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Abstract. The interpolation and comparison of a matrix version of Heron mean,. We shall discuss the complete interpolation and comparison of matrix version of such means by extending the range of α from [0,1] to R + . We shall also discuss some more results involving Heinz means.Mathematics subject classification (2010): 15A60, 47A30, 47A63, 47A64.

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Let A, B ∈ B(H ) be such that 0 < b 1 I ≤ A ≤ a 1 I and 0 < b 2 I ≤ B ≤ a 2 I for some scalars 0 < b i < a i , i = 1, 2 and Φ : B(H ) → B(K ) be a positive linear map. Weshow that for any operator mean σ with the representing function f , the double inequality(1−α)µ and # α (∇ α , resp.) is the weighted geometric (arithmetic, resp.) mean for α ∈ (0, 1).As applications, we present several generalized operator inequalities including Diaz-Metcalf and reverse Ando type inequalities. We also give some related inequalities involving Hadamard product and operator means.