This page intentionally left blank 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 ELSEVIERB.V. ELSEVIERInc. 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ISBN:0-444-51795-2 SeriesISSN:0924-6509 ThepaperusedinthispublicationmeetstherequirementsofANSI/NISOZ39.48-1992(Permanence ofPaper). PrintedinTheNetherlands. TotheMemoryofMyParents This page intentionally left blank 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