Summary: We establish conditions under which the extended Hardy-Littlewood inequality \[ \int_{\mathbb{R}^N} H\left(|x|, u_1(x), \dots u_m(x)\right) dx \leq \int_{\mathbb{R}^N} H\left(|x|, u^*_1(x), \dots u^*_m(x)\right) dx, \] where each \(u_i\) is non-negative and \(u_i^*\) denotes its Schwarz symmetrization, holds. We also determine appropriate monotonicity assumptions on \(H\) such that equality occurs in the above inequality if and only if each \(u_i\) is Schwarz symmetric. We end this paper with some applications of our results in the calculus of variations and partial differential equations.