Coefficient Problems for a Ma-Minda Convex Class Associated with the Normalized Arcsine Mapping

Abstract

Let $\mathcal{C}_{\arcsin}$ denote the Ma--Minda subclass of convex functions generated by the normalized arcsine mapping $φ(z)=1+\frac{2}π\arcsin z.$ For this family, we develop a unified coefficient analysis based on subordination techniques, Carathéodory functions and sharp estimates for Schwarz functions. As consequences, we derive sharp estimates for the initial Taylor coefficients, logarithmic coefficients and certain differences involving the logarithmic and inverse logarithmic coefficients. We further determine the exact bounds for the second Hankel determinant $H_{2,2}(f)$ together with the Hankel determinants $H_{2,1}(F_f/2)$ and $H_{2,1}(F_{f^{-1}}/2)$ associated with the logarithmic coefficients of a function and its inverse. Moreover, sharp estimates are obtained for the initial generalized Zalcman functional and the generalized Fekete--Szegö functional. In every case, the corresponding extremal functions are identified, showing that all of the obtained inequalities are best possible.