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Coefficient estimates for a class of meromorphic bi-univalent functions. (Estimation de coefficients pour une classe de functions méromorphes bi-univalentes.) (English. French summary) Zbl 1283.30026

Let \(\Sigma\) denote the family of meromorphic functions \(g\) of the form \[ g(z)= z+b_0+\sum_{n=1}^\infty b_n z^{-n} \] that are univalent in \(\Delta = \{z:\;1 < |z|<\infty\}\). For \(0\leq \alpha <1, \lambda \geq 1\), let \(B\Sigma(\alpha;\lambda)\) be a subclass of \(\Sigma\), consisting of functions \(g\) such that \[ \mathrm{Re}\left\{(1-\lambda)\frac{g(z)}{z}+\lambda\, g'(z)\right\}>\alpha\quad \text{and}\quad \mathrm{Re}\left\{(1-\lambda)\frac{h(w)}{w}+\lambda\, h'(w)\right\} >\alpha\quad (z, w \in \Delta), \] where \(h\) is the inverse map of \(g\). The family \(B\Sigma(\alpha;\lambda)\) is called meromorphic bi-univalent class of functions. Applying Faber polynomials a coefficient problem for \(g\in \Sigma(\alpha;\lambda)\) is solved.

MSC:

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