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Coefficient conditions for certain univalent functions. (English) Zbl 1158.30008
Let \(\mathcal{A}\) be the class of functions that are analytic in the open unit disc \(\mathcal{U}\) normalized such that \(f(z)=z+a_2z^2+a_3z^3+\cdots.\) Also let \[ {\mathcal T}(\alpha)=\left\{f\in{\mathcal A}: \text{Re } \frac{f(z)}{z}>\alpha, z\in{\mathcal U} \right\}; \]
\[ {\mathcal U}(\alpha)=\left\{f\in{\mathcal A}: \text{Re } f'(z)>\alpha, z\in{\mathcal U} \right\}; \]
\[ {\mathcal {CC}}_\lambda(\alpha;g(z))=\left\{f\in{\mathcal A}: \text{Re } e^{i\lambda}\left(\frac{zf'(z)}{g(z)}-\alpha\right)>0, z\in{\mathcal U} \right\}; \]
\[ {\mathcal {STS}}(\mu_1,\mu_2)=\left\{f\in{\mathcal A}: \frac{\pi\mu_1}{2} <\arg \frac{zf'(z)}{f(z)}<\frac{\pi\mu_2}{2}, z\in{\mathcal U} \right\}; \] where \(0\leq\alpha<1,\) \(-\frac{\pi}{2}<\lambda\leq\frac{\pi}{2},\) \(g(z)\) is starlike and \(-1\leq\mu_1<\mu_2\leq1.\)
In this paper the authors study the above classes, give sufficient conditions, including coefficient inequalities, as well as some interesting corollaries.
MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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