Research Article On Hardy-Pachpatte-Copson s Inequalitiesdownloads.hindawi.com/journals/tswj/2014/607347.pdfInequalities ( )and( )whichlaterwentbythename of Hardy s inequalities led
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Research ArticleOn Hardy-Pachpatte-Copsonrsquos Inequalities
Chang-Jian Zhao1 and Wing-Sum Cheung2
1 Department of Mathematics China Jiliang University Hangzhou 310018 China2Department of Mathematics The University of Hong Kong Pokfulam Road Hong Kong
Correspondence should be addressed to Chang-Jian Zhao chjzhaoaliyuncom
Received 6 December 2013 Accepted 19 January 2014 Published 18 March 2014
Academic Editors B Dragovich and D Xu
Copyright copy 2014 C-J Zhao and W-S CheungThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
We establish new inequalities similar to Hardy-Pachpatte-Copsonrsquos type inequalities These results in special cases yield some ofthe recent results
1 Introduction
The classical Hardyrsquos integral inequality is as follows
Theorem A If 119901 gt 1 119891(119909) ge 0 for 0 lt 119909 lt infin and 119865(119909) =
(1119909) int119909
0
119891(119905)119889119905 then
int
infin
0
119865(119909)119901
119889119909 lt (119901
119901 minus 1)
119901
int
infin
0
119891(119909)119901
119889119909 (1)
unless 119891 equiv 0 The constant is the best possible
Theorem A was first proved by Hardy [1] in an attemptto give a simple proof of Hilbertrsquos double series theorem (see[2]) One of the best known and interesting generalization ofthe inequality (1) given by Hardy [3] himself can be stated asfollows
Theorem B If 119901 gt 1 119898 = 1 119891(119909) ge 0 for 0 lt 119909 lt infin and119865(119909) is defined by
119865 (119909) = int
119909
0
119891 (119905) 119889119905 119898 gt 1
119865 (119909) = int
infin
119909
119891 (119905) 119889119905 119898 lt 1
(2)
then
int
infin
0
119909minus119898
119865(119909)119901
119889119909 lt (119901
|119898 minus 1|)
119901
int
infin
0
119909119901minus119898
119891(119909)119901
119889119909 (3)
unless 119891 equiv 0 The constant is the best possible
Inequalities (1) and (3) which later went by the nameof Hardyrsquos inequalities led to a great many papers dealingwith alternative proofs various generalizations and numer-ous variants and applications in analysis (see [4ndash15]) Inparticular Pachpatte [4] established some generalizations ofHardy inequalities (1) and (3) Very recently Leng and Feng[16] proved some newHardy-type integral inequalities In thepresent paper we establish new inequalities similar to Hardyrsquosintegral inequalities (1) and (3) These results provide somenew estimates to these types of inequalities and in specialcases yield some of the recent results
2 Main Results
Our main results are given in the following theorems
119901 gt 1 119902 lt 1and 120572 gt 0 be constants Let 119908(119909 119910) be positive and locallyabsolutely continuous in (119886 119887) times (119888 119889) Let ℎ(119909 119910) be a positivecontinuous function and let 119867(119909 119910) = int
119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 607347 7 pageshttpdxdoiorg1011552014607347
2 The Scientific World Journal
for (119909 119910) isin (119886 119887) times (119888 119889) Let 119891(119909 119910) be nonnegative andmeasurable on (119886 119887) times (119888 119889) If
119863(119909 119910)
= 1 minus1
1 minus 119902
119867 (119909 119910)
ℎ (119909 119910)
1
119908 (119909 119910)
120597119908 (119909 119910)
120597119909log(
119867(119877 1198771015840
)
119867 (119909 119910))
+119901
1 minus 119902
119867 (119909 119910)
ℎ (119909 119910)times
1
119903 (119909 119910)
120597119903 (119909 119910)
120597119909log(
119867(119877 1198771015840
)
119867 (119909 119910))
ge1
120572
(4)
for almost all (119909 119910) isin (119886 119887) times (119888 119889) and if 119865(119909 119910) is definedby
Remark 2 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduce to119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitable
modifications in Theorem 1 (6) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120572(119901
1 minus 119902)]
119901
times int
119887
119886
[(log(119867 (119877)
119867 (119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(8)
This is just a new inequality established by Pachpatte [4]Moreover we note that the inequality established in
Theorem 1 is the further generalizations of the inequalityestablished by Copson [17]
Taking for 119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120572 = 1 in (8)(8) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
1 minus 119902)
119901
times int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(9)
This is just a new inequality established by Love [7]Let ℎ(119909) = 1 119886 rarr 0 119887 rarr infin and log (119877(119909 minus 119886)) = 1
in (9) then (9) changes to the following result
int
infin
0
119909minus1
119865(119909)119901
119889119909 le (119901
1 minus 119902)
119901
int
infin
0
119909119901minus1
119891(119909)119901
119889119909 (10)
This result is obtained in (3) stated in the Introduction
0 be constants Let 119908(119909 119910) be positive and locally absolutelycontinuous in (119886 119887)times(119888 119889) Let ℎ(119909 119910) be a positive continuousfunction and let 119867(119909 119910) = int
119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin
(119886 119887) times (119888 119889) Let 119891(119909 119910) be nonnegative and measurable on(119886 119887) times (119888 119889) Let
119864 (119909 119910)
= 1 minus1
119902 minus 1
119867 (119909 119910)
ℎ (119909 119910)
1
119908 (119909 119910)
120597119908 (119909 119910)
120597119910log(
119867(119877 1198771015840
)
119867 (119909 119910))
+119901
119902 minus 1
119867 (119909 119910)
ℎ (119909 119910)times
1
119903 (119909 119910)
120597119903 (119909 119910)
120597119910log(
119867(119877 1198771015840
)
119867 (119909 119910))
ge1
120573
(11)
The Scientific World Journal 3
for almost all (119909 119910) isin (119886 119887) times (119888 119889) If 119865(119909 y) is defined by
Remark 4 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduceto 119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitablemodifications in Theorem 3 (13) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120573(119901
119902 minus 1)]
119901
int
119887
119886
[(log(119867(119877)
119867(119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(15)
This is just a new inequality established by Pachpatte [4]On the other hand we note that the inequality established
in Theorem 3 is the further generalizations of the inequalityestablished by Copson [17]
Taking for119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120573 = 1 in (15)(15) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
119902 minus 1)
119901
int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
times 119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(16)
This is just a new inequality established by Love [7]
3 Proof of Theorems
Proof of Theorem 1 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
for (119909 119910) isin (119886 119887) times (119888 119889) Let 119891(119909 119910) be nonnegative andmeasurable on (119886 119887) times (119888 119889) If
119863(119909 119910)
= 1 minus1
1 minus 119902
119867 (119909 119910)
ℎ (119909 119910)
1
119908 (119909 119910)
120597119908 (119909 119910)
120597119909log(
119867(119877 1198771015840
)
119867 (119909 119910))
+119901
1 minus 119902
119867 (119909 119910)
ℎ (119909 119910)times
1
119903 (119909 119910)
120597119903 (119909 119910)
120597119909log(
119867(119877 1198771015840
)
119867 (119909 119910))
ge1
120572
(4)
for almost all (119909 119910) isin (119886 119887) times (119888 119889) and if 119865(119909 119910) is definedby
Remark 2 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduce to119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitable
modifications in Theorem 1 (6) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120572(119901
1 minus 119902)]
119901
times int
119887
119886
[(log(119867 (119877)
119867 (119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(8)
This is just a new inequality established by Pachpatte [4]Moreover we note that the inequality established in
Theorem 1 is the further generalizations of the inequalityestablished by Copson [17]
Taking for 119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120572 = 1 in (8)(8) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
1 minus 119902)
119901
times int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(9)
This is just a new inequality established by Love [7]Let ℎ(119909) = 1 119886 rarr 0 119887 rarr infin and log (119877(119909 minus 119886)) = 1
in (9) then (9) changes to the following result
int
infin
0
119909minus1
119865(119909)119901
119889119909 le (119901
1 minus 119902)
119901
int
infin
0
119909119901minus1
119891(119909)119901
119889119909 (10)
This result is obtained in (3) stated in the Introduction
0 be constants Let 119908(119909 119910) be positive and locally absolutelycontinuous in (119886 119887)times(119888 119889) Let ℎ(119909 119910) be a positive continuousfunction and let 119867(119909 119910) = int
119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin
(119886 119887) times (119888 119889) Let 119891(119909 119910) be nonnegative and measurable on(119886 119887) times (119888 119889) Let
119864 (119909 119910)
= 1 minus1
119902 minus 1
119867 (119909 119910)
ℎ (119909 119910)
1
119908 (119909 119910)
120597119908 (119909 119910)
120597119910log(
119867(119877 1198771015840
)
119867 (119909 119910))
+119901
119902 minus 1
119867 (119909 119910)
ℎ (119909 119910)times
1
119903 (119909 119910)
120597119903 (119909 119910)
120597119910log(
119867(119877 1198771015840
)
119867 (119909 119910))
ge1
120573
(11)
The Scientific World Journal 3
for almost all (119909 119910) isin (119886 119887) times (119888 119889) If 119865(119909 y) is defined by
Remark 4 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduceto 119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitablemodifications in Theorem 3 (13) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120573(119901
119902 minus 1)]
119901
int
119887
119886
[(log(119867(119877)
119867(119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(15)
This is just a new inequality established by Pachpatte [4]On the other hand we note that the inequality established
in Theorem 3 is the further generalizations of the inequalityestablished by Copson [17]
Taking for119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120573 = 1 in (15)(15) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
119902 minus 1)
119901
int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
times 119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(16)
This is just a new inequality established by Love [7]
3 Proof of Theorems
Proof of Theorem 1 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
Remark 4 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduceto 119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitablemodifications in Theorem 3 (13) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120573(119901
119902 minus 1)]
119901
int
119887
119886
[(log(119867(119877)
119867(119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(15)
This is just a new inequality established by Pachpatte [4]On the other hand we note that the inequality established
in Theorem 3 is the further generalizations of the inequalityestablished by Copson [17]
Taking for119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120573 = 1 in (15)(15) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
119902 minus 1)
119901
int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
times 119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(16)
This is just a new inequality established by Love [7]
3 Proof of Theorems
Proof of Theorem 1 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976