On certain multivalently starlike functions (Q804743)

Let A(p) denote the class of functions \(f(z)=z^ p+\sum^{\infty}_{n=p+1}a_ nz^ n\) which are analytic in the open unit disk \(E=\{z:| z| <1\}\). A function f(z)\(\in A(p)\) is called p-valently starlike with respect to the origin if and only if \(Re\frac{zf'(z)}{f(z)}>0\) in \(E\). The author shows that if \(f(z)\in A(p)\) and satisfies the inequality \(1+Re\frac{zf''(z)}{f'(z)}<p+\) in E, then \(f(z)\) is p-valently starlike in E. The proof uses a lemma of T. Sheil- Small for functions in A(1), and generalizes a recent result of R. Singh and S. Singh.