Estimates Logarithmic Coefficient Inequalities for Certain Families of Analytic Functions

The family \(\mathcal {S}\) $\mathcal{S}$ consists of functions that are analytic and univalent in the unit disc \(\mathbb {D}\) $\mathbb{D}$ and of the form \(f(z)= z+\sum _{n=2}^\infty a_n z^n \) $f(z)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$ . By the logarithmic coefficients of f in \(\mathcal {S}\) $\mathcal{S}$ , one means the coefficients of the expansion \( \log (f(z)/z)=2\sum _{n=1}^\infty \gamma _nz^n,\, z\in {\mathbb D}.\) $log(f(z)/z)=2{\sum }_{n=1}^{\infty }{\gamma }_n{z}^n,z\in \mathbb{D}.$ I. M. Milin proposed a system of inequalities for the logarithmic coefficients of the family \(\mathcal {S}\) $\mathcal{S}$ . Among those inequalities, one that is well-known as the Milin conjecture, was the key result in proving the Bieberbach conjecture by L. de Branges in 1984. In this study, we establish the logarithmic coefficient inequalities for a general family of starlike functions which are described by a subordination relation. Then, several special cases are deduced, which include one that corrects an earlier published result.