Coefficients estimates for starlike functions of order \(\alpha\) and type \(\beta\) (Q1061891)

Let \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) be analytic in the unit disc E and let \[ | ((zf'/f)-1)| \{2\beta ((zf'/f)-\alpha)- ((zf'/f)-1)\}| <1 \] for some \(\alpha\),\(\beta\) \((0\leq \alpha <1\), \(0<\beta \leq 1)\) and \(z\in E\). The class of such functions f, denoted by \(S^*(\alpha,\beta)\) was introduced by Juneja and Mogra. For \(f\in S^*(\alpha,\beta)\), the sharp coefficient estimates for functions of the form \(f(z)^ t\), t being a positive integer, are determined. These results generalize earlier results due to MacGregor, Boyd, Mogra and Juneja and others.