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

Abstract

Let $\mathcal{C}(Φ)$ 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 $Φ$ represents the Ma--Minda function in $\mathbb{D}$. For $f\in\mathcal{C}(Φ)$, we find the sharp estimates of $||γ_{2}|-|γ_{1}||$, where $γ_{1}$ and $γ_{2}$ are the first two logarithmic coefficients of $f$. The bound is also obtained for the inverse logarithmic coefficients $Γ_{1}$ and $Γ_{2}$. We then show that, under suitable conditions, the bounds for $||γ_{2}|-|γ_{1}||$ and $||Γ_{2}|-|Γ_{1}||$ 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 $Φ$. Thus our results unify several results studied in the literature.