Logarithmic Coefficients of the Inverse of Univalent Functions

Abstract

Let $${\mathcal {S}}$$ be the class of analytic and univalent functions in the unit disk $$|z|<1$$ , that have a series of the form $$f(z)=z+ \sum _{n=2}^{\infty }a_nz^n$$ . Let F be the inverse of the function $$f\in {\mathcal {S}}$$ with the series expansion $$F(w)=f^{-1}(w)=w+ \sum _{n=2}^{\infty }A_nw^n$$ for $$|w|<1/4$$ . The logarithmic inverse coefficients $$\Gamma _n$$ of F are defined by the formula $$\log \left( F(w)/w\right) = 2\sum _{n=1}^{\infty }\Gamma _n(F)w^n$$ . In this paper, we first determine the sharp bound for the absolute value of $$\Gamma _n(F)$$ when f belongs to $${\mathcal {S}}$$ and for all $$n \ge 1$$ . This result motivates us to carry forward similar problems for some of its important geometric subclasses. In some cases, we have managed to solve this question completely but in some other cases it is difficult to handle for $$n\ge 4$$ . For example, in the case of convex functions f, we show that the logarithmic inverse coefficients $$\Gamma _n(F)$$ of F satisfy the inequality $$\begin{aligned} |\Gamma _n(F)|\le \frac{1}{2n} \quad \text{ for } n\ge 1,2,3 \end{aligned}$$ and the estimates are sharp for the function $$l(z)=z/(1-z)$$ . Although this cannot be true for $$n\ge 10$$ , it is not clear whether this inequality could still be true for $$4\le n\le 9$$ .