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