Hölder mean inequalities for the complete elliptic integrals. (English) Zbl 1258.33011
The authors determine the least constants \(p_1\) and greatest \(p_2\) such that \(M_p(K(x), K(y))\geq K(M_p(x,y))\) or \(M_p(E(x), E(y))\leq E(M_p(x, y))\) hold for all \(p\in [p_1,p_2]\) and all \(x,y\in (0,1)\), where \(M_p\) is the Hölder mean of order \(p\), while \(K\) and \(E\) denote the complete elliptic integrals of the first and second kind, respectively. For \(p\in [0,2]\), the first inequality has been studied by A. Baricz [JIPAM, J. Inequal. Pure Appl. Math. 8, No. 2, Paper No. 40, 9 p. (2007; Zbl 1134.33004)].
Reviewer: József Sándor (Cluj-Napoca)
Citations:
Zbl 1134.33004References:
| [1] | DOI: 10.1016/j.cam.2004.02.009 · Zbl 1059.33029 · doi:10.1016/j.cam.2004.02.009 |
| [2] | DOI: 10.1007/BF02767349 · Zbl 0652.30013 · doi:10.1007/BF02767349 |
| [3] | DOI: 10.1137/0521029 · Zbl 0692.33001 · doi:10.1137/0521029 |
| [4] | DOI: 10.1137/0523025 · Zbl 0764.33009 · doi:10.1137/0523025 |
| [5] | DOI: 10.2307/2154966 · Zbl 0826.33003 · doi:10.2307/2154966 |
| [6] | Anderson G. D., Conformal Invariants, Inequalities, and Quasiconformal Maps (1997) · Zbl 0885.30012 |
| [7] | DOI: 10.1007/s003659900070 · Zbl 0901.33011 · doi:10.1007/s003659900070 |
| [8] | DOI: 10.2140/pjm.2000.192.1 · Zbl 0951.33012 · doi:10.2140/pjm.2000.192.1 |
| [9] | DOI: 10.1016/j.jmaa.2007.02.016 · Zbl 1125.26017 · doi:10.1016/j.jmaa.2007.02.016 |
| [10] | András S., Expo. Math. 28 pp 357– (2010) · Zbl 1204.33004 · doi:10.1016/j.exmath.2009.12.005 |
| [11] | Baricz Á., JIPAM. J. Inequal. Pure Appl. Math. 8 (2007) |
| [12] | DOI: 10.1007/s00209-007-0111-x · Zbl 1125.26022 · doi:10.1007/s00209-007-0111-x |
| [13] | Bowman F., Introduction to Elliptic Functions with Applications (1961) · Zbl 0098.28304 |
| [14] | Bullen P. S., Handbook of Means and Their Inequalities (2003) · Zbl 1035.26024 · doi:10.1007/978-94-017-0399-4 |
| [15] | Byrd P. F., Handbook of Elliptic Integrals for Engineers and Scientists (1971) · Zbl 0213.16602 |
| [16] | DOI: 10.1080/10652460212900 · Zbl 1019.30011 · doi:10.1080/10652460212900 |
| [17] | DOI: 10.1080/10652460701210284 · Zbl 1133.42005 · doi:10.1080/10652460701210284 |
| [18] | Evans, R. J. Ramanujan’s second notebook: Asymptotic expansions for hypergeometric series and related functions. Ramanujan Revisited: Proceedings of the Ramanujan Centenary Conference. June1–51987. Edited by: Andrews, G. E., Askey, R. A., Berndt, B. C., Ramanathan, K. G. and Rankin, R. A. pp.537–560. Boston, MA: Academic Press. University of Illinois |
| [19] | DOI: 10.1080/10652469408819056 · Zbl 0822.33003 · doi:10.1080/10652469408819056 |
| [20] | DOI: 10.1080/17476930903276134 · Zbl 1187.30017 · doi:10.1080/17476930903276134 |
| [21] | DOI: 10.1080/17476939708815054 · Zbl 0951.30010 · doi:10.1080/17476939708815054 |
| [22] | DOI: 10.1090/S0025-5718-05-01734-5 · Zbl 1084.33001 · doi:10.1090/S0025-5718-05-01734-5 |
| [23] | DOI: 10.1080/10652460701722627 · Zbl 1141.26004 · doi:10.1080/10652460701722627 |
| [24] | Qiu S.-L., Acta Math. Sinica 43 pp 283– (2000) |
| [25] | DOI: 10.1155/S1025583499000326 · Zbl 0946.30010 · doi:10.1155/S1025583499000326 |
| [26] | Saigo M., Proc. Amer. Math. Soc. 110 pp 71– (1990) |
| [27] | DOI: 10.1080/1065246042000210403 · doi:10.1080/1065246042000210403 |
| [28] | DOI: 10.1006/jmaa.1994.1137 · Zbl 0802.26009 · doi:10.1006/jmaa.1994.1137 |
| [29] | Vuorinen M., Studia Math. 121 pp 221– (1996) |
| [30] | DOI: 10.1080/10652460500422031 · Zbl 1136.35420 · doi:10.1080/10652460500422031 |
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