  The Aluthge transformation is generalized in the viewpoint of the axiom of
operator means by using double operator integrals. It includes the mean
transformation which is defined by S. H. Lee, W. Y. Lee and Yoon. Next we shall
give some properties of it. Especially, we shall show that the $n$-th iteration
of mean transformation of an invertible matrix converges to a normal matrix.
Inclusion relations among numerical ranges of generalized Aluthge
transformations respect to some operator means are considered.
