The subject of "fairness" in artificial intelligence (AI) refers to assessing AI algorithms for potential bias based on demographic characteristics such as race and gender, and the development of algorithms to address this bias. Most applications to date have been in computer vision, although some work in healthcare has started to emerge. The use of deep learning (DL) in cardiac MR segmentation has led to impressive results in recent years, and such techniques are starting to be translated into clinical practice. However, no work has yet investigated the fairness of such models. In this work, we perform such an analysis for racial/gender groups, focusing on the problem of training data imbalance, using a nnU-Net model trained and evaluated on cine short axis cardiac MR data from the UK Biobank dataset, consisting of 5,903 subjects from 6 different racial groups. We find statistically significant differences in Dice performance between different racial groups. To reduce the racial bias, we investigated three strategies: (1) stratified batch sampling, in which batch sampling is stratified to ensure balance between racial groups; (2) fair meta-learning for segmentation, in which a DL classifier is trained to classify race and jointly optimized with the segmentation model; and (3) protected group models, in which a different segmentation model is trained for each racial group. We also compared the results to the scenario where we have a perfectly balanced database. To assess fairness we used the standard deviation (SD) and skewed error ratio (SER) of the average Dice values. Our results demonstrate that the racial bias results from the use of imbalanced training data, and that all proposed bias mitigation strategies improved fairness, with the best SD and SER resulting from the use of protected group models.