The $k$-golden mean of two positive numbers and its applications

Bull. Korean Math. Soc. Published online 2018 Sep 07

Jin Ho Choi, and Young Ho Kim
Kyungpook National University

Abstract : We define a mean of two positive numbers called the {\it $k$-golden mean}
and study some properties of it.
Especially, we show that the $2$-golden mean refines the harmonic and the geometric means.
As an application, we define the {\it $k$-golden ratio}
and give some properties of it as a generalization of the {\it golden ratio}.
Furthermore, we define the matrix $k$-golden mean of two positive-definite matrices
and give some properties of it.

Keywords : mean, arithmetic mean, geometric mean, harmonic mean, logarithmic mean, golden mean, golden ratio, golden spiral, positive definite matrix