Logarithmic coefficient bounds for the class of Bazilevič functions

Abstract

If \({\mathcal {S}}\) denotes the class of all univalent functions in the open unit disk \({\mathbb {D}}:=\left\{ z\in {\mathbb {C}}:|z|<1\right\} \) with the form \(f(z)=z+\sum \nolimits _{n=2}^{\infty }a_{n}z^n\), then the logarithmic coefficients \(\gamma _{n}\) of \(f\in {\mathcal {S}}\) are defined by

$$\begin{aligned} \log \frac{f(z)}{z}=2\sum _{n=1}^{\infty }\gamma _{n}(f)z^n,\;z\in {\mathbb {D}}. \end{aligned}$$

The logarithmic coefficients were brought to the forefront by I.M. Milin in the 1960’s as a method of calculating the coefficients \(a_{n}\) for \(f\in {\mathcal {S}}\). He concerned himself with logarithmic coefficients and their role in the theory of univalent functions, while in 1965 Bazilevič also pointed out that the logarithmic coefficients are crucial in problems concerning the coefficients of univalent functions. In this paper we estimate the bounds for the logarithmic coefficients \(|\gamma _{n}(f)|\) when f belongs to the class \({\mathcal {B}}(\alpha ,\beta )\) of Bazilevič function of type \((\alpha ,\beta )\).