${Linear preservers of circulant majorization on $M_{n

Let $M_n$ be the set of all $n\times n$ matrices with real entries. We say that $C\in M_n $ is circulant matrix majorized by $D\in M_n $ and write $C\prec_c D$, if $C=QD$ for some circulant doubly stochastic matrix $Q\in M_n $. A linear operator $T:M_n\longrightarrow M_n$ is said to be a preserver of $\prec_c$, if $TC\prec_cTD$ whenever $C\prec_c D$ for $C,D\in M_n $. The main purpose of this paper is to characterize the linear preservers of $\prec_c$ from $ M_n $ to $ M_n$.