The $k$-golden mean of two positive numbers and its applications
Bull. Korean Math. Soc. 2019 Vol. 56, No. 2, 521-533
Published online March 1, 2019
Jin Ho Choi, Young Ho Kim
Kyungpook National University; Kyungpook National University
Abstract : In this paper, we define a mean of two positive numbers called the $k$-$golden~mean$ 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 $k$-$golden~ratio$ and give some properties of it as an generalization of the $golden~ratio$. Furthermore, we define the matrix $k$-golden mean of two positive-definite matrices and give some properties of it. This is an improvement of Lim's results \cite{Lim1} for which the matrix golden mean.
Keywords : mean, arithmetic mean, geometric mean, harmonic mean, logarithmic mean, golden mean, golden ratio, golden spiral, positive definite matrix
MSC numbers : 15B48, 26D07, 47A64, 51M04
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