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Three families of two-parameter means constructed by trigonometric functions. (English) Zbl 1297.26071
Summary: In this paper, we establish three families of trigonometric functions with two parameters and prove their monotonicity and bivariate log-convexity. Based on them, three two-parameter families of means involving trigonometric functions, which include Schwab-Borchardt mean, the first and second Seiffert means, Sándor’s mean and many other new means, are defined. Their properties are given and some new inequalities for these means are proved. Lastly, two families of two-parameter hyperbolic means, which similarly contain many new means, are also presented without proofs.

##### MSC:
 26E60 Means 26D05 Inequalities for trigonometric functions and polynomials 33B10 Exponential and trigonometric functions 26A48 Monotonic functions, generalizations
##### Keywords:
trigonometric function; hyperbolic function; mean; inequality
Full Text:
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