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On the coefficients of \(\mathcal{B}_1(\alpha)\) Bazilevič functions. (English) Zbl 1452.30008

Summary: Denote by \(\mathcal{A}\), the class of functions \(f\), analytic in \(\mathbb{D} =\{z:|z|<1\}\) and given by \(f(z)=z+\sum_{n=2}^\infty a_nz^n\) for \(z\in\mathbb{D}\), and by \(\mathcal{S}\) the subset of \(\mathcal{A}\) whose elements are univalent in \(\mathbb{D}\). The class \(\mathcal{B}_1(\alpha) \subset \mathcal{S}\), of Bazilevič functions is defined by \(Re\frac{zf'(z)}{f(z)} \left(\frac{f(z)}{z}\right)^\alpha >0\), for \(\alpha \geq 0\) and \(z\in\mathbb{D}\). We give sharp bounds for \(|\gamma_n|\), where \(\log \frac{f(z)}{z}=2\sum_{n=1}^\infty \gamma_nz^n\), when \(n=1,2,3\), and \(\alpha \geq 0\), and obtain the sharp bound for \(|\gamma_4|\) when \(0\leq \alpha \leq \alpha^*\) (\(\alpha^*\approx 1.5464\)), together with another bound for \(|\gamma_4|\) when \(\alpha \geq 0\). Sharp bounds for some initial coefficients of the inverse function when \(f\in \mathcal{B}_1(\alpha)\) are also found, which augment known results.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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