  In this paper, we investigate Jordan *-homomorphisms on $C^*$-algebras
associated with the following functional inequality
$\|f(\frac{b-a}{3})+f(\frac{a-3c}{3})+f(\frac{3a+3c-b}{3})\| \leq \|f(a)\|.$ We
moreover prove the superstability and the generalized Hyers-Ulam stability of
Jordan *-homomorphisms on $C^*$-algebras associated with the following
functional equation
$$f(\frac{b-a}{3})+f(\frac{a-3c}{3})+f(\frac{3a+3c-b}{3})=f(a).$$
