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Some new inequalities of Karamata type are established with a convex functionin this paper. The methods of our proof allow us to obtain an extended versionof the reverse of Jensen inequality given by Pe{\v} cari\'c and Mi\'ci\'c.  Applying the obtained results, we give reverses for information inequality(Shannon inequality) in different types, namely ratio type and difference type,under some conditions. Also, we provide interesting inequalities for vonNeumann entropy and quantum Tsallis entropy which is a parametric extension ofvon Neumann entropy. The inequality for von Neumann entropy recovers thenon-negativity and gives a refinement for the weaker version of Fannes'sinequality for only special cases. Finally, we estimate bounds for the Tsallisrelative operator entropy.

Some new Karamata type inequalities and their applications to some entropies