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

Let $\mathrm{\Phi }$ be a univalent function in $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$% , $\mathrm{\Phi }(\mathbb{D})$ is symmetric with respect to the real axis, starlike with respect to $\mathrm{\Phi }(0)=1$, and ${\mathrm{\Phi }}^{\mathrm{\prime }}(0)>0$. Let $\mathcal{C}$ denote the class of Ma-Minda convex functions. In this article, we present the bounds on $\mathrm{\mid }\mathrm{\mid }{a}_3\mathrm{\mid }-\mathrm{\mid }{a}_2\mathrm{\mid }\mathrm{\mid }$ for Taylor’s coefficients of the function $f$ in the class $\mathcal{C}(\mathrm{\Phi })$. 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 $\mathrm{\mid }{a}_3\mathrm{\mid }-\mathrm{\mid }{a}_2\mathrm{\mid }\mathrm{\mid }$ and $\mathrm{\mid }{A}_3\mathrm{\mid }-\mathrm{\mid }{A}_2\mathrm{\mid }\mathrm{\mid }$ are invariant, where $A_2$ and $A_3$ are the first two coefficients of the Taylor series of the inverse functions of $f\in \mathcal{C}(\mathrm{\Phi })\mathrm{.}$ Thus provides examples of invariance and nonvariance among the subclasses of convex functions.