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Hyperbolic linear invariance and hyperbolic \(k\)-convexity. (English) Zbl 0838.30022

Let \(f\) be an analytic locally univalent function that maps the unit disk \(\mathbb{D}\) into itself. The authors introduce the hyperbolically invariant quantity \(\alpha_h(f)\) defined as the supremum of \[ \Biggl|{1- |z|^2\over 2} {f''(z)\over f'(z)}- \overline z+ {(1- |z|^2)\overline{f(z)} f'(z)\over 1- |f(z)|^2}\Biggr|\Biggl/\Biggl(1- {(1- |z|^2) |f'(z)|\over 1- |f(z)|^2}\Biggr) \] over \(z\in \mathbb{D}\) and study its close connection to the hyperbolic \(k\)-convexity.
For instance, they show that always \(\alpha_h(f)\geq 1\), with equality if and only if \(f\) is hyperbolically 2-convex, that is, if \(a\) and \(b\) lie in the image domain then the lens-shaped domain between the two horocyles through \(a\) and \(b\) also lies in the image domain. If \(f\) is univalent then \(\alpha_h(f)\leq 2\). One of the sharp estimates they prove is: If \(f(z)= \beta z+\cdots\) and \(\alpha_h(f)\leq \alpha\) then the hyperbolic distortion satisfies \[ {(1- |z|^2)|f'(z)|\over 1- |f(z)|^2}\leq {\beta(1+ r)^\alpha\over \beta(1+ r)^\alpha+ (1- \beta)(1- r)^\alpha},\qquad |z|\leq r. \]

MSC:

30C99 Geometric function theory
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C25 Covering theorems in conformal mapping theory
30D45 Normal functions of one complex variable, normal families
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