Some Geometric Characterizations of a Certain Class of Log-Harmonic Mappings

This article investigates structural properties of a class of log-harmonic mappings associated with starlike analytic functions in the unit disk. Beginning with a general representation of the log-harmonic mappings, we use sharp inequalities using analytic and dilatation functions to determine growth and distortion bounds for the mappings and their complex derivatives. The exact region where the log harmonic mappings of the form f(z)=zh(z)h′(z)¯, with h being starlike analytic, give the starlike image is determined. Subordination relations for logarithmic derivatives are established, connecting the mappings with the Schwarz lemma and the Carathéodory class. Furthermore, we obtain the growth estimates for the underlying starlike functions h and their derivatives, as well as accurate inequalities governing the arclength of the circle image under log-harmonic mappings. These findings contribute to the geometric function theory of log-harmonic mappings.