tailieunhanh - Ellipses and similarity transformations with norm functions

In this paper, we deal with a conjecture related to the images of ellipses (resp. circles) under similarities that are the special Möbius transformations. We consider ellipses (resp. circles) corresponding to any norm function (except in the Euclidean case) on the complex plane and examine some conditions to confirm this conjecture. | Turk J Math (2018) 42: 3204 – 3210 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Ellipses and similarity transformations with norm functions Nihal Yılmaz ÖZGÜR∗, Department of Mathematics, Faculty of Arts and Sciences, Balıkesir University, Balıkesir, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we deal with a conjecture related to the images of ellipses (resp. circles) under similarities that are the special Möbius transformations. We consider ellipses (resp. circles) corresponding to any norm function (except in the Euclidean case) on the complex plane and examine some conditions to confirm this conjecture. Some illustrative examples are also given. Key words: Möbius transformation, ellipse, norm 1. Introduction Images of circles and ellipses (corresponding to the Euclidean norm or to another norm function on the complex plane C) have been studied extensively under some special transformations such as Möbius transformations or harmonic Möbius transformations (see [1–16] and the references therein). Let us consider the real linear space structure of the complex plane C. In [14], the present author proved that the image of any ellipse Er (F1 , F2 ) = {z ∈ C : ∥z − F1 ∥ + ∥z − F2 ∥ = r} corresponding to any norm function ∥.∥ (except in the Euclidean case) on C under the similarity transformation w = f (z) = αz + β ; α ̸= 0 , α, β ∈ C (which is a special Möbius transformation) is an ellipse corresponding to the same norm function or corresponding to the norm function ∥z∥ϕ = e−iϕ z , () where ϕ = arg(α). For a given norm function ∥.∥, the functions ∥z∥ϕ define new norms for every real number ϕ . Clearly, for the Euclidean norm, all of the norm functions ∥.∥ϕ are equal to the Euclidean norm. For any other norm function, we have ∥.∥kπ = ∥.∥ where k ∈ Z, but we do not know the exact values of ϕ for which .