A monotonicity property of the composition of regularized and inverted-regularized gamma functions with applications

The gamma function and its various modifications such as the (upper) incomplete, regularized and inverted-regularized incomplete gamma functions are of importance in both theory and applications. In this note we observe an ‘if and only if ’ relationship between a certain axiom of insurance risk management and a monotonicity property of the composition of regularized and inverted-regularized incomplete gamma functions, assuming that the risks follow gamma distributions. We derive the monotonicity property by utilizing the above noted relationship and a probabilistic technique. The aforementioned insurance axiom, called consistent no-undercut, is explained in detail and related to several techniques of analysis.