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 The $k$-golden mean of two positive numbers and its applications Bull. Korean Math. Soc. 2019 Vol. 56, No. 2, 521-533 https://doi.org/10.4134/BKMS.b180385Published 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 Downloads: Full-text PDF   Full-text HTML