Convex function (of a complex variable)

A regular univalent function

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c0262301.png" />

in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c0262302.png" /> mapping the unit disc onto some convex domain. A regular univalent function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c0262303.png" /> is a convex function if and only if the tangent to the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c0262304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c0262305.png" />, at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c0262306.png" /> rotates only in one direction as the circle is traversed. The following inequality expresses a necessary and sufficient condition for convexity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c0262307.png" />:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c0262308.png" />

(1)

On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c0262309.png" /> is a convex function if and only if it can be parametrically expressed as follows:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623010.png" />

(2)

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623011.png" /> is a non-decreasing real-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623012.png" /> such that

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623013.png" />

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623014.png" /> are complex constants, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623015.png" />. Formula (2) can be regarded as a generalization of the Christoffel–Schwarz formula for mappings of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623016.png" /> onto convex polygons.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623017.png" /> be the class of all convex functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623018.png" /> normalized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623020.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623022.png" /> be the subclasses of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623023.png" /> consisting of functions that map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623024.png" /> onto convex domains of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623025.png" />-plane with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623026.png" />-fold symmetry of rotation about the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623028.png" />. The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623029.png" /> are compact with respect to uniform convergence on compact sets inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623030.png" />. Their integral representations, in particular formula (2) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623031.png" />, make it possible to develop variational methods for the solution of extremal problems in the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623032.png" /> [2], [3], [4], [5].

Fundamental extremal properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623033.png" /> may be described by the following sharp inequalities:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623034.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623035.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623036.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623037.png" />

The argument of the function is understood to mean the branch that vanishes if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623038.png" />. In all these estimates the equality sign holds for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623040.png" />, only. Sharp bounds are also available for the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623041.png" /> between the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623042.png" /> of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623043.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623044.png" /> on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623046.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623047.png" /> and the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623048.png" /> of the pre-image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623049.png" />, i.e. the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623050.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623051.png" />. The disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623052.png" /> belongs to the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623054.png" />, and the radius of this circle cannot be increased without imposing additional restrictions on the class of functions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623055.png" />, the univalent function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623056.png" /> will be star-shaped in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623057.png" />, i.e. will map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623058.png" /> onto a domain that is star-shaped with respect to the coordinate origin.

Examples of generalizations and modifications of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623059.png" /> and its subclasses include: the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623060.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623061.png" /> univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623062.png" />, regular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623063.png" /> and mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623064.png" /> onto a domain with a convex complement; the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623065.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623066.png" /> regular in the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623067.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623068.png" /> of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623069.png" /> with real coefficients in the Taylor series in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623070.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623071.png" /> regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623072.png" /> is called close-to-convex if there exists a convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623074.png" />, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623075.png" /> such that, everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623076.png" />,

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623077.png" />

It has been proved that all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623078.png" /> in this class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623079.png" /> are univalent, and necessary and sufficient conditions for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623080.png" /> to belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623081.png" /> have been found. The parametric representation of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623082.png" /> with the aid of Stieltjes integrals is

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623083.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623084.png" />

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623086.png" /> are non-decreasing real-valued functions with

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623087.png" />

The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623088.png" /> includes convex, star-shaped and other functions. The Bieberbach conjecture, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623089.png" />, is valid for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623090.png" />. The following sharp estimates are known:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623091.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623092.png" />

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623093.png" />

The argument of a function is understood to mean the branch that vanishes if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623094.png" />. In all these estimates the equality sign holds for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623096.png" />, only. Geometrically, functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623097.png" /> of class K are characterized by the fact that they map the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623098.png" /> onto domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c02623099.png" /> whose exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c026230100.png" /> can be filled by rays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c026230101.png" /> drawn from points on the boundary of the domain, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c026230102.png" />. 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

Comments

With the phrase "sharp estimatesharp estimate" is meant an estimate which cannot be improved (as is usual in complex analysis).

The Bieberbach conjecture has been proved for arbitrary (normalized) univalent functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026230/c026230103.png" />, see Bieberbach conjecture and the references to it.