Two inequalities with an application

Summary An inequality used in Brown and Cohen (1974,Ann. Statist., 2(5), 963–976) and Bhattacharya (1978,Ann. Inst. Statist. Math., A,30, 407–414) is generalized and another useful inequality derived from it. An application of the latter which provides a more elegant approach to and an improvement over a result in Shinozaki (1978,Commun. Statist. Theor. Meth., A,7, 1421–1432), is also presented.