Abstract

By using operator perspectives of regular operator mappings we obtain some operator concave (convex) functions and derive the concavity (convexity) of some trace functions associated with the Fréchet differential mapping of certain power functions. Especially, we explore the concavity of the Fréchet differential mapping x→dg(x)⁎df(x)−1 in positive definite invertible operators with f(t)=tp for 0<p≤1 and g(t)=tq for p≤q≤p+1, and also with f(t)=log⁡t and g(t)=tq for 0<q≤1.

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