Recent zbMATH articles in MSC 30-02https://www.zbmath.org/atom/cc/30-022021-03-30T15:24:00+00:00WerkzeugResearch problems in function theory. Fiftieth anniversary edition. Updated and expanded edition of the 1967 original published by The Athlone Press.https://www.zbmath.org/1455.300022021-03-30T15:24:00+00:00"Hayman, Walter K."https://www.zbmath.org/authors/?q=ai:hayman.walter-kurt"Lingham, Eleanor F."https://www.zbmath.org/authors/?q=ai:lingham.eleanor-fThe first edition of Research Problems in Function Theory was published in [\textit{W. K. Hayman}, Research problems in function theory. London: University of London (1967; Zbl 0158.06301)] and it has served as an important source for function theorists ever since its appearance. This book is an updated version of the original list, where the authors go through each problem in the list detailing the possible progress that has been made in solving them. In the beginning of each of the nine chapters there is an introduction, by a distinguished researcher in the relevant field, going over the key developments and most important problems in the subject area of the chapter.
The first chapter ``Meromorphic functions'' has an introduction by A. Eremenko. He singles out the Nevanlinna inverse problem solved by \textit{D. Drasin} [in: Symp. complex Analysis, Canterbury 1973, 31--41 (1974; Zbl 0291.30015)] and its finite-order counterparts, and mentions two important developments in Nevanlinna theory outside of the problems listed in the book. These are \textit{K. Yamanoi}'s generalization of the second main theorem for small functions [Int. J. Math. 17, No. 4, 417--440 (2006; Zbl 1101.30031)], and a solution of the Gol'dberg conjecture, also by \textit{K. Yamanoi} [Proc. Lond. Math. Soc. (3) 106, No. 4, 703--780 (2013; Zbl 1300.30067)].
In the introduction to the second chapter ``Entire functions'' P. J. Rippon discusses the relationship of the problems listed in the chapter to the theories on asymptotic values of entire functions, ordinary differential equations and complex dynamics.
Chapter three ``Subharmonic and Harmonic Functions'' is introduced by S. J. Gardiner, who points out three important solved problems in the list. They are the problem of Littlewood concerning bounded continuous functions possessing a one-radius mean value property, solved by \textit{W. Hansen} and \textit{N. Nadirashvili} [Acta Math. 171, No. 2, 139--163 (1993; Zbl 0808.31004)], the verification of boundary Harnack principle by \textit{B. E. J. Dahlberg} [Arch. Ration. Mech. Anal. 65, 275--288 (1977; Zbl 0406.28009)] and \textit{J.-M. G. Wu} [Ann. Inst. Fourier 28, No. 4, 147--167 (1978; Zbl 0368.31006)] and the treatment of asymptotic paths of subharmonic functions by \textit{B. Fuglede} [Math. Ann. 213, 261--274 (1975; Zbl 0283.31001)]. On problems still remaining open, Gardiner mentions, firstly, the problem due to Lipman Bers on whether a non-constant harmonic function on the unit ball \(\mathbb{R}^3\) that is smooth up to the boundary can vanish with its normal derivative on a set of positive surface area measure, and secondly, the classification of null quadrature domains in higher dimensions.
The introduction to the fourth chapter ``Polynomials'' has been written by E. Crane. He describes the connection of the problems listed in the chapter to approximations to holomorphic functions, and to the theories of algebraic geometry, holomorphic dynamics and planar electrostatics. Crane points out a conjecture by Sendov on the locations of critical points of polynomials, and concludes by discussing an important conjecture by Stephen Smale on mean values of polynomials, formulated after Hayman's original list of problems first appeared.
The fifth chapter ``Functions in the Unit Disc'' has an introduction by L. R. Sons, who singles out several interesting problems in this chapter including conjectures by Littlewood, K. F. Barth and J. G. Clunie, as well as the determination of the values of Bloch's, Landau's and schlicht Bloch's constants.
Bieberbach's conjecture states that the \(n\)th coefficient in the power series of a univalent function is no greater than \(n\). In the introduction to chapter six ``Univalent and Multivalent Functions'', Ch. Pommerenke discusses the solution to this conjecture and points out the asymptotic version of Bieberbach's conjecture due to \textit{W. K. Hayman} [Proc. Lond. Math. Soc. (3) 5, 257--284 (1955; Zbl 0067.30104)]. In addition, he singles out the BCJK conjecture, according to which if \(\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}\), \(p\in(-\infty,\infty)\), and
\[
B(p)=\sup\{\beta_f(p): f \textrm{ conformal map of }\mathbb{D}\textrm{ into }\mathbb{D}\},
\]
where
\[
\beta_f(p)=\displaystyle \frac{\limsup_{r\to1}\int_{|\zeta|=r}|f'(r\zeta)|^p |d\zeta |}{-\log(1-r)},
\]
then \(B(p)=|p|-1\) for \(|p|\geq 2\) (Brennan conjecture) and \(B(p)=p^2/4\) for \(|p|\leq 2\) (a conjecture by Kraetzer).
Chapter seven ``Miscellaneous'' contains the list of problems which do not belong to any of the other categories. In the introduction to this chapter D. Sixsmith points out a problem due to Fuchs, which asks, given domains
\(D_1, D_2 \subset \{z\in\mathbb{C}:|z|<R\}\) with hyperbolic metrics \(\lambda_1|dz|\) and \(\lambda_2|dz|\), respectively, and denoting by \(\lambda|dz|\) the hyperbolic metric in \(D_1\cap D_2\), what is the least number \(A=A(R)\) such that \(\lambda(z)<A(\lambda_1(z)+\lambda_2(z))\)?
Sixsmith also discusses a problem posed by Rubel, who asked if there exists a sequence \(\{z_n\}_{n=1}^\infty\) of distinct complex numbers such that
\[\sum_{n=1}^\infty \frac{1}{|z_n|} < +\infty\]
and
\[\sum_{n=1}^\infty \frac{1}{z-z_n} \not=0\]
for all \(z\in \mathbb{C}\). This problem has a physical interpretation as follows: If infinitely many electrons are placed in the plane, is there necessarily an equilibrium point?
Chapter eight ``Spaces of Functions'' begins with an introduction due to F. Holland, who points out that problems listed in this chapter are mainly about Banach spaces of analytic functions in the unit disc, with different kinds of growth conditions. Examples of such spaces are the Bergman, Besov and Hardy spaces, for instance. An attempt to solve most, but not all, of the problems in this chapter is likely to require knowledge of both complex analysis and operator theory. Holland has picked several problems which may have an approach from an alternative point of view.
In the introduction to Chapter nine ``Interpolation and Approximation'' J. L. Rovnyak discusses two important problems relevant to this subject. The first one is the characterization of interpolating sequences. Namely, let \(B\) be the set of bounded analytic functions on \(\mathbb{D}\), and let a sequence \(\{z_n\}\) in \(\mathbb{D}\) be called interpolating for \(B\) if for every bounded sequence \(\{w_n\}\subset \mathbb{C}\) there exists \(f\in B\) such that \(f(z_n)=w_n\) for all \(n\in\mathbb{N}\). According to a conjecture by R. C. Buck a sequence is interpolating for \(B\) if it approaches the boundary fast enough. The second problem is the corona problem which asks if \(\mathbb{D}\) is dense in the maximal ideal space \(\mathcal{M}\) of \(B\), where \(\mathcal{M}\) is defined as the set of nonzero homomorphisms from \(B\) into \(\mathbb{C}\). The other alternative is that there are unknown points of \(\mathcal{M}\) beyond the closure of \(\mathbb{D}\) that form a corona about the disc. The corona problem was solved by \textit{L. Carleson} [Ann. Math. (2) 76, 547--559 (1962; Zbl 0112.29702)] who showed that the corona does not exist. However, many interesting related problems still remain open.
At the end of each chapter there is a list of new research problems. The book is concluded by a problem reference table explaining in which edition of the research problems in function theory, or another problem collection, one can find the original statement of each problem, and where information on the progress of solving the problem has been added. The list of references contains over 1000 entries on central problems in function theory. This book is superbly written, it is a pleasure to read, and it will surely prove to be an indispensable tool for all researchers interested in function theory. Inspirational problems are essential in keeping any branch of mathematics active and developing. This up-to-date list of problems will surely keep function theorists intrigued for further 50 years or more.
Reviewer: Risto Korhonen (Joensuu)