Hunduma LegeseNamomsa Tafasa2025-08-172025-08-172024-09https://etd.aau.edu.et/handle/123456789/6915This thesis explores coe_cient estimates for close-to-starlike log-harmonic mappings, On the unit disk, a subclass of univalent log-harmonic functions is de_ned..The results contribute to the understanding of log-harmonic map-pings and propose a Log-Harmonic Coe_cient Conjecture, which parallels classical conjectures in complex analysis. This work lays the foundation for future research aimed at further exploring and validating these conjec- tures and their broader implications. Topics discussed include log-harmonic polynomials, subclasses of log-harmonic mappings, we _rst give a general un- derstanding of how to construct log-harmonic Koebe mappings. additionally, For some special subclasses of log-harmonic mappings, growth and distortion theorems are examined.en-USCoefficients EstimateLog-Harmonic MappingsThesis