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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.ThezbMATHlogoappearingonthebackcoveraswell asthename“zbMATH”areregisteredtrademarksofFIZKarlsruheandareusedwithpermission.The publisherandauthorsacknowledgetheassistanceofzbMATHinthepreparationofthisbook. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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