Research Problems in Function Theory

288 Pages·2019·3.059 MB·English

by  Hayman, Walter K., Lingham, Eleanor F.

Problem Books in Mathematics
Walter K. Hayman
Eleanor F. Lingham
Research Problems
in Function Theory
Fiftieth Anniversary Edition
Problem Books in Mathematics
Series Editor
Peter Winkler
Department of Mathematics
Dartmouth College
Hanover, NH
USA
More information about this series at http://www.springer.com/series/714
Walter K. Hayman Eleanor F. Lingham
(cid:129)
Research Problems
in Function Theory
Fiftieth Anniversary Edition
123
Walter K.Hayman Eleanor F.Lingham
Department ofMathematics DepartmentofEngineeringandMathematics
Imperial CollegeLondon Sheffield Hallam University
London,UK Sheffield,SouthYorkshire, UK
ISSN 0941-3502 ISSN 2197-8506 (electronic)
Problem Booksin Mathematics
ISBN978-3-030-25164-2 ISBN978-3-030-25165-9 (eBook)
https://doi.org/10.1007/978-3-030-25165-9
MathematicsSubjectClassification(2010): 30D20,30D30,30D35,32H50,30C15,30C55
FirsteditionpublishedbyTheAthlonePress,London,1967,under:Hayman,W.K.
©SpringerNatureSwitzerlandAG1967,2019
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Preface
In1967,thefirstauthorwrote‘ResearchProblemsinFunctionTheory’,whichwas
publishedbytheAthlonePressoftheUniversityofLondon.Therehadbeenearlier
problem collections by Littlewood [678] and by Erdős [299] in the 1960s.
Subsequent additions to the 1967 booklet were [504], [505], [36], [180], [86] and
[156]. It was the idea of the second author to give an account of the progress that
has been made on these problems in the intervening half-century, and for the
addition of new problems to what is now known as ‘Hayman’s List’.
We are most grateful to the many mathematicians worldwide who helped to
make this book possible by answering our queries and suggesting corrections,
amendments, omissions and additions. Among these, we would like to single out
the following persons and organisations:
– Alex Eremenko, who provided us with the updated information for most of
Chaps. 1 and 2;
– The nine colleagues who have written prefaces for the chapters: A. Eremenko,
P.J.Rippon,S.J.Gardiner,E.Crane,L.R.Sons,Ch.Pommerenke,D.Sixsmith,
F. Holland and J.L. Rovnyak;
– Ourfriendswhohavespentahugeamountoftimereadingthiswork,including
J.K. Langley, D.A. Brannan, Ch. Pommerenke and J. Becker;
– The four reviewers who have greatly improved this work;
– TheMathematicalResearchInstituteatOberwolfach,whichallowedustomake
a start on writing this book, and the London Mathematical Society, which
supported its completion;
– zbMATH for providing us with access to its database;
– Rémi Lodh and the Springer Press, who agreed to publish it.
Thereadermayfindithelpfultoknowalittleabouthowthisbookisstructured.Itis
the amalgamation of the original edition, the additions and research which has
occurredoverthelastfewdecades.Thisperhapsexplainsitsidiosyncrasies,suchas
whythe‘Miscellaneous’chapteristheseventhofnine,andwhysomeofthemore
important or famous problems are buried in the middle of chapters. Also, as the
language of mathematics has changed over the last half-century, we have adjusted
v
vi Preface
chaptertitlesandproblemstatementsaccordingly,forexample,‘schlicht’hasbeen
replacedby‘univalent’and‘integralfunctions’arenowknownas‘entirefunctions’.
Anyreaderwhoisgreatlyinterestedinaparticularproblemwillfinddirectionhere,
butisremindedofthevalueofalsocheckingtheoriginalstatementandtheprogress
informationinthesubsequentadditions.Forthis,theproblemreferencetableatthe
end ofthis bookwill beuseful.
Any science thrives on its problems, and we hope that this book will keep
function theory flourishing for a while longer.
London, UK Walter K. Hayman
Sheffield, UK Eleanor F. Lingham
April 2019
Contents
1 Meromorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Preface by A. Eremenko. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Progress on Previous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 New Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Preface by P.J. Rippon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Progress on Previous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 New Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3 Subharmonic and Harmonic Functions. . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Preface by S.J. Gardiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Progress on Previous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 New Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1 Preface by E. Crane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Progress on Previous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 New Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Functions in the Unit Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1 Preface by L.R. Sons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Progress on Previous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3 New Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6 Univalent and Multivalent Functions . . . . . . . . . . . . . . . . . . . . . . . . 133
6.1 Preface by Ch. Pommerenke . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2 Progress on Previous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3 New Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
vii
viii Contents
7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.1 Preface by D. Sixsmith. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.2 Progress on Previous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.3 New Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8 Spaces of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.1 Preface by F. Holland. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.2 Progress on Previous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.3 New Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
9 Interpolation and Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
9.1 Preface by J.L. Rovnyak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
9.2 Progress on Previous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.3 New Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Appendix: Tables .. .... .... ..... .... .... .... .... .... ..... .... 241
References.... .... .... .... ..... .... .... .... .... .... ..... .... 245
Chapter 1
Meromorphic Functions
1.1 PrefacebyA.Eremenko
AccordingtoHayman[510],HilbertoncetoldNevanlinna:“Youhavemadeahole
in the wall of Mathematics. Other mathematicians will fill it.” Hayman continues:
“Iftheholemeansthatmanynewproblemswereopenedup,thenthisisindeedthe
case,andIamcertainthatNevanlinnatheorywillcontinuetosolveproblemsasit
hasdoneinthelast50years.”
This chapter is dedicated to the problems on meromorphic functions stated by
various authors during the period 1967–1989 and collected by Hayman and his
collaborators.
Inthisbooka“meromorphicfunction”meansafunctionmeromorphicinthecom-
plexplane.Mostproblemsareabouttranscendentalmeromorphicfunctions(having
anessentialsingularityatinfinity).
The theory of meromorphic functions was mostly created by Nevanlinna in the
1920s,andhewrotetwoinfluentialbooksonit[756, 757].
Thesebooks,especiallythesecondone,containedmanyunsolvedproblems,and
thepresentcollectionmentionsonlyafewofthem.Thischapterreflectsverywell
thedevelopmentofNevanlinnatheoryinthethesecondhalfofthe20thcentury.
HereIwilltrytogiveaverybriefoverviewofthemostimportantproblemsand
theirsolutions.Ofcourse,thisselectionreflectsmyowntaste.
We use the definitions and notation introduced in the beginning of the chapter,
andadd tothisn (r, f),thecounting functionofcriticalpoints ofameromorphic
1
function f, including multiplicities, and the averaged counting function N (r, f),
1
seeUpdate1.33.TheSecondFundamentalTheoremofNevanlinnasaysthat
(cid:2)q
m(r,a , f)+N (r, f)≤2T(r, f)+S(r, f), (1.1)
j 1
j=1
©SpringerNatureSwitzerlandAG2019 1
W.K.HaymanandE.F.Lingham,ResearchProblems
inFunctionTheory,ProblemBooksinMathematics,
https://doi.org/10.1007/978-3-030-25165-9_1