Given a multiplicative function f, we let \(S(x,f)=\sum _{n\le x}f(n)\) be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums associated to their products. More precisely, we let \(f_j\) , \(j=1,2\) be such that \(|f_j(n)|\le \tau (n)^\kappa \) for some fixed positive constant \(\kappa \) , and assume their partial sums satisfy \(\left| S(x_j,f_j)\right| \ge \eta x_j (\log x_j)^{2^\kappa -1}\) for some \(x_1, x_2\gg 1\) and \(\eta >\max _j\{(\log x_j)^{-1/100}\}\) . We then show that there exists \(x\ge \min \{x_1, x_2\}^{\xi ^2}\) such that \(\left| S(x,f_1f_2)\right| \ge \xi x (\log x)^{2^{2\kappa }-1}\) , where \(\xi =C\eta ^{1+2^{\kappa +3}}\) for some positive constant C that depends only on \(\kappa \) .