3-variable additive $\rho$-functional inequalities and equations

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In this paper, we introduce and investigate additive $\rho$-functional inequalities associated with the following additive functional equations \begin{aligned}f(x+y+z) - f(x)-f(y)-f(z) &= & 0, 3f\left(\frac{x+y+z}{3}\right) - f(x)-f(y)-f(z) &= & 0.\end{aligned} Furthermore, we prove the Hyers-Ulam stability of the additive $\rho$-functional inequalities in complex Banach spaces and prove the Hyers-Ulam stability of additive $\rho$-functional equations associated with the additive $\rho$-functional inequalities in complex Banach spaces.