This paper tackles the problem of the worst-class error rate, instead of the standard error rate averaged over all classes. For example, a three-class classification task with class-wise error rates of 10%, 10%, and 40% has a worst-class error rate of 40%, whereas the average is 20% under the class-balanced condition. The worst-class error is important in many applications. For example, in a medical image classification task, it would not be acceptable for the malignant tumor class to have a 40% error rate, while the benign and healthy classes have a 10% error rates. To avoid overfitting in worst-class error minimization using Deep Neural Networks (DNNs), we design a problem formulation for bounding the worst-class error instead of achieving zero worst-class error. Moreover, to correctly bound the worst-class error, we propose a boosting approach which ensembles DNNs. We give training and generalization worst-class-error bound. Experimental results show that the algorithm lowers worst-class test error rates while avoiding overfitting to the training set. This code is available at https://github.com/saito-yuya/Bounding-the-Worst-class-error-A-Boosting-Approach.