Conformal Mappings, Extreme and Support Points and Slit Structures

Chromatic Well 2, July 2, 2015 | © Courtesy of Jonathan Lidbeck/Flickr.
Chromatic Well 2, July 2, 2015 | © Courtesy of Jonathan Lidbeck/Flickr.

On February 17, 2019, Ronen Peretz has published an article entitled “Extreme points and support points of conformal mappings” in Open Mathematics.

A Blog Post by Pablo Markin.

In his article, Ronen Peretz has presented results related to “extending a representation theorem on a conformal mapping that omits two values of equal modulus […] [that] constructs convex combinations with unlimited number of terms” (23), while determining a support point as a slit mapping tip and the properties of support points of polynomial subspaces. This paper, thus, represents “an extension of a result of L. Brickman and D. R. Wilken” (23).

As Peretz indicates, the emergence “of the slit structure of the image is starting to be visible by their support points. An important part of geometric function theory is the solution of extremal problems, such as coefficient problems, integral means problems, distortion problems and many other extremal problems. In order to apply functional analytic tools, it is natural to identify the extreme […] and […] support points.” (24). The author also adds that, according to “the Krein-Milman theorem, there is an extreme point of S, [a normal family and a compact subspace of the holomorphic functions on U, H(U),] among the support points associated with each linear continuous functional on H(U). Knowing properties of support points might allow restricting the search for a solution to a much smaller family of points in S, than the whole of S. This is one aspect of the importance of such results.” (23-24).

Likewise, as this paper argues, “[i]f f S is a support point of f that corresponds to the continuous linear functional L on H(U), then Γ = C−f(U) is an analytic curve, called the slit of f” (30). This article defines space H(U) as “a linear topological locally convex space. The normalized conformal mappings SH(U) is a compact topological subspace of H(U). The topology is that of uniform convergence on compact subsets” (28).

By Pablo Markin

Featured Image Credits: Chromatic Well 2, July 2, 2015 | © Courtesy of Jonathan Lidbeck/Flickr.

Cite this article as: Pablo Markin, "Conformal Mappings, Extreme and Support Points and Slit Structures," in Open Access Publishing, 15/09/2019,

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