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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).

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 30C25 Covering theorems in conformal mapping theory
##### Keywords:
concave univalent function; domain of variability; residuum