Identity Extension for Function on Three Intervals and Application to Csiszar Divergence, Levinson and Ky Fan Inequalities

Using Taylor-type expansions, we obtain identity expressions for functions on three intervals and differences for two pairs of Csiszár ϕ-divergence. With some more assumptions in these identities, inequalities for functions on three intervals and Csiszár ϕ-divergence can be obtained as special cases. They can also deduce the known generalized trapezoid type inequality. Furthermore, we use the identity to obtain a new extension for Levinson inequality; thus, new refinements and reverses for Ky Fan-type inequalities are established, which can be used to compare or estimate the yields in investments. Special cases of Csiszár ϕ-divergence are given, and we obtain new inequalities concerning different pairs of Kullback–Leibler distance, Hellinger distance, α-order entropy and χ2-distance.