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This paper addresses the important yet underexplored task of **multi-class sentiment analysis (MCSA)**, which remains challenging due to the subtle semantic differences between adjacent sentiment categories and the scarcity of high-quality annotated data. To tackle these challenges, we propose **RD-MCSA** (**R**ationales and **D**emonstrations-based **M**ulti-**C**lass **S**entiment **A**nalysis), an In-Context Learning (ICL) framework designed to enhance MCSA performance under limited supervision by integrating classification rationales with adaptively selected demonstrations. First, semantically grounded classification rationales are generated from a representative, class-balanced subset of annotated samples selected using a tailored balanced coreset algorithm. These rationales are then paired with demonstrations chosen through a similarity-based mechanism powered by a **multi-kernel Gaussian process (MK-GP)**, enabling large language models (LLMs) to more effectively capture fine-grained sentiment distinctions. Experiments on five benchmark datasets demonstrate that RD-MCSA consistently outperforms both supervised baselines and standard ICL methods across various evaluation metrics.