In the theory of inequalities there are often encountered inequalities which are first proved for finite series and then established for infinite series or integrals. We shall discuss here the finite analogs of several integral inequalities which appear to have been established directly. There is often the added interest in the finite case of considering whether or not an improvement in the constants is possible (cf. H. Frazer [7]). ; © 1955 Springer. (Received September 3, 1954.) This research was partially supported by the United States Air Force, through the Office of Scientific Research of the Air Research and Development Command. It was performed in part under a contract between the American University and the National Bureau of Standards.