Monotonicity and convexity involving generalized elliptic integral of the first kind
In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind K a ( r ) and its approximation log [ 1 + 2 / ( a r ′ ) ] , and also the convexity (concavity) of their difference for a ∈ ( 0 , 1 / 2 ] . As an application, we give new bounds...
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| Published in | Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 115; no. 2 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
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Springer International Publishing
01.04.2021
Springer Nature B.V |
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| Abstract | In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind
K
a
(
r
)
and its approximation
log
[
1
+
2
/
(
a
r
′
)
]
, and also the convexity (concavity) of their difference for
a
∈
(
0
,
1
/
2
]
. As an application, we give new bounds for generalized Grötzsch ring function
μ
a
(
r
)
and a upper bound for
K
a
(
r
)
. |
|---|---|
| AbstractList | In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind
K
a
(
r
)
and its approximation
log
[
1
+
2
/
(
a
r
′
)
]
, and also the convexity (concavity) of their difference for
a
∈
(
0
,
1
/
2
]
. As an application, we give new bounds for generalized Grötzsch ring function
μ
a
(
r
)
and a upper bound for
K
a
(
r
)
. In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind Ka(r) and its approximation log[1+2/(ar′)], and also the convexity (concavity) of their difference for a∈(0,1/2]. As an application, we give new bounds for generalized Grötzsch ring function μa(r) and a upper bound for Ka(r). |
| ArticleNumber | 46 |
| Author | Zhao, Tie-Hong Chu, Yu-Ming Wang, Miao-Kun |
| Author_xml | – sequence: 1 givenname: Tie-Hong orcidid: 0000-0002-6394-1049 surname: Zhao fullname: Zhao, Tie-Hong organization: Department of Mathematics, Hangzhou Normal University – sequence: 2 givenname: Miao-Kun surname: Wang fullname: Wang, Miao-Kun organization: Department of Mathematics, Huzhou University – sequence: 3 givenname: Yu-Ming surname: Chu fullname: Chu, Yu-Ming email: chuyuming2005@126.com organization: Department of Mathematics, Huzhou University |
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| Cites_doi | 10.1016/j.jmaa.2015.03.043 10.1137/S0036141096310491 10.3934/math.2020290 10.1080/10652469.2011.627511 10.1017/S030821050000264X 10.1016/j.jmaa.2018.03.005 10.1007/s00209-007-0111-x 10.1007/s13398-020-00856-w 10.1007/s13398-020-00784-9 10.1007/s11139-018-0061-4 10.1016/j.jnt.2015.10.013 10.1186/s13660-020-02327-7 10.1090/bproc/41 10.1186/s13660-017-1606-6 10.1007/s10476-014-0101-2 10.1016/j.jmaa.2018.08.061 10.1016/j.jmaa.2015.04.035 10.1186/s13660-017-1578-6 10.1007/s13398-019-00719-z 10.1137/0523025 10.2140/pjm.2000.192.1 10.1016/j.jnt.2016.01.021 10.2298/AADM171015001Y |
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| References | OlvermFWJLozierDWBoisvertRFClarkCWNIST Handbook of Mathematical Functions2010CambridgeCambridge University Press1198.00002 BariczÁTurán type inequalities for generalized complete elliptic integralsMath. Z.20072564895911230889610.1007/s00209-007-0111-x QiFLiW-HA logarithmically completely monotonic function involving the ratio of gamma functionsJ. Appl. Anal. Comput.20155462663433674571447.33002 AlzerHSome beta-function inequalitiesProc. R. Soc. Edinburgh.2003133A4731745200619910.1017/S030821050000264X YangZ-HTianJ-FSharp inequalities for the generalized elliptic integrals of the first kindRamanujan J.201948191116390249710.1007/s11139-018-0061-4 HaiG-JZhaoT-HMonotonicity properties and bounds involving the two-parameter generalized Grötzsch ring functionJ. Inequal. Appl.202020201710.1186/s13660-020-02327-7(Article 66) YangZ-HChuY-MWangM-KMonotonicity criterion for the quotient of power series with applicationsJ. Math. Anal. Appl.20154281587604332700510.1016/j.jmaa.2015.03.043 HuangT-RQiuS-LMaX-YMonotonicity properties and inequalities for the generalized elliptic integral of the first kindJ. Math. Anal. Appl.2019469195116385751210.1016/j.jmaa.2018.08.061 AbramowitzMStegunIAHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables1964WashingtonU.S. Government Printing Office0171.38503 ZhaoT-HWangM-KChuY-MA sharp double inequality involving generalized complete elliptic integral of the first kindAIMS Math.20205545124528414746210.3934/math.2020290 YinLHuangL-GWangY-LLinX-LAn inequality for generalized complete elliptic integralJ. Inequal. Appl.201720176373659910.1186/s13660-017-1578-6( Article 303) YangZ-HTianJ-FWangM-KA positive answer to Bhatia–Li conjecture on the monotonicity for a new mean in its parameterRev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.20201143126409544710.1007/s13398-020-00856-w1439.33002 ChenC-PInequalities and asymptotic expansions for the psi function and the Euler–Mascheroni constantJ. Number Theory.2016163596607345958910.1016/j.jnt.2015.10.013 QianW-MHeZ-HChuY-MApproximation for the complete elliptic integral of the first kindRev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.2020114257404987610.1007/s13398-020-00784-91434.33023 YangZ-HTianJ-FConvexity and monotonicity for elliptic integrals of the first kind and applicationsAppl. Anal. Discret. Math.2019131240260394805410.2298/AADM171015001Y Yang, Z.-H.: A new way to prove L’Hospital monotone rules with applications. arXiv:1409.6408 [math.CA] AndrewsGEAskeyRRoyRSpecial Functions. Encyclopedia of Mathematics and Its Applications1999CambridgeCambridge University Press GuoB-NQiFMonotonicity of functions connected with the gamma function and the volume of the unit ballIntegral Transforms Spec. Funct.2012239701708296888710.1080/10652469.2011.627511 RichardsKCA note on inequalities for the ratio of zero-balanced hypergeometric functionsProc. Am. Math. Soc. Ser. B201961520394686210.1090/bproc/41 WangM-KChuY-MQiuS-LSharp bounds for generalized elliptic integrals of the first kindJ. Math. Anal. Appl.2015429744757334249010.1016/j.jmaa.2015.04.035 HuangT-RHanB-WMaX-YChuY-MOptimal bounds for the generalized Euler–Mascheroni constantJ. Inequal. Appl.201820189380408610.1186/s13660-017-1606-6(Article 118) AndersonGDQiuS-LVamanamurthyMKVuorinenMGeneralized elliptic integrals and modular equationsPac. J. Math.20001921137174103110.2140/pjm.2000.192.1 AndersonGDVamanamurthyMKVuorinenMFunctional inequalities for hypergeometric functions and complete elliptic integralsSIAM J. Math. Anal.199223512524114787510.1137/0523025 YangZ-HQianW-MChuY-MZhangWOn approximating the arithmetic-geometric mean and complete elliptic integral of the first kindJ. Math. Anal. Appl.2018462217141726377431310.1016/j.jmaa.2018.03.005 AndersonGDQiuS-LVamanamurthyMKVuorinenMConformal Invariants, Inequalities, and Quasiconformal Maps1997New YorkWiley0885.30012 AlzerHInequalities for the beta functionAnal. Math.2014401111316863810.1007/s10476-014-0101-2 BorweinJMBorweinPBPi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts. A Study in Analytic Number Theory and Computational Complexity; A Wiley-Interscience Publication1987New YorkWiley ChenC-PSharp inequalities and asymptotic series of a product related to the Euler–Mascheroni constantJ. Number Theory.2016165314323347922610.1016/j.jnt.2016.01.021 QiFBounds for the ratio of two gamma functionsJ. Inequal. Appl.2010Article ID 4930588426110441194.33005 YangZ-HTianJ-FMonotonicity, convexity, and complete monotonicity of two functions related to the gamma functionRev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.2019113436033617399903710.1007/s13398-019-00719-z QiuS-LVamanamurthyMKVuorinenMSome inequalities for the growth of elliptic integralsSIAM J. Math. 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| References_xml | – reference: YangZ-HTianJ-FSharp inequalities for the generalized elliptic integrals of the first kindRamanujan J.201948191116390249710.1007/s11139-018-0061-4 – reference: ZhaoT-HWangM-KChuY-MA sharp double inequality involving generalized complete elliptic integral of the first kindAIMS Math.20205545124528414746210.3934/math.2020290 – reference: BariczÁTurán type inequalities for generalized complete elliptic integralsMath. Z.20072564895911230889610.1007/s00209-007-0111-x – reference: QiFBounds for the ratio of two gamma functionsJ. Inequal. Appl.2010Article ID 4930588426110441194.33005 – reference: HaiG-JZhaoT-HMonotonicity properties and bounds involving the two-parameter generalized Grötzsch ring functionJ. Inequal. Appl.202020201710.1186/s13660-020-02327-7(Article 66) – reference: YangZ-HChuY-MWangM-KMonotonicity criterion for the quotient of power series with applicationsJ. Math. Anal. Appl.20154281587604332700510.1016/j.jmaa.2015.03.043 – reference: ChenC-PSharp inequalities and asymptotic series of a product related to the Euler–Mascheroni constantJ. Number Theory.2016165314323347922610.1016/j.jnt.2016.01.021 – reference: ChenC-PInequalities and asymptotic expansions for the psi function and the Euler–Mascheroni constantJ. Number Theory.2016163596607345958910.1016/j.jnt.2015.10.013 – reference: AbramowitzMStegunIAHandbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables1964WashingtonU.S. Government Printing Office0171.38503 – reference: QiFLiW-HA logarithmically completely monotonic function involving the ratio of gamma functionsJ. Appl. Anal. Comput.20155462663433674571447.33002 – reference: AndersonGDQiuS-LVamanamurthyMKVuorinenMGeneralized elliptic integrals and modular equationsPac. J. Math.20001921137174103110.2140/pjm.2000.192.1 – reference: AndrewsGEAskeyRRoyRSpecial Functions. Encyclopedia of Mathematics and Its Applications1999CambridgeCambridge University Press – reference: AndersonGDQiuS-LVamanamurthyMKVuorinenMConformal Invariants, Inequalities, and Quasiconformal Maps1997New YorkWiley0885.30012 – reference: QianW-MHeZ-HChuY-MApproximation for the complete elliptic integral of the first kindRev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.2020114257404987610.1007/s13398-020-00784-91434.33023 – reference: HuangT-RHanB-WMaX-YChuY-MOptimal bounds for the generalized Euler–Mascheroni constantJ. Inequal. Appl.201820189380408610.1186/s13660-017-1606-6(Article 118) – reference: BorweinJMBorweinPBPi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts. A Study in Analytic Number Theory and Computational Complexity; A Wiley-Interscience Publication1987New YorkWiley – reference: YinLHuangL-GWangY-LLinX-LAn inequality for generalized complete elliptic integralJ. Inequal. 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| Snippet | In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind
K
a
(
r
)
and its approximation
log... In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind Ka(r) and its approximation... |
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| SubjectTerms | Applications of Mathematics Concavity Convexity Integrals Mathematical and Computational Physics Mathematics Mathematics and Statistics Original Paper Theoretical Upper bounds |
| Title | Monotonicity and convexity involving generalized elliptic integral of the first kind |
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