A note on the convexity of the Moore–Penrose inverse

This note is a sequel to an earlier study (Nordström [7]) on convexity properties of the inverse and Moore–Penrose inverse, in which the following question was raised. Given nonnegative definite matrices A and B with Moore–Penrose inverses A⁺ and B⁺, respectively, can one show that (λA+λ‾B)⁺⩽λA⁺+λ‾B⁺ holding for a single λ∈]0,1[is enough to guarantee its validity for all λ∈]0,1[? (The ordering above is the partial ordering, induced by the convex cone of nonnegative definite matrices, and λ‾:=1−λ.) In this note an affirmative answer is provided to this question.