Jensen's type inequalities for two times differentiable functions with applications

In the main body, first of all this work recommends an inequality of Jensen's type involving Green functions for a class of two times differentiable functions. This result enables further to obtain some related interpolating inequalities for a function f such that |f''|^q is either concave or convex for q1. Then manipulation of certain existing results in the corresponding interpolating inequalities gives bounds for the differences of the Jensen-Steffensen and Jensen's inequalities. In the similar fashion, they provide some new variants for the reverse form of aforementioned inequalities. Further, the obtained results about Jensen's inequality yield different novel adaptations of Hölder's inequality, fresh insights into the discrepancy of the well known Hermite-Hadamard inequality, and inequalities for geometric mean, quasi-arithmetic mean, and power mean. As a resultant, this work also suggests graphical interpretation of some results to verify the authenticity and sharpness of the obtained results about Jensen's inequality. Finally, this research work put forward some applications involving Zipf-Mandelbrot entropy and various types of Csiszár divergence from information theory.