Abstract. Let p be a positive number and h a function on R + satisfying h(xy) ≥ h(x)h(y) for any x, y ∈ R + . A non-negative continuous function f onholds for all positive semidefinite matrices A, B of order n with spectra in K, and for any α ∈ (0, 1).In this paper, we study properties of operator (p, h)-convex functions and prove the Jensen, Hansen-Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator (p, h)-convex. In applications, we obtain Choi-Davis-Jensen type inequality for operator (p, h)-convex functions and a relation between operator (p, h)-convex functions with operator monotone functions.

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It is well-known that the Heron mean is a linear interpolation between the arithmetic and the geometric means while the matrix power meaninterpolates between the harmonic, the geometric, and the arithmetic means. In this article, we establish several comparisons between the matrix power mean, the Heron mean, and the Heinz mean. Therefore, we have a deeper understanding about the distribution of these matrix means.

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The in-sphere property for matrix means is studied. It is proved that the matrix power mean satisfies in-sphere property with respect to the Hilbert-Schmidt norm. A new characterization of the matrix arithmetic mean is provided. Some reverse AGM inequalities involving unitarily invariant norms and operator monotone functions are also obtained.