In the big data era, cloud-based machine learning as a service (MLaaS) has attracted considerable attention. However, when handling sensitive data, such as financial and medical data, a privacy issue emerges, because the cloud server can access clients' raw data. A common method of handling sensitive data in the cloud uses homomorphic encryption, which allows computation over encrypted data without decryption. Previous research usually adopted a low-degree polynomial mapping function, such as the square function, for data classification. However, this technique results in low classification accuracy. In this study, we seek to improve the classification accuracy for inference processing in a convolutional neural network (CNN) while using homomorphic encryption. We adopt an activation function that approximates Google's Swish activation function while using a fourth-order polynomial. We also adopt batch normalization to normalize the inputs for the Swish function to fit the input range to minimize the error. We implemented CNN inference labeling over homomorphic encryption using the Microsoft's Simple Encrypted Arithmetic Library for the Cheon-Kim-Kim-Song (CKKS) scheme. The experimental evaluations confirmed classification accuracies of 99.22% and 80.48% for MNIST and CIFAR-10, respectively, which entails 0.04% and 4.11% improvements, respectively, over previous methods.