On a theorem of Kobori

摘要

Let S and S∗ denote the well-known classes of normalized analytic functions which are, respectively, univalent and starlike in |z| < 1. A theorem of Kobori states that, for a function g(z)=z+∑∞k=2 akzk∈L*, the sequence of the nth sections gn(z)=z+∑∞k=21kakzkmust be starlike in |z| < 12. Recently, Silverman [1], Gruenberg et al. [2], and Rønning [3] extended the above result. In this note, we first improve the results of Ilieff [4], Ruscheweyh [5], and Silverman [1]. We then give a remarkable simple proof of a result of Gruenberg et al. [2] and Rønning {3} that S4(z) is starlike in |z| < 12 for g ϵS.