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Inequalities for the Schwab-Borchardt mean and their applications. (English) Zbl 1252.26006

Summary: Inequalities for the Schwab-Borchardt mean are obtained. They contain known results for the trigonometric and hyperbolic functions including those obtained by J. Wilker [Problem 3306, Am. Math. Mon. 96, No. 1, 55 (1989; doi:10.2307/2323259)] and C. Huygens [Oeuvres complètes de Christiaan Huygens publiées par la Société Hollandaise des Sciences. La Haye (Den Haag), M. Nijhoff (1888–1937)]. The main results of this paper can also be utilized to obtain new inequalities for some bivariate means including the logarithmic mean and two means introduced by Seiffert.

MSC:

26D05 Inequalities for trigonometric functions and polynomials
26D07 Inequalities involving other types of functions
33B10 Exponential and trigonometric functions
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