In this paper, we approximate the following additive functional inequality \[ \left\|(\sum\limits_{i=1}^{d+1}f(x_{1i}),\cdots,\sum\limits_{i=1}^{d+1}f(x_{ki}))\right\|_k\leq\left\|\left(mf\left(\frac{\sum^{d+1}_{i=1}x_{1i}}m\right),\cdots,mf\left(\frac{\sum^{d+1}_{i=1}x_{ki}}m\right)\right)\right\|_k \] for all $x_{11},\cdots.x_{kd+1}\in\Cal X$. We investigate homomorphisms in proper multi-$CQ^*$-algebras and derivations on proper multi-$CQ^*$-algebras associated with the above additive functional inequality.