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On the derivative of hyperbolically convex functions. (English) Zbl 1021.30010
A conformal mapping \(f\) of the unit disk D into itself is called hyperbolically convex if the non-euclidean segment between any two points of \(f({\mathbf D})\) also belongs to \(f({\mathbf D})\). Hyperbolically convex functions were first systematically studied by W. Ma and D. Minda [Ann. Pol. Math. 60, 1, 81-100 (1994; Zbl 0818.30010) and Ann. Pol. Math. 71, 273-285 (1999; Zbl 0927.30009)]. Among many other results they obtained the characterization \[ \text{Re} \left[1+ z \frac{ f^{\prime \prime} (z)}{ f'(z)}+ \frac{2zf'(z) \overline{f(z)} }{ 1- |f(z)|^{2}} \right]>0 , \qquad (z\in {\mathbf D}). \] In this paper the authors obtain the exact order of growth for the derivative of these functions and investigate also their boundary behaviour. Precisely the present authors in (1998, 2000), also with A. Vasilev (2001) derived a number of estimates for h-convex functions. The upper bound for the derivative remained an open problem and it was conjectured in the paper published by D. Mejía and C. Pommerenke [J. Geom. Anal. 10, 365-378 (2000; Zbl 0980.30015)] that \[ f'(z) = O \left(\frac{1}{1- |z|} \left(\log \frac{1}{1- |z|} \right)^{-2} \right) \qquad (|z|\rightarrow 1), \] this was proved with the exponent \(-1\) instead of \(-2\). The theorem 1 of the present paper have an interesting demonstration which is “very geometric”. From the second conclusion of theorem 1 follows that the upper conjecture is true. The stronger conjecture given by D. Mejia and C. Pommerenke [Rev. Colomb. Mat. 32, 29-43 (1998; Zbl 0922.30013)] as \[ a_{n}= O(n^{-1}(\log n)^{-2}) \qquad (n\rightarrow \infty) \] however remains open.

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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