On the Coefficients of Starlike Functions

Abstract

Every probability measure $\mu$ on the circle group generates a function f that is starlike univalent on the open unit disc $\Delta$. In this note the relationship between $({c_n})$, the Fourier-Stieltjes coefficients of $\mu$, and $({a_n})$, the Taylor coefficients of f, is examined. A number of theroems are presented which indicate (possibly in the presence of fairly mild restrictions) that the sequences $({c_n})$ and $(n{a_n})$ behave similarly. For example, it is shown that if $f(\Delta )$ is finite, then $({c_n})$ converges to zero if, and only if, $(n{a_n})$ converges to zero, thereby completing a result of Pommerenke.