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Let s* denote the maximal function associated with the rectangular partial sums $s_{mn}(x,y)$ of a given double function series with coefficients $c_{jk}$. The following generalized Hardy-Littlewood inequality is investigated: ${\left|\right|s*\left|\right|}_{p,\mu }\le C_{p,\alpha ,\beta }{{\Sigma }_{j=0}^{\infty }{\Sigma }_{k=0}^{\infty }{\left(j̅\right)}^{p-\alpha -2}{\left(k̅\right)}^{p-\beta -2}{\left|c_{jk}\right|}^p}^{1/p}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{jk}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property of $s_{mn}(x,y)$...