A note on $n$-Jordan homomorphisms

By using a variation of a theorem on $n$-Jordan homomorphisms due to Herstein, we deduce the following G. An's result: Let $A$ and $B$ be two rings where $A$ has a unit and $char(B)> n.$ If every Jordan homomorphism from $A$ into $B$ is a homomorphism (anti-homomorphism), then every $n$-Jordan homomorphism from $A$ into $B$ is an $n$-homomorphism (anti-$n$-homomorphism).