Invariance of the Initial Coefficient Differences of Ma-Minda Convex Functions

Abstract

Let $\Phi $ be a univalent function in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$% , $\Phi (\mathbb{D})$ is symmetric with respect to the real axis, starlike with respect to $\Phi (0)=1$, and $\Phi ^{\prime }(0)>0$. Let $\mathcal{C}% (\Phi )$ denote the class of Ma-Minda convex functions. In this article, we present the bounds on $||a_{3}|-|a_{2}||$ for Taylor's coefficients of the function $f$ in the class $\mathcal{C}(\Phi )$. We also establish the same bounds for the inverse coefficients. All the bounds we study here are sharp. We also present the conditions such that the bounds on $|a_{3}|-|a_{2}||$ and $|A_{3}|-|A_{2}||$ are invariant, where $A_{2}$ and $A_{3}$ are the first two coefficients of the Taylor series of the inverse functions of $f\in \mathcal{C}(\Phi ).$ Thus provides examples of invariance and nonvariance among the subclasses of convex functions.