  Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq
N^2)$ be a prime number and $a$ be an integer with $(a,\,q)=1$, $\chi$ be a
non-principal Dirichlet character modulo $q$. In this paper, we shall prove
that $$ \sum_{n\leq N}f(n)\chi(n+a)\ll {N\over q^{1\over
4}}\log\log(6N)+q^{1\over 4}N^{1\over 2}\log(6N)+{N\over \sqrt{\log\log(6N)}}.
$$
  We shall also prove that \begin{align*} &\sum_{n\leq
N}f(n)\chi(n+a_1)\cdots\chi(n+a_t)\ll {N\over q^{1\over 4}}\log\log(6N)\\
&\quad+q^{1\over 4}N^{1\over 2}\log(6N)+{N\over \sqrt{\log\log(6N)}},
\end{align*} where $t\geq 2$, $a_1,\,\cdots,\,a_t$ are pairwise distinct
integers modulo $q$.
