Generalized cyclic Jensen and information inequalities

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We present Levinson’s type generalizations of cyclic refinements of Jensen’s inequality by employing recent class of functions that further characterize and extend the family of 3-convex functions. We get monotonic cyclic Jensen’s inequalities and particularly the renowned Jensen’s inequality for 3-convex functions at a point (f∈κ1c(I)). As an applications in information theory, we first introduce new Csiszár type cyclic divergence functional for 3-convex functions and establish cyclic-Kullback–Leibler and Hellinger distances. We give monotonicity of cyclic divergence functionals which enable us to construct monotonic Shannon, Relative and Zipf–Mandelbrot entropies.