Group max-min fairness (MMF) is commonly used in fairness-aware recommender systems (RS) as an optimization objective, as it aims to protect marginalized item groups and ensures a fair competition platform. However, our theoretical analysis indicates that integrating MMF constraint violates the assumption of sample independence during optimization, causing the loss function to deviate from linear additivity. Such nonlinearity property introduces the Jensen gap between the model's convergence point and the optimal point if mini-batch sampling is applied. Both theoretical and empirical studies show that as the mini-batch size decreases and the group size increases, the Jensen gap will widen accordingly. Some methods using heuristic re-weighting or debiasing strategies have the potential to bridge the Jensen gap. However, they either lack theoretical guarantees or suffer from heavy computational costs. To overcome these limitations, we first theoretically demonstrate that the MMF-constrained objective can be essentially reformulated as a group-weighted optimization objective. Then we present an efficient and effective algorithm named FairDual, which utilizes a dual optimization technique to minimize the Jensen gap. Our theoretical analysis demonstrates that FairDual can achieve a sub-linear convergence rate to the globally optimal solution and the Jensen gap can be well bounded under a mini-batch sampling strategy with random shuffle. Extensive experiments conducted using six large-scale RS backbone models on three publicly available datasets demonstrate that FairDual outperforms all baselines in terms of both accuracy and fairness. Our data and codes are shared at https://github.com/XuChen0427/FairDual.