On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

We consider the inverse function \(z=g(w)\) of a (normalized) starlike function \(w=f(z)\) of order \(\alpha\) on the unit disk of the complex plane with \(0<\alpha<1.\) Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.