The goal of open relation extraction (OpenRE) is to develop an RE model that can generalize to new relations not encountered during training. Existing studies primarily formulate OpenRE as a clustering task. They first cluster all test instances based on the similarity between the instances, and then manually assign a new relation to each cluster. However, their reliance on human annotation limits their practicality. In this paper, we propose an OpenRE framework based on large language models (LLMs), which directly predicts new relations for test instances by leveraging their strong language understanding and generation abilities, without human intervention. Specifically, our framework consists of two core components: (1) a relation discoverer (RD), designed to predict new relations for test instances based on demonstrations formed by training instances with known relations; and (2) a relation predictor (RP), used to select the most likely relation for a test instance from n candidate relations, guided by demonstrations composed of their instances. To enhance the ability of our framework to predict new relations, we design a self-correcting inference strategy composed of three stages: relation discovery, relation denoising, and relation prediction. In the first stage, we use RD to preliminarily predict new relations for all test instances. Next, we apply RP to select some high-reliability test instances for each new relation from the prediction results of RD through a cross-validation method. During the third stage, we employ RP to re-predict the relations of all test instances based on the demonstrations constructed from these reliable test instances. Extensive experiments on three OpenRE datasets demonstrate the effectiveness of our framework. We release our code at https://github.com/XMUDeepLIT/LLM-OREF.git.