Measure theoretic generalizations of Jensen's inequality by Fink's identity

We generalize integral Jensen's inequality and its converse for real Stieltjes measure utilizing the theory of n-convex function by employing Fink's identity. We also give several versions of discrete Jensen's inequality along with its reverses and its converse for real weights. As an application we give generalized variants of Hermite-Hadamard's inequality. Also we give applications in information theory by giving new estimations of generalized divergence functional, Shannon and relative entropies. Finally we give connections to Zipf-Mandelbrot and hybrid Zipf-Mandelbrot entropies.