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On one-parameter family of bivariate means. (English) Zbl 1238.33011
A one-parameter family of bivariate means is introduced. They are defined in terms of the inverse functions of Jacobian elliptic functions cn and nc, formulas 2.5 and 2.6. It is shown that the new means are symmetric and homogeneous of degree one in their variables. Members of this family of means interpolate an inequality which connects two Schwab-Borchardt means. Computable lower and upper bounds for the new mean are also established. Reviewer’s remark: The formulas 2.5 and 2.6. are not the same as in [p. 596] [M. Abramowitz (ed.) and I. A. Stegun (ed.), Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 ed.. John Wiley & Sons, Inc; Washington, D.C.: National Bureau of Standard, (1984; Zbl 0643.33001)] . The u should be changed to x in both formulas.
MSC:
33E05Elliptic functions and integrals
26D07Inequalities involving other types of real functions
26E60Means
Software:
DLMF
References:
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[2]Carlson B.C.: Algorithms involving arithmetic and geometric means. Am. Math. Mon. 78, 496–505 (1971) · Zbl 0218.65035 · doi:10.2307/2317754
[3]Carlson B.C.: Special Functions of Applied Mathematics. Academic Press, New York (1977)
[4]Neuman, E.: Inequalities for the Schwab-Borchardt mean and their applications. J. Math. Inequal. (2011, in press)
[5]Neuman, E.: Product formulas and bounds for Jacobian elliptic functions with applications. Integral Transforms Spec. Funct. (2011, in press)
[6]Neuman E., Sándor J.: On the Schwab-Borchardt mean. Math. Pannon. 14(2), 253–266 (2003)
[7]Neuman E., Sándor J.: On the Schwab-Borchardt mean II. Math. Pannon. 17(1), 49–59 (2006)
[8]Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds): The NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)
[9]Seiffert H.-J.: Problem 887. Nieuw. Arch. Wisk. 11, 176 (1993)
[10]Seiffert H.-J.: Aufgabe 16. Würzel 29, 87 (1995)