Inequalities and Applications: Conference on Inequalities and Applications, Noszvaj (Hungary), September 2007

354 Pages·2009·2.736 MB·English

by  Catherine Bandle (auth.), Catherine Bandle, László Losonczi, Attila Gilányi, Zsolt Páles, Michael Plum (eds.)

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ISNM
International Series of Numerical Mathematics
Volume 157
Managing Editors:
K.-H. Hoffmann, Bonn
D. Mittelmann, Tempe
Associate Editors:
R. E. Bank, La Jolla
H. Kawarada, Chiba
R. J. LeVeque, Seattle
C. Verdi, Milano
Honorary Editor:
J. Todd, Pasadena
Inequalities
and Applications
Conference on Inequalities and Applications,
Noszvaj (Hungary), September 2007
Catherine Bandle
Attila Gilányi
László Losonczi
Zsolt Páles
Michael Plum
Editors
Birkhäuser
Basel · Boston · Berlin
Editors:
Catherine Bandle Attila Gilányi
Institut Mathematik Zsolt Páles
Universität Basel Department of Analysis
Rheinsprung 21 Institute Mathematics & Informatics
4051 Basel University of Debrecen
Switzerland Pf. 12
Email: [email protected] 4010 Debrecen
Hungary
László Losonczi Email: [email protected]
Department Economic Analysis & [email protected]
Information Technology for Business
University of Debrecen Michael Plum
Kassai út.26 Institut für Analysis
4028 Debrecen Universität Karlsruhe
Hungary Kaiserstrasse 12
Email: [email protected] 76128 Karlsruhe
Germany
Email: [email protected]
2000 Mathematics Subject Classification: 15-06, 15A39,15A42, 15A45, 26D05, 26D07, 26D10,
26D15, 26D20, 26E60, 30A10, 32A22, 34A40, 35J85, 35K85, 35L85, 35R45, 39B72, 41A17,
42A05, 47A30, 47A50, 47A63, 47J20, 49J40, 51M16, 52A40, 58E35, 60E15
Library of Congress Control Number: 2008935754
Bibliographic information published by Die Deutsche Bibliothek.
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISBN 978-3-7643-8772-3 Birkhäuser Verlag AG, Basel - Boston - Berlin
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Contents
Preface ................................................................... ix
Abstracts of Talks ........................................................ xi
Problems and Remarks ..................................................... xli
List of Participants ....................................................... xlvii
Part I: Inequalities Related to Ordinary and Partial Differential Equations
C. Bandle
A Rayleigh-Faber-KrahnInequalitiy and Some Monotonicity Properties
for Eigenvalue Problems with Mixed Boundary Conditions ........... 3
H. Behnke
Lower and Upper Bounds for Sloshing Frequencies ................... 13
B.M. Brown, V. Hoang, Michael Plum and I.G. Wood
On Spectral Bounds for Photonic Crystal Waveguides ................ 23
C. Bu¸se
Real Integrability Conditions for the Nonuniform Exponential
Stability of Evolution Families on Banach Spaces .................... 31
K. Nagatou
Validated Computations for Fundamental Solutions of
Linear Ordinary Differential Equations ............................... 43
Part II: Integral Inequalities
S. Barza and L.-E. Persson
Equivalence of Modular Inequalities of Hardy Type on
Non-negative Respective Non-increasing Functions ................... 53
R.C. Brown and D.B. Hinton
Some One Variable Weighted Norm Inequalities and Their
Applications to Sturm-Liouville and Other Differential Operators .... 61
vi Contents
P. Cerone
Bounding the Gini Mean Difference .................................. 77
B. Gavrea
On Some Integral Inequalities ........................................ 91
M. Johansson
A New Characterizationof the Hardy and Its Limit Po´lya-Knopp
Inequality for Decreasing Functions .................................. 97
A. Cˇivljak, Lj. Dedi´c and M. Mati´c
Euler-Gru¨ss Type Inequalities Involving Measures .................... 109
W.D. Evans, A. Gogatishvili and B. Opic
The ρ-quasiconcaveFunctions and Weighted Inequalities ............. 121
Part III: Inequalities for Operators
S.S. Dragomir
Inequalities for the Norm and Numerical Radius of Composite
Operators in Hilbert Spaces ......................................... 135
F. Kittaneh
Norm Inequalities for Commutators of Normal Operators ............ 147
J. Matkowski
Uniformly Continuous Superposition Operators in the Spaces of
Differentiable Functions and Absolutely Continuous Functions ....... 155
T. Ogita and S. Oishi
Tight Enclosures of Solutions of Linear Systems ..................... 167
Part IV: Inequalities in Approximation Theory
I. Gavrea
Operators of Bernstein-Stancu Type and the Monotonicity of
Some Sequences Involving Convex Functions ......................... 181
A.I. Mitrea and P. Mitrea
Inequalities Involving the Superdense Unbounded Divergence of
Some Approximation Processes ...................................... 193
C.P. Niculescu
An Overview of Absolute Continuity and Its Applications ............ 201
Contents vii
Part V: Generalizations of Convexity and Inequalities for Means
S. Abramovich and S.S. Dragomir
Normalized Jensen Functional, Superquadracity and
Related Inequalities ................................................. 217
P. Burai
Comparability of Certain Homogeneous Means ....................... 229
M. Klariˇci´c Bakula, M. Mati´c and J. Peˇcari´c
On Some General Inequalities Related to Jensen’s Inequality ......... 233
A.W. Marshall and I. Olkin
Schur-Convexity,Gamma Functions, and Moments ................... 245
Z. Dar´oczy and Zs. P´ales
A Characterizationof Nonconvexity and Its Applications
in the Theory of Quasi-arithmetic Means ............................ 251
J. Mrowiec, Ja. Tabor and J´o. Tabor
Approximately Midconvex Functions ................................. 261
Part VI: Inequalities, Stability, and Functional Equations
W. Fechner and J. Sikorska
Sandwich Theorems for Orthogonally Additive Functions ............ 269
R. Ger
On Vector Pexider Differences Controlled by Scalar Ones ............ 283
Gy. Maksa and F. M´esza´ros
A Characterizationof the Exponential Distribution through
Functional Equations ................................................ 291
D. Popa
Approximate Solutions of the Linear Equation ....................... 299
A. Varga and Cs. Vincze
On a Functional Equation Containing Weighted
Arithmetic Means ................................................... 305

Preface
Inequalitiesarefoundinalmostallfieldsofpureandappliedmathematics.Because
of their various applications in areas such as the natural and engineering sciences
as well as economics, new types of interesting inequalities are discovered every
year. In the theory of differential equations, in the calculus of variations and in
geometry, fields which are dominated by inequalities, efforts are made to extend
and improve the classical ones.
Thestudyofinequalitiesreflectsthedifferentaspectsofmodernmathematics.
On one hand, there is the systematic search for the basic principles, such as the
deeper understanding of monotonicity and convexity. On the other hand, finding
thesolutionstoaninequalityrequiresoftennewideas.Someofthemhavebecome
standard tools in mathematics. In view of the wide-ranging research related to
inequalities,severalrecentmathematicalperiodicalshavebeendevotedexclusively
to this topic.
Apossible wayto speedup the communicationbetweengroupsofspecialists
of the seemingly unconnected areas is to bring them together from many parts of
theglobe.DuetotheeffortsofJ´anosAcz´el,GeorgAumann,EdwinF.Beckenbach,
RichardBellmanand WolfgangWalter,the firstGeneralInequalities meeting was
organizedin Oberwolfach,Germany in 1976.Then six meetings wereorganizedin
Oberwolfach between 1978 and 1995 and one in Noszvaj, Hungary in 2002.
The Conference on Inequalities and Applications ’07 also took place at the
De La Motte Castle in Noszvaj, Hungary from September 9 to 15, 2007. It was
organized by the Department of Analysis of the University of Debrecen.
The members of the Scientific Committee were Catherine Bandle (Basel),
William Norrie Everitt (Birmingham, honorary member), La´szl´o Losonczi (De-
brecen), Zsolt P´ales (Debrecen), Michael Plum (Karlsruhe) and Wolfgang Walter
(Karlsruhe, honorary member).
TheorganizingCommitteeconsistedofZolt´anDaro´czy(honorarychairman),
Attila Gila´nyi (chairman), Miha´ly Bessenyei (scientific secretary), Zolt´an Boros,
Gyula Maksa, Szabolcs Baja´k and Fruzsina M´esz´aros.There were 66 participants
from 16 countries.
The talks at the symposium focused on the following topics: convexity and
its generalizations; mean values and functional inequalities; matrix and operator
inequalities;inequalitiesforordinaryandpartialdifferentialoperators;integraland
differential inequalities; variational inequalities; numerical methods. A number of
x Preface
sessions were, as usual, devoted to problems and remarks. The scientific program
was complemented by several social events, such as a harpsichord recital of some
masterpieces of Bach and Haydn, performed by A´gnes V´arallyay.
This volume contains 33 research papers, about half of the works presented
at the meeting. The material is arranged into six chapters ranging from Inequali-
ties related to ordinary and partial differential equations to Inequalities, stability,
and functional equations. The contributions given here reflect the ramification of
inequalitiesintomanyareasofmathematics,andalsopresentasynthesisofresults
in both theory and practice.
The editors of the volume are thankful to Mrs. Phyllis H. Brown for the
artistic drawings made at the conference, which are illustrations to the six chap-
ters.They thank Miha´lyBessenyeifor enthusiasticallycompilingthe reportofthe
meeting,AndreaPa´kozdyforthepreparationofthemanuscriptsandthepublisher,
Birkh¨auser Verlag, for the careful typesetting and technical assistance.
The organizationofthe meeting and the publicationof the proceedings were
partially supported by the Hungarian Scientific ResearchFund Grants NK–68040
and K–62316.
The Editors