This paper develops an external-field optimization framework for logarithmic coefficients of multiplier-defined univalent functions and establishes optimality principles with applications to geometric image analysis. Using the Herglotz representation, logarithmic coefficients are expressed as nonlinear moment functionals of probability measures on the unit circle, where multiplier coefficients act as an external field imposing admissibility constraints. A general Euler–Lagrange condition is derived, yielding a nonlinear equilibrium equation that characterizes extremal solutions. When the external-field potential admits a unique maximizer, the extremal measure collapses to a one-point distribution, leading to explicit optimal special-function solutions. The theoretical framework is applied to a dataset of citrus lesion images. After segmentation and conformal normalization of lesion regions, approximate logarithmic coefficients are computed from boundary harmonic expansions. A distortion index and harmonic separation criterion are introduced, and a coefficient separation theorem is verified numerically, demonstrating that geometric differences in lesion morphology correspond to measurable differences in logarithmic coefficient distributions. The results provide a mathematically rigorous connection between nonlinear external-field optimization, conformal special-function representations, and shape-based image descriptors. This approach offers a theoretically grounded and conformally invariant methodology for analyzing geometric irregularity in biomedical and agricultural imaging.