zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Bounds for symmetric elliptic integrals. (English) Zbl 1041.33015
In this paper the author derives lower and upper bounds for the four standard incomplete symmetric elliptic integrals. The bounding functions are expressed in terms of the elementary transcendental functions. All elliptic integrals delt with in this paper can be represented by the $R$-hypergeometric functions introduced in [{\it B. C. Carlson}, Special Functions of Applied Mathematics, Academic Press, New York (1977; Zbl 0394.33001)]. Sharp bounds for the ratio of the complete elliptic integrals of the first and second kind are also obtained. These results can be used to obtain bounds for the product of these integrals. It is shown that an iterative numerical algorithm for computing the ratios and products of complete integrals has the second order of convergence.

33E05Elliptic functions and integrals
Full Text: DOI
[1] Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M.: Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. anal. 21, No. 2, 536-549 (1990) · Zbl 0692.33001
[2] Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M.: Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. anal. 23, No. 2, 512-524 (1992) · Zbl 0764.33009
[3] Carlson, B. C.: Some series and bounds for incomplete elliptic integrals. J. math. Phys. 40, 125-134 (1961) · Zbl 0113.28103
[4] Carlson, B. C.: A hypergeometric mean value. Proc. amer. Math. soc. 16, No. 4, 759-766 (1965) · Zbl 0137.26802
[5] Carlson, B. C.: Some inequalities for hypergeometric functions. Proc. amer. Math. soc. 17, No. 1, 32-39 (1966) · Zbl 0137.26803
[6] Carlson, B. C.: Inequalities for a symmetric elliptic integral. Proc. amer. Math. soc. 25, No. 3, 698-703 (1970) · Zbl 0194.08603
[7] Carlson, B. C.: Special functions of applied mathematics. (1977) · Zbl 0394.33001
[8] Carlson, B. C.: Numerical computation of real or complex elliptic integrals. Numer. algorithms 10, 13-26 (1995) · Zbl 0827.65024
[9] Carlson, B. C.; Gustafson, J. L.: Asymptotic expansion of the first elliptic integral. SIAM J. Math. anal. 16, No. 5, 1072-1092 (1985) · Zbl 0593.33002
[10] Carlson, B. C.; Gustafson, J. L.: Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. anal. 25, No. 2, 288-303 (1994) · Zbl 0794.41021
[11] Neuman, E.: The weighted logarithmic mean. J. math. Anal. appl. 188, No. 3, 885-900 (1994) · Zbl 0823.33002
[12] E. Neuman, J. Sándor, On the Schwab-Borchardt mean, submitted.
[13] Vamanamurthy, M. K.; Vuorinen, M.: Inequalities for means. J. math. Anal. appl. 183, 155-166 (1994) · Zbl 0802.26009