New Jensen-Type Inequalities and their Applications

Convex analysis is fundamental to proving inequalities that have a wide variety of applications in economics and mathematics. In this paper we provide Jensen-type inequalities for functions that are, intuitively, "very" convex. These inequalities are simple to apply and can be used to generalize and extend previous results or to derive new results. We apply our inequalities to quantify the notion "more risk averse" provided by J. W. Pratt [Risk aversion in the small and in the large, in: Uncertainty in Economics, P. Diamond and M. Rothschild (eds.), Elsevier, Amsterdam (1978) 59--79]. We also apply our results in other applications from different fields, including risk measures, Poisson approximation, moment generating functions, log-likelihood functions, and Hermite-Hadamard type inequalities.