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Partial sums of generalized class of analytic functions involving Hurwitz-Lerch zeta function. (English) Zbl 1220.30025
Summary: Let \(f_n(z) = z + \sum^n_{k=2} a_kz^k\) be the sequence of partial sums of the analytic function \(f(z) = z + \sum^{\infty}_{k=2} a_kz^k\). We determine sharp lower bounds for
\[ \operatorname{Re} \frac{f(z)}{f_n(z)}, \quad \operatorname{Re} \frac{f_n(z)}{f(z)}, \quad \operatorname{Re} \frac{f'(z)}{f'_n(z)},\quad\text{and}\quad \operatorname{Re} \frac{f'_n(z)}{f'(z)}. \]
The main result not only provides a unification of results discussed in the literature, but gives also certain new results.

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