Bull. Korean Math. Soc. 2019; 56(2): 521-533
Online first article September 7, 2018 Printed March 1, 2019
https://doi.org/10.4134/BKMS.b180385
Copyright © The Korean Mathematical Society.
Jin Ho Choi, Young Ho Kim
Kyungpook National University; Kyungpook National University
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
Supported by: The second named author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIP) (2016R1A2B1006974)
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