Background: Topological indices have a significant role in determining a molecule’s topology in the light of theoretical chemistry. Objectives: This study aimed to find new bounds for some eccentric-based topological indices. Methods: We used comparison theorem for integrals. Results: We found new bounds for the eccentric-based harmonic index and atom-bond connectivity index. Our results are better than previous results in the literature. Conclusion: This study has contributed to the chemical graph theory literature by giving a new method to find eccentric-based topological indices with lower and upper bounds.