On a half-discrete Hilbert-type inequality related to hyperbolic functions
By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the...
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| Published in | Journal of inequalities and applications Vol. 2021; no. 1; pp. 1 - 16 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Cham
Springer International Publishing
10.09.2021
Springer Nature B.V SpringerOpen |
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| ISSN | 1029-242X 1025-5834 1029-242X |
| DOI | 10.1186/s13660-021-02688-7 |
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| Abstract | By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper. |
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| AbstractList | By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper. Abstract By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper. |
| ArticleNumber | 153 |
| Author | You, Minghui |
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| References | ChenZ.XuJ.New extensions of Hardy–Hilbert’s inequality with multiple parametersActa Math. Hung.200711743834001164.26020 RassiasM.T.YangB.C.On half-discrete Hilbert’s inequalityAppl. Math. Comput.2013220759330918321329.26041 YangB.C.DebnathL.On a new generalization of Hardy–Hilbert’s inequality and its applicationJ. Math. Anal. Appl.199923248449716895740935.26009 YouM.H.On a class of Hilbert-type inequalities in the whole plane related to exponent functionJ. Inequal. Appl.202120214217995 YangB.C.On a dual Hardy–Hilbert’s inequality and its generalizationAnal. Math.200531215116121515241100.26014 ZhongJ.H.ChenQ.A half-discrete Hilbert-type inequality with the deceasing and homogeneous kernel of degree 0J. Zhejiang Univ. Sci. Ed.201542177813362486 RichardC.F.J.Introduction to Calculus and Analysis1989New YorkSpringer KrnićM.PečarićJ.PerićI.Advances in Hilbert-Type Inequalities2012ZagrebElement Press1301.26004 RassiasM.T.YangB.C.RaigorodskiiA.On a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic secant function related to the Hurwitz zeta functionTrigonometric Sums and Their Applications2020BerlinSpringer229259 WangZ.X.GuoD.R.Introduction to Special Functions2012BeijingHigher Education Press KrnićM.PečarićJ.General Hilbert’s and Hardy’s inequalitiesMath. Inequal. Appl.200584295121379041079.26018 KrnićM.PečarićJ.Extension of Hilbert’s inequalityJ. Math. Anal. Appl.200632415016022624621112.26019 KrnićM.PečarićJ.VukovićP.Discrete Hilbert-type inequalities with general homogeneous kernelsRend. Circ. Mat. Palermo201160116117128257021231.26018 YangB.C.On Hilbert’s integral inequalityJ. Math. Anal. Appl.1998220277878516149680911.26011 RassiasM.T.YangB.C.On a Hilbert-type integral inequality related to the extended Hurwitz zeta function in the whole planeActa Appl. Math.20191601678039346581429.26037 HongY.HeB.YangB.C.Necessary and sufficient conditions for the validity of Hilbert-type inequalities with a class of quasi-homogeneous kernels and its applications in operator theoryJ. Math. Inequal.201812377778838573621403.26020 YangB.C.A new Hilbert-type integral inequality with the homogeneous kernel of degree 0J. Zhejiang Univ. Sci. Ed.20123943903923012253 YangB.C.A mixed Hilbert-type inequality with a best constant factorInt. J. Pure Appl. Math.200520331932821520331100.26015 YouM.H.On a new discrete Hilbert-type inequality and applicationMath. Inequal. Appl.20151841575157834146181331.26048 YangB.C.On an extension of Hardy–Hilbert’s inequalityChin. Ann. Math., Ser. A200223224725419006421023.26013 GaoM.Z.YangB.C.On the extended Hilbert’s inequalityProc. Am. Math. Soc.1998126375175914591220935.26011 YangB.C.On an extension of Hilbert’s integral inequality with some parametersAust. J. Math. Anal. Appl.200411182077658 RassiasM.T.YangB.C.On a Hilbert-type integral inequality in the whole plane related to the extended Riemann zeta functionComplex Anal. Oper. Theory20191341765178239570131435.26031 LiuQ.A Hilbert-type integral inequality under configuring free power and its applicationsJ. Inequal. Appl.201920193934716 HeB.YangB.C.ChenQ.A new multiple half-discrete Hilbert-type inequality with parameters and a best possible constant factorMediterr. J. Math.2015121227124434168581327.26021 YangB.C.A note on Hilbert’s integral inequalitiesChin. Q. J. Math.1998134838517058120970.26011 MoH.M.YangB.C.On a new Hilbert-type integral inequality involving the upper limit functionsJ. Inequal. Appl.202020204062070 YangB.C.WuS.H.WangA.Z.On a reverse half-discrete Hardy–Hilbert’s inequality with parametersMathematics2019711 YangB.C.The Norm of Operator and Hilbert-Type Inequalities2009BeijingScience Press KrnićM.PečarićJ.VukovićP.A unified treatment of half-discrete Hilbert-type inequalities with a homogeneous kernelMediterr. J. Math.2013101697171631193281280.26037 YangB.C.DebnathL.Half-Discrete Hilbert-Type Inequalities2014SingaporeWorld Scientific1296.26007 RassiasM.T.YangB.C.RaigorodskiiA.Two kinds of the reverse Hardy-type integral inequalities with the equivalent forms related to the extended Riemann zeta functionAppl. Anal. Discrete Math.20181222732963875322 YangB.C.On a new extension of Hardy–Hilbert’s inequality with some parametersActa Math. Hung.200510843373501091.26021 KuangJ.C.DebnathL.On new generalizations of Hilbert’s inequality and their applicationsJ. Math. Anal. Appl.2002245124826517565880968.26010 RassiasM.T.YangB.C.A Hilbert-type integral inequality in the whole plane related to the hypergeometric function and the beta functionJ. Math. Anal. Appl.201542821286130833349801316.33004 YangB.C.DebnathL.On the extended Hardy–Hilbert’s inequalityJ. Math. Anal. Appl.2002272118719919307101009.26016 RassiasM.T.YangB.C.On an equivalent property of a reverse Hilbert-type integral inequality related to the extended Hurwitz zeta functionJ. Math. Inequal.201913231533439549391425.26015 LiuQ.LongS.C.A Hilbert-type integral inequality with the kernel of hyperbolic secant functionJ. Zhejiang Univ. Sci. Ed.20134032552593113888 HongY.LiaoJ.Q.YangB.C.A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applicationsJ. Inequal. Appl.202020204101559 YangB.C.On new extensions of Hilbert’s inequalityActa Math. Hung.2004104429129920827781062.26023 RassiasM.T.YangB.C.RaigorodskiiA.On a more accurate reverse Hilbert-type inequality in the whole planeJ. Math. Inequal.20201441359137441935411461.26015 YouM.H.On an extension of the discrete Hilbert inequality and applicationsJ. Wuhan Univ. Natur. Sci. Ed.2021672179184 YangB.C.ChenQ.A half-discrete Hilbert-type inequality with a homogeneous kernel and an extensionJ. Inequal. Appl.2011201128871701279.26060 HardyG.H.LittlewoodJ.E.PólyaG.Inequalities1952LondonCambridge University Press0047.05302 YangB.C.A half-discrete Hilbert’s inequalityJ. Guangdong Univ. Educ.2011313171249.26044 M.Z. Gao (2688_CR7) 1998; 126 J.C. Kuang (2688_CR15) 2002; 245 Y. Hong (2688_CR28) 2020; 2020 B.C. Yang (2688_CR11) 2009 B.C. Yang (2688_CR4) 1999; 23 H.M. Mo (2688_CR29) 2020; 2020 M.T. Rassias (2688_CR21) 2019; 13 Y. Hong (2688_CR27) 2018; 12 M. Krnić (2688_CR40) 2013; 10 B.C. Yang (2688_CR38) 2005; 20 B.C. Yang (2688_CR35) 2011; 2011 M. Krnić (2688_CR10) 2012 M.T. Rassias (2688_CR22) 2019; 160 Z. Chen (2688_CR13) 2007; 117 B.C. Yang (2688_CR14) 2005; 31 M. Krnić (2688_CR8) 2006; 324 B. He (2688_CR36) 2015; 12 Q. Liu (2688_CR42) 2013; 40 M.T. Rassias (2688_CR26) 2018; 12 Z.X. Wang (2688_CR45) 2012 M. Krnić (2688_CR20) 2005; 8 M.T. Rassias (2688_CR24) 2019; 13 M.T. Rassias (2688_CR25) 2020; 14 B.C. Yang (2688_CR41) 2012; 39 M.T. Rassias (2688_CR33) 2013; 220 B.C. Yang (2688_CR12) 2005; 108 G.H. Hardy (2688_CR1) 1952 B.C. Yang (2688_CR3) 2002; 272 M.H. You (2688_CR6) 2021; 67 B.C. Yang (2688_CR19) 1998; 220 B.C. Yang (2688_CR2) 2002; 23 C.F.J. Richard (2688_CR44) 1989 M.H. You (2688_CR31) 2021; 2021 B.C. Yang (2688_CR17) 2004; 1 M. Krnić (2688_CR9) 2011; 60 B.C. Yang (2688_CR18) 1998; 13 M.H. You (2688_CR16) 2015; 18 B.C. Yang (2688_CR37) 2019; 7 J.H. Zhong (2688_CR43) 2015; 42 B.C. Yang (2688_CR32) 2011; 31 M.T. Rassias (2688_CR34) 2020 M.T. Rassias (2688_CR23) 2015; 428 Q. Liu (2688_CR30) 2019; 2019 B.C. Yang (2688_CR5) 2004; 104 B.C. Yang (2688_CR39) 2014 |
| References_xml | – reference: YangB.C.On a dual Hardy–Hilbert’s inequality and its generalizationAnal. Math.200531215116121515241100.26014 – reference: YangB.C.On an extension of Hardy–Hilbert’s inequalityChin. Ann. Math., Ser. A200223224725419006421023.26013 – reference: RassiasM.T.YangB.C.A Hilbert-type integral inequality in the whole plane related to the hypergeometric function and the beta functionJ. Math. Anal. Appl.201542821286130833349801316.33004 – reference: ZhongJ.H.ChenQ.A half-discrete Hilbert-type inequality with the deceasing and homogeneous kernel of degree 0J. Zhejiang Univ. Sci. Ed.201542177813362486 – reference: HongY.HeB.YangB.C.Necessary and sufficient conditions for the validity of Hilbert-type inequalities with a class of quasi-homogeneous kernels and its applications in operator theoryJ. Math. Inequal.201812377778838573621403.26020 – reference: YangB.C.A half-discrete Hilbert’s inequalityJ. Guangdong Univ. Educ.2011313171249.26044 – reference: HardyG.H.LittlewoodJ.E.PólyaG.Inequalities1952LondonCambridge University Press0047.05302 – reference: GaoM.Z.YangB.C.On the extended Hilbert’s inequalityProc. Am. Math. Soc.1998126375175914591220935.26011 – reference: YouM.H.On a new discrete Hilbert-type inequality and applicationMath. Inequal. Appl.20151841575157834146181331.26048 – reference: YangB.C.ChenQ.A half-discrete Hilbert-type inequality with a homogeneous kernel and an extensionJ. Inequal. Appl.2011201128871701279.26060 – reference: YangB.C.DebnathL.Half-Discrete Hilbert-Type Inequalities2014SingaporeWorld Scientific1296.26007 – reference: RichardC.F.J.Introduction to Calculus and Analysis1989New YorkSpringer – reference: KrnićM.PečarićJ.VukovićP.Discrete Hilbert-type inequalities with general homogeneous kernelsRend. Circ. Mat. Palermo201160116117128257021231.26018 – reference: RassiasM.T.YangB.C.RaigorodskiiA.Two kinds of the reverse Hardy-type integral inequalities with the equivalent forms related to the extended Riemann zeta functionAppl. Anal. Discrete Math.20181222732963875322 – reference: YangB.C.On Hilbert’s integral inequalityJ. Math. Anal. Appl.1998220277878516149680911.26011 – reference: YangB.C.DebnathL.On a new generalization of Hardy–Hilbert’s inequality and its applicationJ. Math. Anal. Appl.199923248449716895740935.26009 – reference: RassiasM.T.YangB.C.RaigorodskiiA.On a more accurate reverse Hilbert-type inequality in the whole planeJ. Math. Inequal.20201441359137441935411461.26015 – reference: MoH.M.YangB.C.On a new Hilbert-type integral inequality involving the upper limit functionsJ. Inequal. Appl.202020204062070 – reference: WangZ.X.GuoD.R.Introduction to Special Functions2012BeijingHigher Education Press – reference: YangB.C.A new Hilbert-type integral inequality with the homogeneous kernel of degree 0J. Zhejiang Univ. Sci. Ed.20123943903923012253 – reference: YouM.H.On an extension of the discrete Hilbert inequality and applicationsJ. Wuhan Univ. Natur. Sci. Ed.2021672179184 – reference: YouM.H.On a class of Hilbert-type inequalities in the whole plane related to exponent functionJ. Inequal. Appl.202120214217995 – reference: HongY.LiaoJ.Q.YangB.C.A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applicationsJ. Inequal. Appl.202020204101559 – reference: RassiasM.T.YangB.C.On half-discrete Hilbert’s inequalityAppl. Math. Comput.2013220759330918321329.26041 – reference: YangB.C.On new extensions of Hilbert’s inequalityActa Math. Hung.2004104429129920827781062.26023 – reference: ChenZ.XuJ.New extensions of Hardy–Hilbert’s inequality with multiple parametersActa Math. Hung.200711743834001164.26020 – reference: KrnićM.PečarićJ.General Hilbert’s and Hardy’s inequalitiesMath. Inequal. Appl.200584295121379041079.26018 – reference: YangB.C.A note on Hilbert’s integral inequalitiesChin. Q. J. Math.1998134838517058120970.26011 – reference: YangB.C.The Norm of Operator and Hilbert-Type Inequalities2009BeijingScience Press – reference: YangB.C.DebnathL.On the extended Hardy–Hilbert’s inequalityJ. Math. Anal. Appl.2002272118719919307101009.26016 – reference: YangB.C.On an extension of Hilbert’s integral inequality with some parametersAust. J. Math. Anal. Appl.200411182077658 – reference: HeB.YangB.C.ChenQ.A new multiple half-discrete Hilbert-type inequality with parameters and a best possible constant factorMediterr. J. Math.2015121227124434168581327.26021 – reference: KrnićM.PečarićJ.VukovićP.A unified treatment of half-discrete Hilbert-type inequalities with a homogeneous kernelMediterr. J. Math.2013101697171631193281280.26037 – reference: RassiasM.T.YangB.C.On a Hilbert-type integral inequality in the whole plane related to the extended Riemann zeta functionComplex Anal. Oper. Theory20191341765178239570131435.26031 – reference: KrnićM.PečarićJ.Extension of Hilbert’s inequalityJ. Math. Anal. Appl.200632415016022624621112.26019 – reference: RassiasM.T.YangB.C.On a Hilbert-type integral inequality related to the extended Hurwitz zeta function in the whole planeActa Appl. Math.20191601678039346581429.26037 – reference: LiuQ.LongS.C.A Hilbert-type integral inequality with the kernel of hyperbolic secant functionJ. Zhejiang Univ. Sci. 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| Snippet | By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type... Abstract By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a... |
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| SubjectTerms | Analysis Applications of Mathematics Bernoulli number Half-discrete Hilbert-type inequality Hyperbolic functions Inequalities Inequality Mathematical functions Mathematics Mathematics and Statistics Rational fraction expansion |
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| Title | On a half-discrete Hilbert-type inequality related to hyperbolic functions |
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