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On bounded functions, \(\mu\)-areally mean p-sheeted. (Russian) Zbl 0586.30017

Let f(z) denote the regular or meromorphic function in the region D. Let n(w) denote the number of roots of the equation \(f(z)=\rho e^{i\theta}\) in the region D. Let \[ p(\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}n(\rho e^{i\theta})d\theta \quad and\quad W_{\mu}(R)=\int^{R}_{0}p(\rho)d(\rho^{2\mu}). \] If \(W_{\mu}(R)\leq p\) \(R^{2\mu}\), for some non-negative p and \(\mu\) and for every \(R>0\), then f(z) is named a \(\mu\)-areally mean p-sheeted function. Denote by \(\hat E\) the circle \(| z| <1\) with the finite number of concentric circular cuts and by \(E^*\)-the circle \(| z| <1\) with the finite number of radial cuts in the ring \(0<| z| <1.\)
The following theorems are the fundamental results of the present paper. Theorem 1: Let the regular and bounded function \(f(z),| f(z)| <1\), in the region \(\hat E\) have the expansion of the form f(z)\(=a_ pz^ p+..\). in a neighbourhood of the point \(z=0\). We suppose that f(z) is a \(\mu\)-areally mean p-sheeted function and has the following property: when the function f(z) maps \(\hat E,\) then the sum of the numbers of circles around \(w=0\) of the images of all the cuts, which are situated inside of the circle \(| z| =r\), is positive. Then \(| a_ p| \leq 1\) with the equality for the function \(f(z)=e^{i\alpha}z^ p\), where \(\alpha\) denotes any real number. Theorem 2: Let a regular and bounded function f(z), \(| f(z)| <1\), in the region \(E^*\), have the expansion f(z)\(=a_ pz^ p+..\). in the neighbourhood of the point \(z=0\). We suppose that f(z) is \(\mu\)-areally mean p-sheeted function. Then \(| a_ p| \geq d^ 2\), where d\(=\lim \inf_{| z| \to 1}| f(z)|\), with equality for the function \(f(z)=e^{i\alpha}z^ p\).
Reviewer: L.Mikołajczyk

MSC:

30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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