Implicit averaging functions

We propose a new type of weighted averaging functions given implicitly as a solution to an algebraic equation. The main ingredient is incorporation of weights performed by using a function different to the product. Strict triangular norms are one example of such functions. We show that the weights should be normalized in this case differently to the usual sum to one condition. We establish the conditions under which the implicit averages are well defined, are idempotent, monotone and bounded by the minimum and maximum of their arguments. We present many examples of implicit averages, some of which recover the existing families of functions while others define new families. In particular we provide new generalizations of power means, density based averages and weighted OWA functions.