TAILIEUCHUNG - A new characteristic of Mobius transformations by use of apollonius points of pentagons

In this paper, we give a new characterization of Mobius transformations. To this end, a new concept of “Apollonius points of pentagons” is used. | Turk J Math 28 (2004) , 299 – 305. ¨ ITAK ˙ c TUB A New Characteristic of M¨ obius Transformations by Use of Apollonius Points of Pentagons ¨ ur Serap Bulut, Nihal Yılmaz Ozg¨ Abstract In this paper, we give a new characterization of M¨ obius transformations. To this end, a new concept of “Apollonius points of pentagons” is used. Key Words: M¨ obius transformations; Apollonius points of pentagons 1. Introduction Throughout the paper, unless otherwise stated, let w = f(z) be a nonconstant meromorphic function on the complex plane C. Let us consider the following Property 1: Property 1. w = f(z) maps circles in the z-plane onto circles in the w-plane, including straight lines among circles. The well known principle of circle transformation (see [1], [3]) reads as follows: Theorem w = f(z) satisfies Property 1 iff w = f(z) is a M¨ obius transformation. In [2], Haruki and Rassias introduced the definition of the Apollonius point of a triangle, afterwards in [5], Piyapong Niamsup extended this definition to the (k, l)Apollonius point of a triangle. Then, by means of these definitions, two new invariant characteristic properties of M¨ obius transformations were obtained. We recall that the following two definitions from [2] and [5], respectively. 2000 Mathematics Subject Classification: 30C35, 32A20 299 ¨ ¨ BULUT, YILMAZ OZG UR Definition [2] Let 4ABC be an arbitrary triangle and L be a point on the complex plane. We denote by a = BC , b = CA, c = AB, x = AL, y = BL, z = CL. If ax = by = cz holds, then L is said to be an Apollonius point of 4ABC. Definition [5] Let 4ABC be an arbitrary triangle and L be a point on C. We denote by a = BC, b = CA, c = AB, x = AL, y = BL, z = CL. If ax = k(by) = l(cz) holds, where k, l > 0, then L is said to be a (k, l)-Apollonius point of 4ABC. The purpose of this paper is to give a new characterization of M¨ obius transformations. To do this, we introduce the notions of an Apollonius point and of a (λ1 , λ2 , λ3 , λ4

• Toán học
• Mobius transformations
• Apollonius points of pentagons
• Meromorphic function
• Straight lines among circles
• Invariant characteristic properties
• a real variable
• Topological properties
• Conformal capacity
• generalized derivatives
• Möbius transformation
• Norm function
• Resp
• circles
• Special Möbius transformation
• Map circles