Summary: M. Atteia and M. Raïssouli [J. Convex Anal. 8, 223–240 (2001; Zbl 1003.90030)] constructed a self-dual operation from an algorithm using the sum and infimal convolution operators. This operation permits the authors to define the functionals “convex geometric mean” and “convex square root”.
We study the differentiability at a given point of the geometric mean of two convex functionals. This study permits us to obtain some important results, firstly we give a characterization of a hessian which generalizes the quadratic case in [loc. cit.]. Furthermore, we obtain conditions such that the square root of a hessian is also one. Secondly, we can make the hessian of the arithmetico-geometric mean functional explicit.