Abstract
In this article, we show two fundamental features of the restriction of Möbius operations to the real numbers, that is, a Cauchy type inequality and a criterion for convergence of series.
Highlights
Introduction and preliminariesMöbius addition is defined on the complex open unit disk D = {z ∈ C; |z| < 1} by a ⊕ b = a + b (a, b ∈ D), 1 + ab which appears in a wide variety of fields of mathematics
We show two fundamental features of the restriction of Möbius operations to the real numbers, that is, a Cauchy type inequality and a criterion for convergence of series
We show an inequality of Cauchy type for Möbius operations
Summary
We show two fundamental features of the restriction of Möbius operations to the real numbers, that is, a Cauchy type inequality and a criterion for convergence of series. The addition ⊕s and scalar multiplication ⊗s on the open interval (–s, s) in the real line are defined by the equations a On the interval (–s, s), ⊕s is commutative, associative, and the operations ⊕s, ⊗s together with the ordinary addition and multiplication have the following properties: (r1r2) ⊗s a = r1 ⊗s (r2 ⊗s a), (r1 + r2) ⊗s a = r1 ⊗s a ⊕s r2 ⊗s a, r ⊗s (a ⊕s b) = r ⊗s a ⊕s r ⊗s b for any –s < a, b < s, r1, r2, r ∈ R.
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