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.

We study an optimal harvesting for a nonlinear age-spatial-structured population dynamic model, where the dynamic system contains an external mortality rate depending on the total population size. The total mortality consists of two types: the natural, and external mortality and the external mortality reflects the effects of external environmental causes. We prove the existence and uniqueness of solutions for the population dynamic model. We also derive a sufficient condition for optimal harvesting and some necessary conditions for optimality in an optimal control problem relating to the population dynamic model. The results may be applied to an optimal harvesting for some realistic biological models.