On some starlikeness conditions for analytic functions (Q1202801)

Let \(A(p)\) denote the class of functions \(f(z)=z^ p+\sum_{n=p+1}^ \infty a_ n z^ n\) which are analytic in the open unit disc \(E\) of the complex plane. \(f(z)\in A(p)\) is called \(p\)-valently starlike if \[ \text{Re}(zf'(z)/f(z))>0 \quad\text{in\;} E. \] In this paper the authors obtain results of the following type. If \(p\geq 2\) and \(f(z)\in A(p)\) satisfies \[ | \arg f^{(p)}(z)| < (3/4)\pi \] and \(f^{(p- 1)}(z)/z\) is typically real then \(f\) is \(p\)-valently starlike.