Sharp bounds for higher-order logarithmic coefficients in the class S

Determine the sharp bounds for the logarithmic coefficients γ_n of normalized univalent functions on the unit disk (the class S), for all n ≥ 3, where the logarithmic coefficients are defined by F_f(z) = log(f(z)/z) = 2∑_{n=1}^∞ γ_n(f) z^n for functions f(z) = z + ∑_{k=2}^∞ a_k z^k in S.

Background

The paper defines the logarithmic coefficients γn via the expansion F_f(z) = log(f(z)/z) = 2∑{n=1}^∞ γ_n(f) z^n for f in the class S of normalized univalent functions. While sharp bounds are known for γ_1 and γ_2 (|γ_1| ≤ 1 and |γ_2| ≤ 1/2 + 1/e), no sharp bounds are known for higher indices.

The authors emphasize the central role of logarithmic coefficients in geometric function theory and their connection to major results such as Milin’s conjecture and the Bieberbach conjecture, highlighting the importance of establishing sharp bounds for γ_n with n ≥ 3.