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Partial sums of certain meromorphic functions. (English) Zbl 1054.30011
Let \(\Sigma\) be the class of functions \(f\) of the form \[ f(z)= 1/z+ \sum^\infty_{k=1} a_k z^k\tag{1} \] which are holomorphic in the set \({\mathcal D}= \{z: 0< | z|< 1\}\). Let \(\Sigma^*(\alpha)\) and \(\Sigma_k(\alpha)\), \(0\leq\alpha< 1\), be the subclasses of \(\Sigma\) consisting of all functions which are, respectively, starlike and convex of order \(\alpha\) in \({\mathcal D}\). We also denote by \(\Sigma_c(\alpha)\subset\Sigma\) the subclass of functions \(f\) which satisfies \(-\text{Re}\{z^2f'(z)\}> \alpha\), \(z\in{\mathcal U}= {\mathcal D}\cup\{0\}\). We note that every function \(f\in\Sigma_c(\alpha)\) is close-to-convex of order \(\alpha\) in \({\mathcal D}\). We also know that: a sufficient condition for a function \(f\) of the form (1) to be in \(\Sigma^*(\alpha)\) is that \[ \sum^\infty_{k=1} (k+\alpha)| a_k|\leq 1-\alpha\tag{2} \] and to be in \(\Sigma_k(\alpha)\) is that \[ \sum^\infty_{k=1} k(k+\alpha)| a_k|\leq 1-\alpha.\tag{3} \] In the present paper, motivated essentially by the work of H. Silverman [J. Math. Anal. Appl. 209, No. 1, 221–227 (1997; Zbl 0894.30010)], the authors investigated the ratio of a function \(f\) of the form (1) to its sequence of partial sums \(f_n(z)= z^{-1}+ \sum^n_{k=1} a_k z^k\) when the coefficients satisfy either condition (2) or (3). In particular, they obained lower bounds for \(\text{Re}\{f(z)/f_n(z)\}\), \(\text{Re}\{f_n(z)/f(z)\}\), \(\text{Re}\{f'(z)/f_n'(z)\}\) and \(\text{Re}\{f_n'(z)/f'(z)\}\) and also considered the partial sums of certain integral operators.

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