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We present several generalizations and refinements of the Bellman inequality involving the interpolation paths by the Callebautinequality and apply it to the operator geometric means acting on a Hilbert space. We also give some reverse operator Bellmaninequalities. Finally, we get a reverse Choi-Davis-Jensen inequality concerning operator mean and use it for obtaining agenernalization of the reverse operator Bellman type inequality. AMS Subject Classification: 15A42; 47A63; 47A30

This note aims to generalize the reverse weighted arithmetic–geometric mean inequality of n positive invertible operators due to Lawson and Lim. In addition, we make comparisons between the weighted Karcher mean and Lawson–Lim geometric mean for higher powers.