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 $f(0)=0=f'(0)-1$. The logarithmic coefficients $\gamma _n$ of $f\in \mathcal {S}$ are defined by $\log \frac{f(z)}{z}= 2\sum _{n=1}^{\infty } \gamma _n z^n.$ Let $\mathcal {F}_1 (\mathcal {F}_2 ~ \text{ and } \mathcal {F}_3~ \text{ resp. })$ denote the class of functions $f\in \mathcal {A}$ such that $\text {Re}\,(1-z)f'(z)>0~ (~ \text {Re}\,(1-z^2)f'(z)>0 \quad \text{ and } \quad \text {Re}\,(1-z+z^2)f'(z)>0 ~ \text{ resp. }) ~ \text{ in } \mathbb {D}.$ The classes $\mathcal {F}_1, \mathcal {F}_2$ and $\mathcal {F}_3$ are subclasses of the class of close-to-convex functions. In the present paper, we determine the sharp upper bound for $|\gamma _1|$, $|\gamma _2|$ and $|\gamma _3|$ for functions f in the classes $\mathcal {F}_1, \mathcal {F}_2$ and $\mathcal {F}_3$.