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On estimates of functionals in some classes of functions with positive real part. (English) Zbl 0890.30009
Let $$\mathcal {P}$$ denote the well-known class of functions of the form $$p(z)=1+q_1z+\dots +q_n z^n+\dots$$ holomorphic in the unit disc $$\mathbb{D}$$ with $$\text{Re } p(z)>0$$ in $$\mathbb{D}$$. The subclasses $$\mathcal {P}(B,b,\alpha ;F)$$ and $$\mathcal {P}(B,b,\alpha )$$ of the class $$\mathcal P$$ are studied. For $$\mathcal P(B,b,\alpha ;F)$$ the set of values of the $$k$$th coefficient, $$k=1,2,\dots$$, is described in the case $$F=F_n=\bigcup _{k=1}^n F_n^k$$, $$n=1,2,\dots$$, where $$F_n^k= \{z\in \mathbb{T}: z=e^{\frac {2k\pi {i}}{n}} e^{\rho i}$$, $$-\frac {\alpha \pi }n\leq \rho \leq \frac {\alpha \pi }n\}$$, $$\mathbb{T}$$ – unit circle. In $$\mathcal P(B,b,\alpha )$$, the set of values of the $$k$$th coefficient is described, the sharp two-sided estimates of $$\text{Re }p(z)$$ and $$\text{Im } p(z)$$ in a given point $$z\in \mathbb{D}$$ are found and the non-compactness of $$\mathcal P(B,b,\alpha )$$ in the topology given by uniform convergence on compact subsets of $$\mathbb{D}$$ is proved.
##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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##### References:
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