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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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