Coefficient estimates and Hankel determinant for the class of leaf shaped starlike function

RESUMO

In this paper, we study a Ma--Minda type subclass of starlike functions associated with the normalized arcsine function $\varphi(z)=1+\frac{2}{\pi}\arcsin z.$ Using the Carathéodory function approach together with sharp coefficient estimates for Schwarz functions, we obtain explicit sharp bounds for the initial Taylor coefficients and logarithmic coefficients of functions in this class. We also derive a sharp estimate for the second Hankel determinant \(H_{2,2}(f)\), as well as for 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. In addition, we establish a sharp bound for a difference involving logarithmic coefficients. The extremal functions for all the main results are identified, confirming the sharpness of the estimates.