Mathematical inequalities

606 Pages·2005·2.272 MB·English

by  B. G. Pachpatte

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Mathematical Inequalities
North-Holland Mathematical Library
BoardofHonoraryEditors:
M. Artin, H.Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H.Halberstam,
L.V.Hörmander,J.H.B.Kemperman,W.A.J.Luxemburg,F.Peterson,I.M.Singer
andA.C.Zaanen
BoardofAdvisoryEditors:
A.Björner,R.H.Dijkgraaf,A.Dimca,A.S.Dow,J.J.Duistermaat,E.Looijenga,
J.P. May, I. Moerdijk, S.M. Mori, J.P. Palis, A. Schrijver, J. Sjöstrand,
J.H.M.Steenbrink,F.TakensandJ.vanMill
VOLUME67
Amsterdam–Boston–Heidelberg–London–NewYork–Oxford–
Paris–SanDiego–SanFrancisco–Singapore–Sydney–Tokyo
Mathematical Inequalities
B.G. Pachpatte
57ShriNiketanColony
NearAbhinayTalkies
Aurangabad431001
Maharashtra
India
2005
Amsterdam–Boston–Heidelberg–London–NewYork–Oxford–
Paris–SanDiego–SanFrancisco–Singapore–Sydney–Tokyo
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Preface
Inequalitiesplayanimportantroleinalmostallbranchesofmathematicsaswell
asinotherareasofscience.Thebasicwork“Inequalities”byHardy,Littlewood
and Pólya appeared in 1934 and the books “Inequalities” by Beckenbach and
Bellman published in 1961 and “Analytic Inequalities” by Mitrinovic´ published
in 1970 made considerable contributions to this field and supplied motivations,
ideas,techniquesandapplications.Since1934anenormousamountofefforthas
beendevotedtothediscoveryofnewtypesofinequalitiesandtotheapplicationof
inequalitiesinmanypartsofanalysis.Theusefulnessofmathematicalinequalities
is felt from the very beginning and is now widely acknowledged as one of the
majordrivingforcesbehindthedevelopmentofmodernrealanalysis.
The theory of inequalities is in a process of continuous development state
and inequalities have become very effective and powerful tools for studying a
wide range of problems in various branches of mathematics. This theory in re-
centyearshasattractedtheattentionofalargenumberofresearchers,stimulated
newresearchdirectionsandinfluencedvariousaspectsofmathematicalanalysis
andapplications.Amongthemanytypesofinequalities,thoseassociatedwiththe
namesofJensen,Hadamard,Hilbert,Hardy,Opial,Poincaré,Sobolev,Levinand
Lyapunovhavedeeprootsandmadeagreatimpactonvariousbranchesofmath-
ematics.Thelastfewdecadeshavewitnessedimportantadvancesrelatedtothese
inequalitiesthatremainactiveareasofresearchandhavegrownintosubstantial
fieldsofresearchwithmanyimportantapplications.Thedevelopmentofthethe-
ory related to these inequalities resulted in a renewal of interest in the field and
hasattractedinterestfrommanyresearchers.Ahostofnewresultshaveappeared
intheliterature.
The present monograph provides a systematic study of some of the most fa-
mousandfundamentalinequalitiesoriginatedbytheabovementionedmathemati-
cians and brings together the latest, interesting developments in this important
research area under a unified framework. Most of the results contained here are
onlyrecentlydiscoveredandarestillscatteredoveralargenumberofnonspecial-
istperiodicals.Thechoiceofmaterialcoverssomeofthemostimportantresults
vii
viii Preface
in the field which have had a great impact on many branches of mathematics.
This work will be of interest to mathematical analysts, pure and applied mathe-
maticians, physicists, engineers, computer scientists and other areas of science.
For researchers working in these areas, it will be a valuable source of reference
andinspiration.Itcouldalsobeusedasatextforanadvancedgraduatecourse.
Theauthoracknowledgeswithgreatpleasurehisgratitudeforthefinecooper-
ation and assistance provided by the staff of the book production department of
ElsevierScience.Ialsoexpressdeepappreciationtomyfamilymembersfortheir
encouragement,understandingandpatienceduringthewritingofthisbook.
B.G.Pachpatte