Additive functional inequality in C*-algebras

In this paper, we introduce an additive functional inequality \begin{eqnarray}\label{1} \left\| 2 g\left(\frac{\lambda u+ y}{2}\right) - \lambda g(u) - g(y) \right\|\le \left\| s( g\left(\lambda u+ y\right)- \lambda g(u) - g(y) ) \right\| \end{eqnarray} for all $\lambda\in \mathbb{C}$, all unitary elements $u$ in a unital $C^*$-algebra $P$ and all $y\in P$, where $|s|<1$. Using both the direct method and the fixed point method, we establish the Hyers-Ulam stability of inequality (\ref{1}) in unital $C^*$-algebras. Furthermore, we apply these results to the study of $C^*$-algebra homomorphisms and $C^*$-algebra derivations in unital $C^*$-algebras.