For any unitarily invariant norm | · |, the Heinz inequalities for operators assert that 2 |A , and X any operators on a complex separable Hilbert space such that A, B are positive and ν ∈ [0, 1]. In this paper, we obtain a family of refinements of these norm inequalities by using the convexity of the function f (ν) = |A ν XB 1-ν + A 1-ν XB ν | and the Hermite-Hadamard inequality.