergent Taylor series in the open unit disc. We characterize the finitely generated closed and prime ideals in W+. Finally, we solve a problem of Rubel and McVoy by showing that W+ is not coherent.

The main purpose of this article is to outline for non-compact Riemann surfaces the development of a theory of T-invariant algebras similar to that developed by T. W. Gamelin [7] in the case of the plane. The main idea is to introduce, using the global local uniformizer of R. C. Gunning and R. Narashimhan, a Cauchy transform operator for the Riemann surface which operates on measures and solves an inhomogeneous -equation. This, in turn, can be used, analogously to the Cauchy transform on the plane, to develop meromorphic (rational) approximation theory. We sketch the path of the development but omit most of the details when they are very much similar to the planar case. Our presentation follows closely that of [7]. The original motivation for this study was to obtain more information on Gleason parts useful in the study of Carleman (tangential) approximation theory (see [5]).

We have defined and studied some pseudogroups of local diffeomorphisms which generalise the complex analytic pseudogroups. A 4-dimensional (or 8-dimensional) manifold modelled on these ‘Further pseudogroups’ turns out to be a quaternionic (respectively octonionic) manifold.

We characterise compact Further manifolds as being products of compact Riemann surfaces with appropriate dimensional spheres. It then transpires that a connected compact quaternionic (H) (respectively O) manifold X, minus a finite number of circles (its ‘real set’), is the orientation double covering of the product Y × P2, (respectively Y×P6), where Y is a connected surface equipped with a canonical conformal structure and Pn is n-dimensonal real projective space.

A corollary is that the only simply-connected compact manifolds which can allow H (respectively O) structure are S4 and S2 × S2 (respectively S8 and S2×S6).

Previous authors, for example Marchiafava and Salamon, have studied very closely-related classes of manifolds by differential geometric methods. Our techniques in this paper are function theoretic and topological.

For α≥0 and β≥0 we denote by K (α, β) the Kaplan classes of functions f analytic and non-zero in the open unit disk U = {z: |z| < 1} such that f ∈ K(α, β), if, and only if, for θ1 < θ2 < θ1 + 2π and 0 < r < 1,

Where

In this paper we determine the lower bound on |z| = r < 1 for the functional Re{αp(z) + β zp′(z)/p(z)}, α ≧0, β ≧ 0, over the class Pk (A, B). By means of this result, sharp bounds for |F(z)|, |F',(z)| in the family and the radius of convexity for are obtained. Furthermore, we establish the radius of starlikness of order β, 0 ≦ β < 1, for the functions F(z) = λf(Z) + (1-λ) zf′ (Z), |z| < 1, where ∞ < λ <1, and .

Convergent iterative sequences are constructed for the polynomials fm = z + zm, m ≧ 2, with initial point the lemniscate {z: |fm (z)| ≦1}. In the particular case m = 2 convergent iterative sequences are constructed also for f-1m, (z) with an arbitrary initial point. The method is based on a certain variational principle which allows reducing the problem to the well known situation of an analytic function mapping a simply connected domain into a proper subset of itself and possessing a fixed point in the domain.

Methods from the theory of orthogonal polynomials are extended to L-polynomials . By this means the authors and W. B. Jones (J. Math. Anal. Appl. 98 (1984), 528–554) solved the strong Hamburger moment problem, that is, given a double sequence , to find a distribution function ψ(t), non-decreasing, with an infinitenumber of points of increase and bounded on −∞ < t < ∞, such that for all integers . In this article further menthods such as analogues of the Lioville-Ostrogradski formula and of the Christoffel-Darboux formula are developed to investigated When the moment porblem has a unique solution. This will be the case if and only if a sequence of nested disks associated with the sequence has only a point as its intersection (the so called limit point case).

Suppose that f(z) is non-constant and meromorphic in the plane and that, for some k≥= 1, a0(z),…, ak(z) are meromorphic in the plane with

for j' = 0,…, k. Here, using standard notation from [3], S(r,f) denotes any quantity satisfying S(r,f) = o(T(r,f)) as r→ ∞, possibly outside a set of finite linear measure. Then, setting

we have ([3, p. 57])

Theorem A. Suppose that f(z) is non-constant and meromorphic in the plane, and thatψ (z) given by (1.2) and (1.1) and is non-constant. Then

where N0(r, l/ψ') counts only zeros of ψ' which are not zeros of ψ − 1, and thecounting functions count points without regard to multiplicity.

Let S(m, M) be the set of functions regular and satisfying │zf′(z)/f(z) – m│< M in │z│ <1, where│m –│ <M;≦ m; and let S*(p) be the set of starlike functions of order p, 0≦ p <1. In this paper we obtain integral operators which map S(m, M) into S(mM) and S* (p) × S(mM) into S*(p). Our results improve and generalize many recent results.

For a non-constant entire or rational function f normalized by f(0) = 0, f′(0) = 1, f″(0) ≠ 0, which is not a Möbius tranformation, the existence of a sequence is established which has the properties . The result certainly implies f(0)= |f(0)|= 1, so these conditions cannot be omitted. The condition f″ (0)≠ 0 can be replaced by f(k)(0) ≠ 0 for some k ≥ 2.

Given a measurable function k non-negative a.e. on the circle |z| = 1, when is the outer function Tk (see(1.3)) continuous on the disk |z| < 1 and further, Dirichlet-finite: We shall show, among other results, that the answer is in the positive if , with ess inf k > 0.

In this paper the authors obtain a series transformation of concentric circles into concentric rectangles.

A function

analytic and univalent in U = {z: |z| < 1} is said to be starlike there, if f(U) is f starshaped with respect to the origin, that is, if w ε f(U) implies tw ε f(U) for 0 ≤t ≤ 1. We denote by S* the class of all such functions. The Koebe function; k(z) = z(l – z)-2, z ε U, maps U onto the complex plane minus a slit along the I negative real axis from - ¼ to ∞, and thus belongs to the class S*. Recently Leung [4] has shown that, if

then, for f ε S*,

for every p > 0.

The iterative behaviour of polynomials is contrasted with that of small transcendental functions as regards the existence of unbounded domains of normality for the sequence of iterates.

In a manuscript discovered in 1976 by George E. Andrews, Ramanujan states a formula for a certain continued fraction, without proof. In this paper we establish formulae for the convergents to the continued fraction, from which Ramanujan's result is easily deduced.

The classes of prestarlike functions Rα, α ≧ – 1, were studied recently by St. Ruscheweyh. The author generalizes and extends these classes. In particular the author obtains the radius of Ra+1 for the class Rα, α ≧ –1.