A sharp inequality of Halász type for the mean value of a multiplicative arithmetic function

Let g(n) be a complex valued multiplicative function such that |g(n)| ≤ 1. In this paper we shall be concerned with the validity of the inequalityunder the weak condition g(p)∈ for all primes p, where is a fixed subset of the closed unit disc Thus our point of view is similar to that of Halasz [Hz 2] in that we seek a general inequality in terms of simple quantities, albeit g(p) may have a quite irregular distribution. We are not concerned here with the problem of asymptotic formulae for the sum on the left of (1) studied by (among others) Delange [D], Halasz [Hz 1] and Wirsing [W].