# Convex function (of a complex variable)

A regular univalent function

in the unit disc mapping the unit disc onto some convex domain. A regular univalent function is a convex function if and only if the tangent to the image of , , at the point rotates only in one direction as the circle is traversed. The following inequality expresses a necessary and sufficient condition for convexity of :

 (1)

On the other hand, is a convex function if and only if it can be parametrically expressed as follows:

 (2)

where is a non-decreasing real-valued function on such that

and are complex constants, . Formula (2) can be regarded as a generalization of the Christoffel–Schwarz formula for mappings of the disc onto convex polygons.

Let be the class of all convex functions in normalized by the conditions , ; let , be the subclasses of consisting of functions that map onto convex domains of the -plane with a -fold symmetry of rotation about the point , . The classes are compact with respect to uniform convergence on compact sets inside . Their integral representations, in particular formula (2) for , make it possible to develop variational methods for the solution of extremal problems in the classes [2], [3], [4], [5].

Fundamental extremal properties of may be described by the following sharp inequalities:

The argument of the function is understood to mean the branch that vanishes if . In all these estimates the equality sign holds for the function , , only. Sharp bounds are also available for the ratio between the curvature of the boundary of the domain on the class , at the point and the curvature of the pre-image of , i.e. the circle at the point . The disc belongs to the domains , , and the radius of this circle cannot be increased without imposing additional restrictions on the class of functions. If , the univalent function will be star-shaped in , i.e. will map onto a domain that is star-shaped with respect to the coordinate origin.

Examples of generalizations and modifications of the class and its subclasses include: the class of functions univalent in , regular for and mapping onto a domain with a convex complement; the class of functions regular in the annulus and normalized in a certain manner, each one of them mapping this annulus univalently onto a domain such that the finite component of its complement is convex and its union with this component is convex as well; and the class of functions in with real coefficients in the Taylor series in a neighbourhood of the point . The concept of a convex function can be extended to multi-valent functions (cf. [2], Appendix).

Of independent interest is the following generalization of a convex function [6]: A function regular in the disc is called close-to-convex if there exists a convex function , , on such that, everywhere in ,

It has been proved that all functions in this class are univalent, and necessary and sufficient conditions for a function to belong to have been found. The parametric representation of functions with the aid of Stieltjes integrals is

where and are non-decreasing real-valued functions with

The class includes convex, star-shaped and other functions. The Bieberbach conjecture, , is valid for functions . The following sharp estimates are known:

The argument of a function is understood to mean the branch that vanishes if . In all these estimates the equality sign holds for the function , , only. Geometrically, functions of class K are characterized by the fact that they map the disc onto domains whose exterior can be filled by rays drawn from points on the boundary of the domain, . The concept of a close-to-convex function has been extended to multi-valent functions [7].

#### References

 [1] I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , 1–3 , Teubner (1958–1959) (Translated from Russian) [2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) [3] V.A. Zmorovich, "On some variational problems in the theory of univalent functions" Ukrain. Math. Zh. , 4 : 3 (1952) pp. 276–298 (In Russian) [4] I.A. Aleksandrov, V.V. Chernikov, "Extremal properties of star-shaped mappings" Sibirsk. Mat. Zh. , 4 : 2 (1963) pp. 261–267 (In Russian) [5] V.A. Zmorovich, "On certain classes of analytic functions, univalent in an annulus" Mat. Sb. , 32 (74) : 3 (1953) pp. 633–652 (In Russian) [6] W. Kaplan, "Close-to-convex schlicht functions" Michigan Math. J. , 1 (1952) pp. 169–185 [7] D. Styer, "Close-to-convex multivalued functions with respect to weakly starlike functions" Trans. Amer. Math. Soc. , 169 (1972) pp. 105–112