Based on a re-examination of the convexity results for the inverse of positive definite matrices, and the Moore-Penrose inverse of nonnegative definite matrices, a more general concept of strong convexity is defined which provides additional information on the convexity behaviour and geometry of matrix functions known to be strictly convex. Convexity results which include characterisations of the null spaces are proved and further used in order to obtain short proofs on sums of reproducing kernels, to derive a number of matrix-mean-type inequalities, and, subsequently, a quasilinear extremal representation of a weighted harmonic matrix mean is derived from these inequalities. The main part of the paper deals with a detailed study of convexity of Moore-Penrose inverse of nonnegative definite matrices providing extensions and generalisations of all the earlier work in this area, many of the obtained results being straightforward extensions to the case of bounded linear operators acting on an infinite-dimensional Hilbert space.