Multiplicative functions with small increments. III (Q1187305)

[For Parts I, II cf. ibid. 55, No. 1/2, 97-101 (1990; Zbl 0709.11050); 56, No. 1/2, 159-164 (1990; Zbl 0722.11045).] Let \(f\) be a multiplicative function and put \(g(n)=a_ 0f(n)+a_ 1f(n+1)+\dots +a_ kf(n+k)\). Assume that \[ \sum_{n\leq x}| g(n)|^ \alpha = O(x\rho(x)^ \alpha) \] with a slowly varying function \(\rho\). It is proved that either the same relation holds with \(f\) in the place of \(g\) or \(f\) is of the form \(f(n)=n^ su(n)\), where \(\hbox{Re }s\leq k\) and \(u\) is a periodic multiplicative function.