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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.)
##### Keywords:
analytic functions; univalent functions; partial sums
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##### References:
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