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Generalization of partial sums of certain analytic and univalent functions. (English) Zbl 1152.30308
Summary: Let \(\phi(z)\) be a fixed analytic and univalent function of the form \(\phi(z) = z + \sum^{\infty}_{k=2} c_k z^k\) and \(\mathcal H_{\phi}(c_k,\delta)\) be the subclass consisting of analytic and univalent functions \(f(z)\) which satisfy the inequality \(\sum^{\infty}_{k=2} c_k |a_k| < \delta\). In this paper, we study the ratio of a function of the form \(f_n(z) = z + \sum^n_{k=2} a_kz^k\) to its sequence of partial sums of the form \(f_n(z) = z + \sum^n_{k=2}a_k z^k\) where the coefficients of \(f(z)\) satisfy the above condition. Also, we determine sharp lower bounds for \(\text{Re} \{f(z)/f_n(z)\},\text{Re}\{f_n(z)/f(z)\}\) and \(\text{Re}\{f^{\prime}(z)/f^{\prime}_n(z)\}\) and \(\text{Re}\{f^{\prime}_n(z)/f^{\prime}(z)\}\).

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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