Partial sums of starlike and convex functions
Let \(S\) be the class of functions \(f:f(z)= z+\sum^\infty_{k=2} a_kz^k\) that are analytic in the unit disc \(E\). Let \(S^*(\alpha)\) and \(K(\alpha)\), respectively, be the subclasses of \(S\) which consist of starlike and convex functions of order \(\alpha, 0\leq\alpha <1\). A sufficient condition for \(f\) to be in \(S^*(\alpha)\) is that \(\sum^\infty_{k=2} (k-\alpha) | a_k |\leq 1-\alpha\) and to be in \(K(\alpha)\) is that \(\sum^\infty_{k=2} K(k-\alpha) | a_k|\leq 1- \alpha.\) In this paper, the ratio of a function \(f\) to its sequence of partial sums \(f_n(z)= z+ \sum^n_{k=2} a_kz^k\) is studied when the coefficients of \(f\) are sufficiently small to satisfy one of these conditions. Sharp lower bounds for \(\text{Re} \left\{{f(z) \over f_n(z)}\right\}\), \(\text{Re} \left\{{f_n(z) \over f(z)}\right\}\), \(\text{Re} \left\{{f'(z) \over f_n'(z)}\right\}\) and \(\text{Re} \left\{ {f_n'(z) \over f'(z)}\right\}\) are determined when the coefficients \(\{a_k\}\) are small.
scientific article; zbMATH DE number 1031564