Liver transplantation often faces fairness challenges across subgroups defined by sensitive attributes such as age group, gender, and race/ethnicity. Machine learning models for outcome prediction can introduce additional biases. Therefore, we introduce Fairness through the Equitable Rate of Improvement in Multitask Learning (FERI) algorithm for fair predictions of graft failure risk in liver transplant patients. FERI constrains subgroup loss by balancing learning rates and preventing subgroup dominance in the training process. Our results show that FERI maintained high predictive accuracy with AUROC and AUPRC comparable to baseline models. More importantly, FERI demonstrated an ability to improve fairness without sacrificing accuracy. Specifically, for the gender, FERI reduced the demographic parity disparity by 71.74%, and for the age group, it decreased the equalized odds disparity by 40.46%. Therefore, the FERI algorithm advanced fairness-aware predictive modeling in healthcare and provides an invaluable tool for equitable healthcare systems.