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On the residuum of concave univalent functions. (English) Zbl 1164.30355
A function \(f\) meromorphic in the unit disk \(D\) belongs to the family \(Co(p)\) if
(i) \(f\) has a simple pole in the point \(p\in(0,1)\);
(ii) \(f(0)=0\) and \(f'(0)=1\);
(iii) \(f\) maps \(D\) conformally onto a set whose complement with respect to \(\overline{\mathbb C}\) is convex.
The main Theorem determines the exact set of variability of the residuum of \(f\in Co(p)\) at \(p\). Namely, for \(a\in\mathbb C\) there exists a function \(f\in Co(p)\) such that \(\text{res}(f(z),z=p)=a\) if and only if \[ \left|a+\frac{p^2}{1-p^4}\right|\leq\frac{p^4}{1-p^4}. \] The author describes all the functions \(f\in Co(p)\) with residuum in the boundary of the above set of variability. Earlier S. Zemyan determined the corresponding set for the class \(S_p\) of injective functions \(f\) defined by properties (i)-(ii) without the concavity requirement (iii).

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C25 Covering theorems in conformal mapping theory