Hardy Type Inequalities on Time Scales

309 Pages·2016·3.19 MB·English

by  Ravi P. Agarwal

Ravi P. Agarwal · Donal O'Regan
Samir H. Saker
Hardy Type
Inequalities
on Time
Scales
Hardy Type Inequalities on Time Scales
Ravi P. Agarwal • Donal O’Regan (cid:129) Samir H. Saker
Hardy Type Inequalities
on Time Scales
123
RaviP.Agarwal DonalO’Regan
DepartmentofMathematics SchoolofMathematics
TexasA&MUniversity–Kingsville StatisticsandAppliedMathematics
Kingsville,TX,USA NationalUniversityofIreland
Galway,Ireland
SamirH.Saker
DepartmentofMathematics
MansouraUniversity
Mansoura,Egypt
ISBN978-3-319-44298-3 ISBN978-3-319-44299-0 (eBook)
DOI10.1007/978-3-319-44299-0
LibraryofCongressControlNumber:2016950725
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RaviP.Agarwal:ToSadhna, Sheba,
andDanah
DonalO’Regan:ToAlice,Aoife,Lorna,
Daniel,andNiamh
SamirH.Saker:ToMona,Meran,Maryam,
Mennah,andAhmed
Preface
Neitherofuscompletelyunderstoodwhattheotherwasdoing,
butwerealizedthatourjointeffortwillgivethetheorem,andto
bealittleimpudentandconceited,probabilisticnumbertheory
wasborn!Thiscollaborationisagoodexampletoshowthat
twobrainscanbebetterthanone,sinceneitherofuscouldhave
donetheworkalone.
PaulErdo˝s(1933–1996)
Hardy’sinterestininequalities(inbothdiscreteandcontinuousforms)wasduring
the period 1906–1928. As a result of his work, the subject was changed radically,
andwhathadpreviouslybeenacollectionofisolatedformulasbecameasystematic
discipline.TheclassicalbookInequalitiesbyHardyetal.[77]containstwochapters
devoted to Hardy- and Hilbert-type inequalities and the growth of Hardy-type
inequalitiesintheliteraturestimulatedthisbook.
The book is devoted to dynamic inequalities of Hardy type and extensions and
generalizationsviaconvexityonatimescaleT.Inparticular,thebookcontainsthe
timescaleversionsofclassicalHardy-typeinequalities,Hardy-andLittlewood-type
inequalities, Hardy-Knopp-type inequalities via convexity, Copson-type inequali-
ties,Walsh-typeinequalities,Liendeler-typeinequalities,Levinson-typeinequalities
and Pachpatte-type inequalities, Bennett-type inequalities, Chan-type inequalities,
and Hardy type inequalities with two different weight functions. These dynamic
inequalities contain the classical continuous and discrete inequalities as special
caseswhenTDRandTDNandcanbeextendedtodifferenttypesofinequalities
ondifferenttimescalessuchasTDhN,h>0,TDqN forq>1,etc.
The book consists of seven chapters and is organized as follows. In Chap.1,
we present preliminaries, basic concepts, and the basic inequalities that will be
needed in the book. Next, we present time scale versions of classical Hardy-
type inequalities and Hardy- and Littlewood-type inequalities. We also prove
extensions of Hardy-type inequalities and its general form via convexity on time
scales. Chapter 2 is devoted to Copson-type inequalities on an arbitrary time
scaleT.Inparticular,weconsiderclassicalformsofCopson-typeinequalitiesand
their converses which are extensions of Hardy-type inequalities. In Chap.3, we
present Leindler-type inequalities and their extensions on an arbitrary time scale
T. We also consider the dual of these inequalities and their converses in this
chapter. Chapter 4 is devoted to Littlewood-type inequalities on an arbitrary time
scale T. First, we consider the generalized form of Littlewood-type inequalities
vii
viii Preface
with decreasing functions. Next, we consider a generalization of Littlewood-type
inequalitiesthatwasconsideredbyBennett,andweendthechapterwiththesneak-
out principle on time scales. Chapter 5 is concerned with weighted Hardy-type
inequalitiesonanarbitrarytimescaleT.Theresultscanbeconsideredasextensions
of the results due to Copson, Bliss, Flett and Bennett, Leindler, Chen, and Yang.
In Chap.6, we discuss Levinson-type inequalities on time scales. Also we include
somedynamicinequalitiesontimescalesofChanandPachpattetype.Theproofsof
themainresultsincludethedefinitionofthelogarithmicfunctionontimescalesand
itsdeltaderivativeandtheapplicationofJensen’sinequality.Chapter7isdevoted
toHardy-Knopp-typeinequalitiesonanarbitrarytimescaleT.Aone-dimensional,
two-dimensional, and multidimensional versions of Hardy-Knopp inequalities are
considered and extended on time scales via convexity. The refinement inequalities
ofHardy-Knopptypewhichdependsontheapplicationsofsuperquadraticfunctions
andthecorrespondingrefinementJensen’sinequalitywillalsobediscussed.
In this book, we followed the history and development of these inequalities.
Each section is self-contained, and one can see the relationship between the time
scaleversionsoftheinequalitiesandtheclassicalones.Tothebestoftheauthors’
knowledge, this is the first book devoted to Hardy-type inequalities and their
extensionsontimescales.
Wewishtoexpressourthankstoourfamiliesandfriends.
Kingsville,TX,USA RaviP.Agarwal
Galway,Ireland DonalO’Regan
Mansoura,Egypt SamirH.Saker
Contents
1 HardyandLittlewoodTypeInequalities.................................. 1
1.1 PreliminariesandBasicInequalitiesonTimeScales.................. 1
1.2 Hardy-TypeInequalities ................................................ 14
1.3 Hardy-LittlewoodTypeInequalities ................................... 22
1.4 AnExtensionofHardy’sTypeInequality ............................. 36
1.5 GeneralizationsofHardy’sInequalityviaConvexity ................. 40
2 Copson-TypeInequalities................................................... 49
2.1 Copson-TypeInequalitiesI............................................. 49
2.2 Copson-TypeInequalitiesII............................................ 54
2.3 ConversesofCopson-TypeInequalities................................ 62
3 Leindler-TypeInequalities.................................................. 69
3.1 Leindler-TypeInequalitiesI............................................ 69
3.2 Leindler-TypeInequalitiesII ........................................... 74
3.3 ConversesofLeindler-TypeInequalitiesI............................. 79
3.4 ConversesofLeindler-TypeInequalitiesII............................ 84
4 Littlewood-BennettTypeInequalities ..................................... 91
4.1 Littlewood-TypeInequalities........................................... 91
4.2 Littlewood-BennettTypeInequalitiesI................................ 94
4.3 Littlewood-BennettTypeInequalitiesII ............................... 104
4.4 Sneak-OutPrincipleonTimeScales................................... 112
5 WeightedHardyTypeInequalities......................................... 121
5.1 WeightedHardy-TypeInequalitiesI ................................... 121
5.2 WeightedHardy-TypeInequalitiesII .................................. 136
6 Levinson-TypeInequalities ................................................. 153
6.1 Levinson-TypeInequalitiesI ........................................... 153
6.2 Levinson-TypeInequalitiesII .......................................... 180
6.3 Pachpatte-TypeInequalitiesI........................................... 189
6.4 YangandHwang-TypeInequalities.................................... 197
ix
x Contents
6.5 Chan-TypeInequalities ................................................. 206
6.6 Pachpatte-TypeInequalitiesII.......................................... 212
7 Hardy-KnoppTypeInequalities ........................................... 221
7.1 Hardy-KnoppTypeInequalities........................................ 221
7.2 Hardy-KnoppTypeInequalitieswithTwoFunctions ................. 233
7.3 Hardy-KnoppTypeInequalitieswithKernels......................... 238
7.4 Hardy-KnoppTypeInequalitiesviaSuperquadracity................. 265
7.5 RefinementsofHardy-KnoppTypeInequalities ...................... 274
7.6 Diamond-˛TypeInequalities .......................................... 287
References......................................................................... 295
Index............................................................................... 303
Chapter 1
Hardy and Littlewood Type Inequalities
Ascienceissaidtobeusefulifitsdevelopmenttendsto
accentuatetheexistinginequalitiesinthedistributionofwealth,
ormoredirectlypromotesthedestructionofhumanlife.
GodfreyHaroldHardy(1877–1947).
This chapter considers time scale versions of classical Hardy-type inequalities
and time scale versions of Hardy and Littlewood type inequalities. We present
extensionsofHardy-typeinequalitiesontimescales.Thesedynamicinequalitiesnot
only contain the integral and discrete inequalities but can be extended to different
types of time scales. The chapter is divided into five sections and is organized as
follows.
InSect.1.1,wepresentsomepreliminaries,definitionsandconceptsconcerning
timescalecalculusandbasicdynamicinequalitiesthatwillbeneededintheproofs
of the main results. In Sect.1.2, we give two proofs of Hardy-type inequalities on
time scales. Section 1.3 presents the time scale version of Hardy-Littlewood type
inequalities. In Sect.1.4, we give the time scale version of an extension of Hardy
typeinequalitiesandinSect.1.5,wewillusetheconceptofconvexitytoestablished
ageneralizationofHardy’sinequalityontimescales.
1.1 PreliminariesandBasicInequalitieson TimeScales
In recent years the study of dynamic inequalities on time scales has received a
lot of attention in the literature and has become a major field in pure and applied
mathematics [10, 145]. The idea goes back to its founder Stefan Hilger [80] who
initiated the study of dynamic equations on time scales to avoid proving results
twice—oncefordifferentialequationsandonceagainfordifferenceequations.The
books on the subject of time scale, i.e., measure chain, by Bohner and Peterson
[33, 34] summarize and organize much of time scale calculus. The three most
popular examples of calculus on time scales are differential calculus, difference
calculus, and quantum calculus (see Kac and Cheung [88]), i.e, when T D R,
TDNandTDqN0 Dfqt W t2N0gwhereq>1.Weassumethatthereaderhasa
goodbackgroundintimescalecalculus.
©SpringerInternationalPublishingSwitzerland2016 1
R.P.Agarwaletal.,HardyTypeInequalitiesonTimeScales,
DOI10.1007/978-3-319-44299-0_1