A geometric mean of parameterized arithmetic and harmonic means of convex functions (Q1938303)

Summary: The notion of the geometric mean of two positive reals was extended by Ando in 1978 to the case of positive semidefinite matrices \(A\) and \(B\). Moreover, an interesting generalization of the geometric mean \(A \sharp B\) of \(A\) and \(B\) to convex functions was introduced by \textit{M. Atteia} and \textit{M. Raïssouli} [J. Convex Anal. 8, No. 1, 223--240 (2001; Zbl 1003.90030)] 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 [loc. cit.]. 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.