A new refinement of the classical arithmetic mean and geometric mean inequality is given. Moreover, a new interpretation of the classical mean is given and this refinement theorem is generalized.

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In 1999 Ando and Zhan proved a subadditivity inequality for operator concave functions. We extend it to all concave functions: Given positive semidefinite matrices A, B and a non-negative concave function f on [0, ∞),for all symmetric norms (in particular for all Schatten p-norms). The case f (t) = √ t is connected to some block-matrix inequalities, for instance the operator norm inequalityfor any partitioned Hermitian matrix.

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