A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

Abstract

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 interestinggeneralization of the geometric mean $A\hspace{0.17em}\#\hspace{0.17em}B$ of $A$ and $B$ to convex functionswas introduced by Atteia and Raïssouli (2001) with a different viewpoint of convexanalysis. The present work aims at providing a further development of the geometricmean of convex functions due to Atteia and Raïssouli (2001). A new algorithmicself-dual operator for convex functions named “the geometric mean of parameterizedarithmetic and harmonic means of convex functions” is proposed, and its essentialproperties are investigated.