A stronger version of the Bogoliubov inequality is used to derive an upper bound for the anomalous average ¦<>x)>|s of an interacting nonrelativislic Bose fielda(x) at a finite temperature. This bound is ¦a(x)2|s pR, whereR satisfies 1 -R = (RT/2T c v/2, withv the dimensionality, andT c the critical temperature in the absence of interactions. The formation of nonzero averages is closely related to the Bose-Einstein condensation and ¦|·2 is often believed to coincide with the mean densitypa of the condensate. We have found nonrigorous arguments supporting the inequality po ¦|·2, which parallels the result of Griffiths in the case of spin systems.