Federated Learning (FL) enables collaborative model training without data sharing, yet participants face a fundamental challenge, e.g., simultaneously ensuring fairness across demographic groups while protecting sensitive client data. We introduce a differentially private fair FL algorithm (FedPF) that transforms this multi-objective optimization into a zero-sum game where fairness and privacy constraints compete against model utility. Our theoretical analysis reveals an inverse relationship: privacy mechanisms that protect sensitive attributes can reduce the statistical power available for detecting and correcting demographic biases under finite samples in federated settings. We further show that our theoretical bounds are consistent with a non-monotonic fairness-utility relationship, which is empirically validated by experiments where moderate fairness constraints improve generalization before excessive enforcement degrades performance. Compared with mainstream algorithms, even under strict privacy constraints, FedPF still maintains the lowest discrimination level among all tested algorithms while retaining high utility. Experimental validation demonstrates up to 42.9 % discrimination reduction across three datasets while maintaining competitive accuracy, but more importantly, reveals that achieving strong privacy and fairness simultaneously requires carefully balanced tradeoffs rather than optimizing either objective in isolation. Furthermore, hardware-level simulations demonstrate that FedPF maintains a low computational footprint, making it suitable for resource-constrained edge devices. The source code for our proposed algorithm is publicly accessible at https://github.com/szpsunkk/FedPF.