An improvement of sufficient conditions for starlike functions (Q1178752)

Let \(A_ p\) be the class of functions of the form \(f(z)=z^ p+\sum_{n=p+1}^ \infty a_ nz^ n\) (\(p\in\mathbb{N}\)) which are analytic in the unit disk \(U\). Let \(S^*(p)\) denote the subclass of \(A_ p\) consisting of functions which are \(p\)-valently starlike in \(U\). The authors derive certain corollaries by applying their main theorem, as for example: If \(f(z)\in A_ p\) satisfies \[ | (zf''(z)/f'(z))+1- p|<(\sqrt{2}/8)(5p+4\sqrt{p}+4) \qquad (z\in U), \] then \(f(z)\in S^*(p)\). This is (for \(p=1\)) an improvement of a result by \textit{R. Singh} and \textit{S. Singh} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 35, 145-148 (1981; Zbl 0559.30007)].