An inequality for multiplicative functions (Q1196910)

Let \(k\) be a positive integer and \(f\) a multiplicative function with \(0<f(p)\leq 1/k\) for all primes \(p\). The author shows that, for any squarefree integer \(n\), \[ \sum_{d\mid n} f(d)\leq(k+1)\sum_{d\mid n; d\leq n^{1/(k+1)}}f(d). \] This improves some inequalities of \textit{K. Alladi}, \textit{P. Erdős} and \textit{J. D. Vaaler} [J. Number Theory 31, 183-190 (1989; Zbl 0664.10025)].