Derivatives of Inner Functions

176 Pages·2013·1.557 MB·English

by  Mashreghi, Javad

Fields Institute Monographs
VOLUME 31
The Fields Institute for Research in Mathematical Sciences
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Javad Mashreghi
Derivatives of Inner Functions
123
TheFieldsInstituteforResearch
intheMathematicalSciences
JavadMashreghi
De´partementdemathe´matiquesetde
statistique
Universite´Laval
Que´bec,QC
Canada
ISSN1069-5273 ISSN2194-3079(electronic)
ISBN978-1-4614-5610-0 ISBN978-1-4614-5611-7(eBook)
DOI10.1007/978-1-4614-5611-7
SpringerNewYorkHeidelbergDordrechtLondon
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Ouvrez une ´ecole, vous fermerez une prison
Victor Hugo
It was in 1934 that Pahlavi High School was established in my hometown,
Kashan. In 1946, the school moved to a new building constructed over a
rather vast area and featuring an awesome architectural design. The school
was renamed Imam Khomeini High School after the 1979 revolution. Over
the years, numerous bright minds were trained in the stimulating environ-
mentofthisschool.Beforelong,theimpressiveschoolbuildinghadbecomea
reminderofallthegreatintellectualswhohadeitherstudiedortaughtthere.
To the utter regret of the latter, however, the building was completely de-
molishedin 1995,only to giveway to the current, incomplete one. I dedicate
this monographto all the caring and respectedmen, teachersand employees
alike,whokeptthe flame ofeducationalightformanyyearsinthis institute.

Preface
Infinite Blaschke products were introduced by W. Blaschke in 1915 [9]. In
1929,R. Nevanlinna introducedthe class ofbounded analytic functions with
almost everywhere unimodular boundary values [35]. However, the term in-
nerfunction wascoinedmuchlaterbyA.Beurlinginhisseminalworkonthe
invariantsubspacesoftheshiftoperator[8].Thefirstextensivestudiesofthe
properties of inner functions were made by O. Frostman [22], W. Seidel [43]
andF.Riesz[40].Theireffortstounderstandthe zerosandboundarybehav-
iorofboundedanalyticfunctionsledtothecelebratedcanonicalfactorization
theorem. The special factorization that we need says that each inner func-
tion is the product of a Blaschke product and a zero free inner function, the
so called singular inner function, which is generated by a singular measure
residing on the unit circle. Roughly speaking, we can say that the Blaschke
product is formed with the zeros of an inner function inside the open unit
disc, and the singular part stems from its zeros on the boundary.
InJuly2011,E.FricainandIorganizedaconferenceonBlaschke products
and their applications in the Fields Institute (Toronto). There were several
interesting talks about the boundary behavior of inner functions, in partic-
ular Blaschke products, and their derivatives. I felt the need to gather some
classical results in a short monograph for graduate students and as a handy
reference for experts. However, the literature is very vast and it is a difficult
task to choose among various important results. For example, the book of
P. Colwell [16] can provide a panoramic picture of this subject. Hence, I re-
strictedmyselfjusttotheintegralmeansofthe derivativesand,evenforthis
narrow subject, I was very selective.
The Fields Institute exclusively supported our conference on Blaschke
products, and its direction constantly helped us for the production of the
proceedings and this monograph. In particular, I owe profound thanks to
CarlRiehm,theManagingEditorofPublications,forhiscare,guidance,and
enthusiastic support. Last but not the least, I would like to deeply thank
vii
viii Preface
Joseph Cima (University of North Carolina), Ian Graham (University of
Toronto), and Armen Edigarian (Jagiellonian University) who kindly read
the manuscript and made many valuable suggestions. Their remarks enor-
mously improved the quality of text.
Montreal, QC Javad Mashreghi
Contents
1 Inner Functions........................................... 1
1.1 The Poisson Integral of a Measure ........................ 1
1.2 The Hardy Space Hp(D) ................................ 6
1.3 Two Classes of Inner Functions........................... 8
1.4 The Canonical Factorization ............................. 12
1.5 A Characterizationof Blaschke Products .................. 17
1.6 The Nevanlinna Class N and Its Subclass N+ ............. 20
1.7 Bergman Spaces........................................ 23
2 The Exceptional Set of an Inner Function ................ 27
2.1 Frostman Shifts and the Exceptional Set E ................ 27
ϕ
2.2 Capacity .............................................. 30
2.3 Hausdorff Dimension.................................... 32
2.4 E Has Logarithmic Capacity Zero........................ 35
ϕ
2.5 The Cluster Set at a Boundary Point ..................... 37
3 The Derivative of Finite Blaschke Products............... 39
3.1 Elementary Formulas for B(cid:2) ............................. 39
3.2 The Cardinality of the Zeros of B(cid:2) ........................ 40
3.3 A Formula for |B(cid:2)|...................................... 42
3.4 The Locus of the Zeros of B(cid:2) in D ........................ 46
3.5 B Has a Nonzero Residue ............................... 48
4 Angular Derivative ....................................... 51
4.1 Elementary Formulas for B(cid:2) and S(cid:2) ....................... 51
4.2 Some Estimations for Hp-Means.......................... 54
4.3 Some Estimations for Ap-Means.......................... 57
4.4 The Angular Derivative ................................. 60
4.5 The Carath´eodory Derivative ............................ 62
4.6 Another Characterizationof the Carath´eodoryDerivative.... 68
ix