On the coefficients of meromorphic univalent functions
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- by D. K. Thomas PDF
- Proc. Amer. Math. Soc. 47 (1975), 161-166 Request permission
Abstract:
Let $f \in \Sigma$, the class of all analytic univalent functions defined in $\gamma = \{ z:|z| > 1\}$. For $f,g \in \Sigma$ define $h$ in $\gamma$ by $h(z) = f{(z)^{1 - \alpha }}g{(z)^\alpha },0 < \alpha < 1$. If $h(z) = z + \Sigma _{n = 0}^\infty {c_n}{z^{ - n}}$, it is shown that $\Sigma _{n = 1}^\infty n|{c_n}{|^2} < \infty$. This result is used to show that if ${B_\alpha }$ denotes the class of all meromorphic Bazilevič functions of type $\alpha$ and $f \in {B_\alpha }$ with $f(z) = z + \Sigma _{n = 0}^\infty {a_n}{z^{ - n}}$, then $n{a_n} = O(1)$ as $n \to \infty$, the result being best possible.References
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Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 47 (1975), 161-166
- DOI: https://doi.org/10.1090/S0002-9939-1975-0357765-2
- MathSciNet review: 0357765