A geometric mean of parameterized arithmetic and harmonic means of convex functions

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matrices A and B. Moreover, an interesting generalization of the geometric mean A # B of A and B to convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named "the geometric mean of parameterized arithmetic and harmonic means of convex functions" is proposed, and its essential properties are investigated.