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A hyperbolic-distance inequality for holomorphic maps. (English) Zbl 1423.30032

Let \(\mathbb{D}\) denote the open unit disk in the complex plane with the hyperbolic metric \(\rho\). The authors prove the following main result.
Theorem. Suppose that \(f\) is a holomorphic self-map of \(\mathbb{D}\) and \(a,b,z \in \mathbb{D}\), with \(a \ne b.\) Then \[ \rho(f(z),z) \le K\left(\rho(f(a),a) + \rho(f(b),b)\right), \] where \[ K=\frac{\exp(\rho(z,a)+\rho(a,b)+\rho(b,z))}{\rho(a,b)}. \]

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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References:

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