Some new Karamata type inequalities and their applications to some entropies

Some new inequalities of Karamata type are established with a convex function in this paper. The methods of our proof allow us to obtain an extended version of 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 von Neumann entropy and quantum Tsallis entropy which is a parametric extension of von Neumann entropy. The inequality for von Neumann entropy recovers the non-negativity and gives a refinement for the weaker version of Fannes's inequality for only special cases. Finally, we estimate bounds for the Tsallis relative operator entropy.