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Jensen's inequality for nomalized positive linear maps between the algebras of bounded linear operators on a Hilbert space is one of the most important inequalities in the functional analysis. In this book we establish some operator versions of Bellman's inequality. We also give some Bellman inequalities involving sesquilinear forms and invariant norms. In continuation, we refine the Jensen's operator inequality and use it for a refinement of the Bellman operator inequality. Also, we investigate a notion of relative operator entropy, which develops the theory started by J.I. Fujii and E. Kamei. Afterwards, some inequalities concerning the classical Shannon entropy are drawn from it.