  Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq
N^2)$ be a positive integer and $a$ be an integer with $(a,\,q)=1$. In this
paper, we shall prove that $$\sum_{\substack{n\leq N\\
(n,\,q)=1}}f(n)e({a\bar{n}\over q})\ll\sqrt{\tau(q)\over
q}N\log\log(6N)+q^{{1\over 4}+{\epsilon\over 2}}N^{1\over 2}(\log(6N))^{1\over
2}+{N\over \sqrt{\log\log(6N)}},$$ where $\bar{n}$ is the multiplicative
inverse of $n$ such that $\bar{n}n\equiv 1\,({\rm mod}\,q),\,e(x)=\exp(2\pi
ix),\,\tau(q)$ is the divisor function.
