  In this article we consider means of positive operators on a Hilbert space.
We extend the theory of matrix power means to arbitrary operator means in the
sense of Kubo-Ando. The basis of the extension is relying on ideas coming from
differential geometry. We consider generalized Karcher equations for positive
operators and show that such equations admit unique positive solutions that can
be obtained as a limit of one-parameter families of operator means called
induced operator means. These means are themselves unique fixed points of one
parameter families of strict contractions induced, through Kubo-Ando theory of
operator means, by semigroups of holomorphic functions mapping the upper
half-plane into itself. These semigroups of holomorphic functions are
considered with Koenigs function corresponding to Schroeder's functional
equation. Koenigs function in this setting provides us with a logarithm map
corresponding to every 2-variable operator mean. The semigroups of 2-variable
means behave as geodesics and we exactly classify the cases when they are
indeed geodesics of affinely connected manifolds, thereby providing the cases
when these generalized Karcher equations are exactly Karcher equations in the
geometric sense. This is achieved by studying the arising holonomy groups. The
unique solutions of these generalized Karcher equations are called lambda
extensions and have numerous desirable properties which are inherited from the
induced operator means themselves.
