In this paper, we solve the following additive $\rho$-functional inequalities \begin{equation}abel{e1} \|f(x+y)-f(x)-f(y)\|eq\Big\|\rho\Big(2f\Big(\frac{x+y}2\Big)-f(x)+-f(y)\Big)\Big\|, \end{equation} where $\rho$ is a fixed complex number with $|\rho|<1$, and \begin{equation}abel{e2} \Big\|2f\Big(\frac{x+y}2\Big)-f(x)-f(y)\Big\|eq\|\rho(f(x+y)-f(x)-f(y))\|, \end{equation} where $\rho$ is a fixed complex number with $|\rho|<\frac12$, and prove the Hyers--Ulam stability of the additive $\rho$-functional inequalities \eqref{e1} and \eqref{e2} in $\beta$-homogeneous complex Banach spaces and prove the Hyers--Ulam stability of additive $\rho$-functional equations associated with the additive $\rho$-functional inequalities \eqref{e1} and \eqref{e2} in $\beta$-homogeneous complex Banach spaces.