Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-08T21:35:55.018Z Has data issue: false hasContentIssue false
  • English
  • Français

On generalized complete elliptic integrals and modular functions

Part of: Elementary classical functions

Published online by Cambridge University Press:  12 April 2012

B. A. Bhayo  and
M. Vuorinen
B. A. Bhayo
Affiliation:
Department of Mathematics, University of Turku, 20014 Turku, Finland ([email protected]; [email protected])
M. Vuorinen
Affiliation:
Department of Mathematics, University of Turku, 20014 Turku, Finland ([email protected]; [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.

Keywords

modular equationgeneralized elliptic integralgeometric function theory

MSC classification

Secondary: 33C99: None of the above, but in this section 33B99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 591 - 611
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Abramowitz, M. and Stegun, I., Handbook of mathematical functions with formulas, graphs and mathematical tables (National Bureau of Standards, 1964; Russian transl., Nauka, 1979).Google Scholar
2.Alzer, H. and Qiu, S.-L., Monotonicity theorems and inequalities for the complete elliptic integrals, J. Computat. Appl. Math. 172 (2004), 289–312.CrossRefGoogle Scholar
3.Anderson, G. D., Qiu, S.-L., Vamanamurthy, M. K. and Vuorinen, M., Generalized elliptic integrals and modular equation, Pac. J. Math. 192 (2000), 1–37.CrossRefGoogle Scholar
4.Anderson, G. D., Vamanamurthy, M. K. and Vuorinen, M., Dimension-free quasiconformal distortion in n-space, Trans. Am. Math. Soc. 297 (1986), 687–706.Google Scholar
5.Anderson, G. D., Vamanamurthy, M. K. and Vuorinen, M., Inequalities for the extremal distortion function, Lecture Notes in Mathematics, Volume 1351, pp. 1–11 (Springer, 1988).Google Scholar
6.Anderson, G. D., Vamanamurthy, M. K. and Vuorinen, M., Topics in special functions: a volume dedicated to Olli Martio on the occasion of his 60th birthday(ed. Heinonen, J., Kilpeläinen, T. and Koskela, P., Report 83, pp. 5–26 (University of Jyväskylä, Department of Mathematics and Statistics, 2001).Google Scholar
7.Anderson, G. D., Vamanamurthy, M. K. and Vuorinen, M., Topics in special functions, II, Conform. Geom. Dyn. 11 (2007), 250–270.CrossRefGoogle Scholar
8.Anderson, G. D., Vamanamurthy, M. K. and Vuorinen, M., Conformal invariants, inequalities and quasiconformal maps (Wiley, 1997).Google Scholar
9.Anderson, G. D. and Vuorinen, M., Reflections on Ramanujan's mathematical gems, in Mathematics Newsletter (Special Issue Commemorating ICM 2010 in India), Volume 19, Sp. No. 1, pp. 87–108 (Ramanujan Mathematical Society, 2010; available at www.ramanujanmathsociety.org/mnl/August2010/mnl-spl-fullfiles.pdf.)Google Scholar
10.Balasubramanian, R., Ponnusamy, S. and Vuorinen, M., Functional inequalities for quotients of hypergeometric functions, J. Math. Analysis Applic. 218 (1998), 256–268.CrossRefGoogle Scholar
11.Baricz, Á., Functional inequalities involving special functions, J. Math. Analysis Applic. 319 (2006), 450–459.CrossRefGoogle Scholar
12.Baricz, Á., Functional inequalities involving special functions, II, J. Math. Analysis Applic. 327 (2007), 1202–1213.CrossRefGoogle Scholar
13.Baricz, Á., Turán type inequalities for generalized complete elliptic integrals, Math. Z. 256 (2007), 895–911.CrossRefGoogle Scholar
14.Baricz, Á., Turán type inequalities for hypergeometric functions, Proc. Am. Math. Soc. 136 (2008), 3223–3229.CrossRefGoogle Scholar
15.Baricz, Á., Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, Volume 1994 (Springer, 2010).CrossRefGoogle Scholar
16.Berndt, B. C., Bhargava, S. and Garvan, F. G., Ramanujan's theories of elliptic functions to alternative bases, Trans. Am. Math. Soc. 347 (1995), 4163–4244.Google Scholar
17.Bhayo, B. A. and Vuorinen, M., Inequalities for eigenfunctions of the p-Laplacian, preprint (arXiv:1101.3911 [math.CA]).Google Scholar
18.Hakula, H., Rasila, A. and Vuorinen, M., On moduli of rings and quadrilaterals: algorithms and experiments, SIAM J. Sci. Comput. 33 (2011), 279–302.CrossRefGoogle Scholar
19.Heikkala, V., Lindén, H., Vamanamurthy, M. K. and Vuorinen, M., Generalized elliptic integrals and the Legendre M-function, J. Math. Analysis Applic. 338 (2008), 223–243.CrossRefGoogle Scholar
20.Heikkala, V., Vamanamurthy, M. K. and Vuorinen, M., Generalized elliptic integrals, Computat. Meth. Funct. Theory 9 (2009), 75–109.CrossRefGoogle Scholar
21.Kühnau, R. (ed.), Handbook of complex analysis: geometric function theory, Volume 1 (North-Holland, Amsterdam, 2002).Google Scholar
22.Kühnau, R. (ed.), Handbook of complex analysis: geometric function theory, Volume 2 (North-Holland, Amsterdam, 2005).Google Scholar
23.Lehto, O. and Virtanen, K. I., Quasiconformal mappings in the plane, 2nd edn, Die Grundlehren der mathematischen Wissenschaften, Band 126 (Springer, 1973).CrossRefGoogle Scholar
24.Lindqvist, P., Some remarkable sine and cosine functions, Ric. Mat. 44 (1995), 269–290.Google Scholar
25.Ponnusamy, S. and Vuorinen, M., Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), 278–301.CrossRefGoogle Scholar
26.Ruskeepää, H., Mathematica® navigator, 3rd edn (Academic Press, 2009).Google Scholar
27.Vuorinen, M., Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, Volume 1319 (Springer, 1988).CrossRefGoogle Scholar
28.Vuorinen, M., Special functions and conformal invariants, in Analysis and its applications, pp. 255–268 (Allied Publishers, New Delhi, 2001).Google Scholar
29.Wang, G., Zhang, X., Qiu, S. and Chu, Y., The bounds of the solutions to generalized modular equations, J. Math. Analysis Applic. 321 (2006), 589–594.CrossRefGoogle Scholar
30.Wang, G., Zhang, X. and Chu, Y., Inequalities for the generalized elliptic integrals and modular functions, J. Math. Analysis Applic. 331 (2007), 1275–1283.CrossRefGoogle Scholar
31.Zhang, X., Wang, G. and Chu, Y., Some inequalities for the generalized Grötzsch function, Proc. Edinb. Math. Soc. 51 (2008), 265–272.CrossRefGoogle Scholar
32.Zhang, X., Wang, G. and Chu, Y., Remarks on generalized elliptic integrals, Proc. R. Soc. Edinb. A 139 (2009), 417–426.CrossRefGoogle Scholar