Sharp inequalities for Logarithmic Coefficients for Certain Classes of Univalent Functions

Let $\mathcal{S}$ denote the class of functions $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ that are analytic and univalent in the open unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$.

In this paper, we determine the sharp bounds of the Toeplitz determinants whose entries are the logarithmic coefficients of $f \in \mathcal{S}$.

Furthermore, we investigate the corresponding Toeplitz determinants for the logarithmic coefficients of the associated inverse functions.

These sharp bounds are established for functions belonging to several well-known subclasses of $\mathcal{S}$, namely, the classes $\mathcal{S}^*(\alpha)$ of starlike functions of order $\alpha$, $\mathcal{C}(\alpha)$ of convex functions of order $\alpha$, $\mathcal{S}^*_{\alpha}$ and $\mathcal{C}_{\alpha}$ of strongly starlike and strongly convex functions of order $\alpha$, and $\mathcal{R}(\alpha)$ of functions with bounded turning.

As special cases of our main results, we obtain the exact bounds of these determinants for the classical classes of starlike, convex, and bounded turning functions.