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.


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