Logarithmic Coefficient Estimates via the Caratheodory-Herglotz Moment Representation

Abstract

This paper develops a framework for estimating logarithmic coefficients of analytic functions under quadratic multiplier conditions.
By combining the Caratheodory--Herglotz representation with moment methods, sharp upper bounds are established
for the third, fourth, and fifth logarithmic coefficients across several important subclasses,including starlike, convex, and close-to-convex functions. The analysis reveals that while most multipliers
achieve their extremal values at boundary parameters, certain mixed-sign multipliers admit interior maximizers,
highlighting richer extremal behavior. A potential-theoretic interpretation is provided, showing that
the logarithmic coefficients act as moment functionals encoding boundary distortion, while the annular capacity remains invariant. Explicit extremal functions are constructed to verify the sharpness of the results,
and order-four two-sided distortion inequalities are derived. The study not only unifies classical results but also introduces new phenomena and establishes a pathway
toward higher-order and capacity-based generalizations.