A number of authors, notably \textit{E. Wirsing} [Math. Ann. 143, 75-102 (1961; Zbl 0104.042)] and \textit{B. V. Levin} and \textit{A. S. Fajnlejb} [Usp. Mat. Nauk 22, No.3 (135), 119-197 (1967; Zbl 0204.065)], have obtained asymptotic formulae as well as quantitative estimates for the means \((l/x)\sum_{n\leq x}f(n)\) of multiplicative functions f, for which the behavior of \(f(p^ m)\) on average is known with sufficient accuracy. The present author proves a result of this type, which generalizes some of the earlier results. Its precise formulation is too complicated as to be stated here. It should be noted, however, that the author's result does not contain the deeper and better known mean value theorems of \textit{E. Wirsing} [Acta Math. Acad. Sci. Hung. 18, 411-467 (1967; Zbl 0165.059)] and \textit{G. Halász} [ibid. 19, 365-403 (1968; Zbl 0165.058)], in which, apart from some boundedness conditions, no a priori knowledge of the behavior of \(f(p^ m)\) is assumed.