Let $M$ be the Hardy–Littlewood maximal function and let $ M_{(\alpha)}$ be the $\ell^\alpha$-maximal operator defined by $$M_{(\alpha)} \bar{f}(x)=\|{M\bar {f}\|_{\ell^\alpha}=\Bigg(\sum_{i=1}^\infty Mf_i(x)^\alpha\Bigg)^{1/\alpha},$$ where $\bar f=(f_i)_i$, $M\bar f=(Mf_i)_i$ and $\alpha >1$. The purpose of this work is to study modular inequalities of the form $$\int_{{\mathbb R}^n} P \big(\big|\overline M_{(\alpha)}f(x)\big| \big)\,dx \le \sum_j\int_{{\mathbb R}^n } Q(|f_j(x)|)\, dx,$$ where $P$ and $Q$ are modular functions. Our results apply to operators of the form $$\overline T_{(\alpha)} \bar f(x)= \Bigg(\sum_{i=1}^\infty |T_i f_i(x)|^\alpha\Bigg)^{1/\alpha},$$ where $T_i$ satisfies similar properties to $M$.