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Partial sums of starlike and convex functions. (English) Zbl 0894.30010
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.