Logarithmic coefficients for certain subclasses of close-to-convex functions

Abstract

Let $\mathcal{S}$ denote the class of functions analytic and univalent (i.e. one-to-one) in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ normalized by $f(0)=0=f^{\prime }(0)-1$. The logarithmic coefficients ${\gamma }_n$ of $f\in \mathcal{S}$ are defined by $\log \frac{f(z)}{z}=2{\sum }_{n=1}^{\infty }{\gamma }_n{z}^n.$ In the present paper, we determine the sharp upper bounds for $|{\gamma }_1|$, $|{\gamma }_2|$ and $|{\gamma }_3|$ when $f$ belongs to some familiar subclasses of close-to-convex functions.

17 pages