In this chapter, we introduce and solve the following additive-additive \((s,t)\) -functional inequality: \(\displaystyle \|3 g\left (\frac {x+y+z}{3}\right)-g(x)-g(y)-g(z)\| \) \(\displaystyle +\|3h\left (\frac {x+y+z}{3}\right)+ h(x-2y+z) + h(x+y-2z)-3 h(x) \| \) \(\displaystyle \le \left \|s\left (g\left (x+y+z\right) -g(x) -g(y)-g(z)\right)\right \| \) \(\displaystyle + \left \|t \left (h(x+y+z) + h(x-2y+z) + h(x+y-2z)-3 h(x) \right) \right \|, \) where s and t are fixed nonzero complex numbers with \(|s| <1\) and \( |t| <1\) . Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of ternary derivations and ternary homomorphisms in \(C^*\) -ternary algebras, associated to the additive-additive \((s,t)\) -functional inequality (1) and the following functional inequality: \(\displaystyle \| g([x,y, z])-[g(x), y,z] - [x, g(y), z] - [x,y,g(z)] \| \) \(\displaystyle +\| h([x,y,z]) - [h(x), h(y), h(z)] \| \le \varphi (x,y,z). \)