不等式

图书标准信息

• 作者 G.Hardy、J.E.Littlewood G.Polya 著
• 出版社 世界图书出版公司
• 出版时间 2004-04
• 版次 1
• ISBN 9787506266062
• 定价 59.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 324页

【内容简介】

　　Itisoftenreallydifficulttotracetheoriginofafamiliarinequality.Itisquitelikelytooccurfirstasanauxiliaryproposition，oftenwithoutexplicitstatement，inamemoirongeometryorastronomy;itmayhavebeenrediscovered，manyyearslater，byhalfadozendifferentauthors;andnoaccessiblestatementofitmaybequitecomplete.Wehavealmostalwaysfound，evenwiththemostfamousinequalities，thatwehavealittlenewtoadd.Wehavedoneourbesttobeaccurateandhavegivenallreferenceswecan，butwehaveneverundertakensystematicbibliographicalresearch.Wefollowthecommonpractice，whenaparticularinequalityishabituallyassociatedwithaparticularmathematiciansname;wespeakoftheinequalitiesofSchwarz，HSlder，andJensen，thoughalltheseinequalitiescanbetracedfurtherback;andwedonotenumerateexplicitlyalltheminoradditionswhicharenecessaryforabsolutecompleteness.Wehavereceivedagreatdealofassistancefromfriends.MessrsG.A.Bliss，L.S.Bosanquet，R.Courant，B.Jessen，V.Levin，R.Rado，I.Schur，L.C.Young，andA.Zygmundhaveallhelpeduswithcriticismsororiginalcontributions.DrBosanquet，DrJessen，andProf.Zygmundhavereadthoproofs，andcorrectedmanyinaccuracies.Inparticular，ChapterIIIhasbeenverylargelyrewrittenastheresultofDrJessenssuggestions.Wehopethatthebookmaynowbereasonablyfreefromerror，inspiteofthemassofdetailwhichitcontains.
　　本书为英文版。

【目录】

CHAPTERⅠINTRODUCTION
1.1Finite,infinite,andintegralinequalities
1.2Notations
1.3Positiveinequalities
1.4Homogeneousinequalities
1.5Theaxiomaticbasisofalgebraicinequalities
1.6Comparablefunctions
1.7Selectionofproofs
1.8Selectionofsubjects
CHAPTERⅡELEMENTARYMEANVALUES
2.1Ordinarymeans
2.2Weightedmeans
2.3Limitingcasesofa
2.4Cauchysinequality
2.5Thetheoremofthearithmeticandgeometricmeans
2.6Otherproofsofthetheoremofthemeans
2.7Holdersinequalityanditsextensions
2.8Holdersinequalityanditsextensionscont
2.9Generalpropertiesofthemeansa
2.10Thesumsra
2.11Minkowskisinequality
2.12AcompaniontoMinkowskisinequality
2.13Illustrationsandapplicationsofthefundamentalinequalities
2.14Inductiveproofsofthefundamentalinequalities
2.15ElementaryinequalitiesconnectedwithTheorem37
2.16ElementaryproofofTheorem3
2.17Tchebyehefsinequality
2.18Muirheadstheorem
2.19ProofofMuirheadstheorem
2.20Analternativetheorem
2.21Furthertheoremsonsymmetricalmeans
2.22Theelementarysymmetricfunctionsofnpositivenumbers
2.23Anoteondefiniteforms
2.24AtheoremconcerningstrictlypositiveformsMiscellaneoustheoremsandexamples
CHAPTERⅢMEANVALUESWITHANARBITRARYFUNCTIONANDTHETHEORYOFCONVEXFUNCTIONS
3.1Definitions
3.2Equivalentmeans
3.3Acharacteristicpropertyofthemeans
3.4Comparability
3.5Convexfunctions
3.6Continuousconvexfunctions
3.7Analternativedefinition
3.8Equalityinthefundamentalinequalities
3.9RestatementsandextensionsofTheorem85
3.10Twicedifferentiableconvexfunctions
3.11Applieationsofthepropertiesoftwicedifferentiableconvexfunctions
3.12Convexfunctionsofseveralvariables
3.13GeneralisationsofHoldersinequality
3.14Sometheoremsconcerningmonotonicfunctions
3.15Sumswithanarbitraryfunction:generalisa.tionsofJensensinequality
3.16GeneralisationsofMinkowskisinequality
3.17Comparisonofsets
3.18Furthergeneralpropertiesofconvexfunctions
3.19Furtherpropertiesofcontinuousconvexfunctions
3.20DiscontinuousconvexfunctionsMiscellaneoustheoremsandexamples
……
CHAPTERⅣVARIOUSAPPLICATIONSOFTHECALCULUS
CHAPTERⅤINFINITESERIES
CHAPTERⅥINTEGRALS
CHAPTERⅦSOMEAPPLICATIONSOFTHECALCULUSOFVARIATIONS
CHARTERⅧSOMETHEOREMSCONCERNINGBILINEARANDMULTILINEARFORMS
CHAPTERⅨHILBERTSINEQUALITYANDITSANALOGUESANDEXTENSIONS
CHAPTERⅩREARRANGEMENTS
APPENDIXⅠOnstrictlypositiveforms
APPENDIXⅡThorinsproofandextensionofTheorem295
APPENDIXⅢOnHilbertsinequality
BIBLIOGRAPHY