Index of a bivariate mean and applications

Exploring some results of (Raïssouli in J. Math. Inequal. 10(1):83-99, 2016 ) from another point of view, we introduce here some power-operations for (bivariate) means. As application, we construct some classes of means in one or two parameters including some standard means. We also define a law between means which allows us to obtain, among others, a simple relationship involving the three familiar means, namely the first Seiffert mean, the second Seiffert mean, and the Neuman-Sándor mean. At the end, more examples of interest are discussed and open problems are derived as well.