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Zbl 1238.33011
Neuman, Edward
On one-parameter family of bivariate means. (English)
[J] Aequationes Math. 83, No. 1-2, 191-197 (2012). ISSN 0001-9054; ISSN 1420-8903/e

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] [{\it M. Abramowitz} (ed.) and {\it 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.
[Thomas Ernst (Uppsala)]

MSC 2000:: *33E05 Elliptic functions and integrals
26D07 Inequalities involving other types of real functions
26E60 Means

Keywords: bivariate means; Schwab-Borchardt mean; inverse Jacobian elliptic functions; completely symmetric elliptic integral; inequalities

Citations: Zbl 0643.33001

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