Functional Inequalities for Complete Elliptic Integrals and Their Ratios

Some functional inequalities satisfied by complete elliptic integrals of the first kind are obtained. These inequalities are sharp and generalize the functional identity of Landen. A related inequality is given for certain quotients of such integrals.

  • [AS]  M. Abramowitz and , I. A. Stegun, Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun. Third printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1965xiv+1046 31:1400 Google Scholar

  • [A]  Glen D. Anderson, Derivatives of the conformal capacity of extremal rings, Ann. Acad. Sci. Fenn. Ser. A I Math., 10 (1985), 29–46 88c:30027 0593.31010 CrossrefGoogle Scholar

  • [AV]  G. D. Anderson and , M. K. Vamanamurthy, Inequalities for elliptic integrals, Publ. Inst. Math. (Beograd) (N.S.), 37(51) (1985), 61–63 87a:33001 0573.33001 Google Scholar

  • [AVV1]  G. D. Anderson, M. K. Vamanamurthy and , M. Vuorinen, Dimension-free quasiconformal distortion in n-space, Trans. Amer. Math. Soc., 297 (1986), 687–706 87j:30039 0632.30022 ISIGoogle Scholar

  • [AVV2]  G. D. Anderson, M. K. Vamanamurthy and , M. Vuorinen, Sharp distortion theorems for quasiconformal mappings, Trans. Amer. Math. Soc., 305 (1988), 95–111 89c:30049 0639.30019 CrossrefISIGoogle Scholar

  • [AVV3]  G. D. Anderson, M. K. Vamanamurthy and , M. Vuorinen, Special functions of quasiconformal theory, Exposition. Math., 7 (1989), 97–136 90k:30032 0686.30015 Google Scholar

  • [AVV4]  G. D. Anderson, M. K. Vamanamurthy and , M. Vuorinen, Inequalities for the extremal distortion functionComplex analysis, Joensuu 1987, Lecture Notes in Math., Vol. 1351, Springer, Berlin, 1988, 1–11, Proc. Thirteenth Rolf Nevanlinna Colloquium, New York 90e:30022 0658.30014 CrossrefGoogle Scholar

  • [BB1]  J. M. Borwein and , P. B. Borwein, The arithmetic-geometric mean and fast computation of elementary functions, SIAM Rev., 26 (1984), 351–366 10.1137/1026073 86d:65029 0557.65009 LinkISIGoogle Scholar

  • [BB2]  Jonathan M. Borwein and , Peter B. Borwein, Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1987xvi+414 89a:11134 0611.10001 Google Scholar

  • [BO]  F. Bowman, Introduction to elliptic functions with applications, Dover Publications Inc., New York, 1961ii+115 24:A2060 0098.28304 Google Scholar

  • [BF]  Paul F. Byrd and , Morris D. Friedman, Handbook of elliptic integrals for engineers and physicists, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Bd LXVII, Springer-Verlag, Berlin, 1954xiii+355, New York 15,702a 0055.11905 CrossrefGoogle Scholar

  • [CG]  B. C. Carlson and , John L. Gustafson, Asymptotic expansion of the first elliptic integral, SIAM J. Math. Anal., 16 (1985), 1072–1092 10.1137/0516080 87d:33002 0593.33002 LinkISIGoogle Scholar

  • [C]  A. Cayley, An Elementary Treatise on Elliptic Functions, Deighton, Bell, and Co., Cambridge, U.K., 1876 09.0327.01 Google Scholar

  • [E]  A. Enneper, Elliptische Functionen, Theorie und Geschichte, Louis Nebert, Halle, Germany, 1890 Google Scholar

  • [F]  J. S. Frame, 1988, Letter to G. D. Anderson, dated October 19, 20 Google Scholar

  • [FR]  Carl-Erik Fröberg, Complete elliptic integrals; Lund University, Department of Numerical Analysis, Table No. 2, CWK Gleerup, Lund, 1957, 82–, Sweden 19,68c Google Scholar

  • [G1]  F. W. Gehring, Symmetrization of rings in space, Trans. Amer. Math. Soc., 101 (1961), 499–519 24:A2677 0104.30002 CrossrefGoogle Scholar

  • [G2]  F. W. Gehring, Inequalities for condensers, hyperbolic capacity, and extremal lengths, Michigan Math. J., 18 (1971), 1–20 10.1307/mmj/1029000582 44:2915 0228.30014 CrossrefISIGoogle Scholar

  • [LF]  A. V. Lebedev and , R. M. Fedorova, A guide to mathematical tables, English edition prepared from the Russian by D. G. Fry, Pergamon Press, New York, 1960xlvi+586, Elmsford 22:5139a 0137.11102 Google Scholar

  • [LEVU]  Matti Lehtinen and , Matti Vuorinen, On Teichmüller's modulus problem in the plane, Rev. Roumaine Math. Pures Appl., 33 (1988), 97–106 89g:30035 0681.30013 Google Scholar

  • [LV]  O. Lehto and , K. I. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag, New York, 1973viii+258, Berlin, Grundlehren Math. Wiss. 126, Second edition 49:9202 CrossrefGoogle Scholar

  • [TO]  John Todd, Basic Numerical Mathematics. Vol. 1, International Series of Numerical Mathematics, Vol. 14, Birkhäuser Verlag, Basel, 1979, 253– 81i:65001 0454.65001 CrossrefGoogle Scholar

  • [TR]  F. Tricomi, Lectures on the Use of Special Functions by Calculations with Electronic Computers, Lecture Series 47, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, MD, 1966 Google Scholar

  • [VU1]  Matti Vuorinen, On Teichmüller's modulus problem in ${\bf R}\sp n$, Math. Scand., 63 (1988), 315–333 90k:30038 0665.30020 CrossrefISIGoogle Scholar

  • [VU2]  Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, Vol. 1319, Springer-Verlag, Berlin, 1988xx+209, New York 89k:30021 0646.30025 CrossrefGoogle Scholar

  • [WW]  E. T. Whittaker and , G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, London, 1958 Google Scholar