Analytic inequalities. Recent advances

314 Pages·2012·3.235 MB·English

by  Pachpatte B.G.

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Analytic Inequalities
Recent Advances
†
B.G. Pachpatte
57,ShriNiketenColoney
Aurangabad
India
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Tothememoryofmyfather.
Preface
Analyticinequalitiesarewidelyacknowledgedasoneofthemajordrivingforcesbehindthe
developmentofvariousbranchesofmathematicsandmanyappliedsciences. Thestudyof
inequalitieshasincreasedenormouslyoverrecentdecadesasithasbeendemonstratedthat
they have applications in many diverse fields of mathematics. There exists, for example,
averyrichliteraturerelatedtotheCˇebysˇev, Gru¨ss, Trapezoid, Ostrowski, Hadamardand
Jensen inequalities. This monograph is an attempt to organize recent progress related to
theseinthehopethatitwillfurtherbroadendevelopmentsandthescopeofapplications.It
doesnotintendtobecomprehensive,butratheritismeanttobearepresentativeoverview
oftherecentresearchrelatedtothefundamentalinequalitiesnotedabove.
Alargepartofthematerialincludedinthebookcanonlybefoundintheresearchliterature
although it should be understandable to any reader with a reasonable background in real
analysisanditsrelatedareas.Itwillbeavaluablesourceofreferenceinthefieldforalong
timetocome. Allresultsarepresentedinanelementarywayanditcouldalsoserveasa
textbookforanadvancedgraduatecourse.
TheauthorisgratefultoProfessorJanvanMillandArjenSevensterfortheopportunityto
publishthisbookandtheirinvaluableprofessionalcooperationfortheworkreportedhere.
I am also indebted to the editorial and production staff of the publisher for the care they
have taken with this book. I would like to thank my family members for providing their
strongsupportandconstantencouragementduringthewritingofthismonograph.
B.G.Pachpatte
vii
viii AnalyticInequalities:RecentAdvances
Publisher’sNote
Unfortunatelytheauthorpassedawayduringthelastphaseofthemanuscriptpreparation
andwasthereforeunabletoseethefinalrealizationofthistopicalandinterestingwork.
Contents
Preface vii
Introduction 1
1. Gru¨ss-andCˇebysˇev-typeinequalities 7
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Gru¨ss-typeinequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Cˇebysˇev-typeinequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 InequalitiesoftheGru¨ss-andCˇebysˇev-type . . . . . . . . . . . . . . . . 21
1.5 MoreinequalitiesoftheGru¨ss-andCˇebysˇev-type . . . . . . . . . . . . . 31
1.6 DiscreteInequalitiesoftheGru¨ss-andCˇebysˇev-type. . . . . . . . . . . . 45
1.7 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.8 Miscellaneousinequalities . . . . . . . . . . . . . . . . . . . . . . . . . 61
1.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2. MultidimensionalGru¨ss-Cˇebysˇevand-Trapezoid-typeinequalities 71
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.2 SomeGru¨ss-typeinequalitiesininnerproductspaces . . . . . . . . . . . 71
2.3 Gru¨ss-andCˇebysˇev-typeinequalitiesintwoandthreevariables . . . . . . 80
2.4 Trapezoid-typeinequalitiesintwovariables . . . . . . . . . . . . . . . . 93
2.5 SomemultivariateGru¨ss-typeintegralinequalities. . . . . . . . . . . . . 100
2.6 MultivariateGru¨ss-andCˇebysˇev-typediscreteinequalities . . . . . . . . 107
2.7 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.8 Miscellaneousinequalities . . . . . . . . . . . . . . . . . . . . . . . . . 125
2.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
ix