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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.

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