On the Moduli Difference of Initial Logarithmic and Inverse Logarithmic Coefficients of Ma-Minda Convex Functions

Abstract

Let $\mathcal{C}(\Phi )$ denote a subclass of class $\mathcal{C}$ of convex functions in the unit disc $\mathbb{D}=\{z:|z|<1,z\in \mathbb{C}\}$ defined by \begin{equation*} {Re}\left \{1+ \frac{ zf^{\prime \prime}(z)}{f^{\prime}(z)}\right \} \prec Φ(z), \end{equation*} where $\Phi$ represents the Ma--Minda function in $\mathbb{D}$. For $f\in \mathcal{C}(\Phi )$, we find the sharp estimates of $||{\gamma }_2|-|{\gamma }_1||$, where ${\gamma }_1$ and ${\gamma }_2$ are the first two logarithmic coefficients of $f$. The bound is also obtained for the inverse logarithmic coefficients ${\Gamma }_1$ and ${\Gamma }_2$. We then show that, under suitable conditions, the bounds for $||{\gamma }_2|-|{\gamma }_1||$ and $||{\Gamma }_2|-|{\Gamma }_1||$ are identical, which gives the invariance result between these functionals. We also present bounds for the differences for various classes of univalent functions by choosing particular function $\Phi$. Thus our results unify several results studied in the literature.

arXiv comment15 Pages