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Hyperbolic curvature and conformal mapping. (English) Zbl 0594.30008

The authors investigate the connection between curvature and the second derivative in conformal mapping. Their main result is that if f is a conformal mapping of a hyperbolic simply connected region D into itself and \(\gamma\) is a smooth curve in D, then \((*)\quad \kappa_ D(f(z),f\circ \gamma)\geq \kappa_ D(z,\gamma)\) whenever \(\kappa_ D(z,\gamma)\geq 2\). Here \(\kappa_ D(z,\gamma)\) denotes the geodesic curvature of \(\gamma\) at z relative to the hyperbolic metric \(\lambda_ D(z)| dz|\) on D. They show that (*) is equivalent to the Schiffer-Tammi inequality for conformal mappings of the unit disc B into itself which is in turn equivalent to the classical result \(| a_ 2| \leq 2\) for the class S of normalized univalent functions in B.
In addition, the authors characterize those paths \(\gamma\) in D such that \(f\circ \gamma\) is Euclidean convex for all conformal mappings f of D, and obtain a refinement in case f(D) is required to be convex. In the special case \(D=B\) they obtain a nice geometric description of such paths \(\gamma\). [In the case \(D=B\), the unit disc, the inequality (*) and its equivalence with the Schiffer-Tammi inequality was established by M. M. Haifawi, Rend, Circ. Mat. Palermo, II. Ser. 16, 57-63 (1967; Zbl 0181.083).]
Reviewer: D.Minda

MSC:

30C35 General theory of conformal mappings
53A30 Conformal differential geometry (MSC2010)

Citations:

Zbl 0181.083
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