An invariance of geometric mean with respect to Lagrangian means

The invariance of the geometric mean G with respect to the Lagrangian mean-type mapping (Lf ,Lg), i.e. the equation G ◦ (Lf ,Lg) = G, is considered. We show that the functions f and g must be of high class regularity. This fact allows to reduce the problem to a differential equation and determine the second derivatives of the generators f and g. © 2006 Elsevier Inc. All rights reserved.