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[edit]

[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