In this paper, we introduce and solve the following additive (ρ1,ρ2)-functional inequalities $$\matrix{{\left\| {f(x + y + z) - f(x) - f(y) - f(z)} \right\|} \hfill \cr {\;\;\;\;\;\; \le \left\| {{\rho _1}(f(x + z) - f(x) - f(z))} \right\| + \left\| {{\rho _2}(f(y + z) - f(y) - f(z))} \right\|,} \hfill \cr} $$ where ρ1 and ρ2 are fixed nonzero complex numbers with ∣ρ1∣ + ∣ρ2∣ < 2. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the above additive (ρ1,ρ2)-functional inequality in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of hom-derivations in C*-ternary algebras.

中文翻译：

---

C*-三元代数中的同级导数

在本文中，我们引入并解决了以下加法(ρ1,ρ2)-泛函不等式$$\matrix{{\left\| {f(x + y + z) - f(x) - f(y) - f(z)} \right\|} \hfill \cr {\;\;\;\;\;\; \le \left\| {{\rho _1}(f(x + z) - f(x) - f(z))} \right\| + \左\| {{\rho _2}(f(y + z) - f(y) - f(z))} \right\|,} \hfill \cr} $$ 其中 ρ1 和 ρ2 是固定的非零复数，∣ρ1 ∣ + ∣ρ2∣ < 2。利用不动点法和直接法，我们证明了上述加法(ρ1,ρ2)-泛函不等式在复杂Banach空间中的Hyers-Ulam稳定性。此外，我们证明了 C*-三元代数中hom-导数的Hyers-Ulam 稳定性。