In this paper a new set of inequalities and bounds for the incomplete gamma function are obtained. These inequalities and bounds are based on continued fraction expansions of the incomplete gamma function (Sections 2 and 3). Comparisons between the two sets of inequalities and some other known inequalities are made (Section 4). Bounds are also obtained for the Mills' ratio for the normal integral (Section 5) and an analogue of Mills' ratio (Section 6) for the gamma distribution. Some other applications of these bounds to distribution theory problems arising in multiple decision theory are described (Section 6).
"A System of Inequalities for the Incomplete Gamma Function and the Normal Integral." Ann. Math. Statist. 36 (1) 139 - 149, February, 1965. https://doi.org/10.1214/aoms/1177700278