Convexity radius

convexity limit, , of a function

The least upper bound of the radii of the spheres , each one of which is mapped into a convex domain; here, the function is defined on a domain of a metric space with metric and assumes values in a linear space. The convexity radius at a point with respect to some class of mappings of the domain is, by definition, the number

If is an affine mapping of a Euclidean space , , then . With respect to the class of all normalized univalent conformal mappings , , , of the unit disc of the complex plane, the convexity radius equals ; if one imposes the additional condition of convexity of the domains , i.e. for convex functions (cf. Convex function (of a complex variable)), . See also Univalent function.

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2]	I.A. Aleksandrov, "On bounds for convexity and starlikeness of functions univalent and regular in a disc" Dokl. Akad. Nauk SSSR , 116 : 6 (1957) pp. 903–905 (In Russian)
[3]	A. Marx, "Untersuchungen über schlichte Abbildungen" Math. Ann. , 107 (1932) pp. 40–67