Jordan *-homomorphisms on $C^{\ast }$-algebras

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