Monotonicity of Euclidean curvature under locally univalent functions (Q2785280)

Suppose that \(K(z,\gamma)\) denotes the Euclidean curvature of the \(C^2\langle a,b\rangle\) smooth curve \(\gamma\) and \(f\) is a function holomorphic in a neighborhood of \(\gamma\) and locally univalent. It is known [\textit{B. G. Flinn} and \textit{B. G. Osgood}, Bull. Lond. Math. Soc. 18 272-276 (1986; Zbl 0594.30008)] that if \(f(z)\) is univalent in the unit disk \(\mathbb{D}\), \(f(0) \subset D\), \(f(0)=0\), \(|f'(0) |<1\), then \(K(0,\gamma)\leq K(0,f (\gamma))\) for any curve through the origin, \(K(0,\gamma)\geq 4\). An objective of this paper is to generalize the above result for locally univalent mappings \(f\) of \(D\) into \(D\). It is shown that \(K(0,\gamma)\leq K(0,f(\gamma))\) provided \(K(0, \gamma)\geq 4\tanh {\rho(f)\over 2}\) the latter being an upper bound of the hyperbolic linear invariant norm of \(f\).