The $k$-golden mean of two positive numbers and its applications
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In this paper, 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 an 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. This is an improvement of Lim's results \cite{Lim1} for which the matrix golden mean.
Abstract
In this paper, 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 an 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. This is an improvement of Lim's results \cite{Lim1} for which the matrix golden mean.
참고문헌 (4)
- B. C. Carlson, "The logarithmic mean", Amer. Math. Monthly, 79, pp.615~618 (1972)
- Y. Lim, "The matrix golden mean and its applications to Riccati matrix equations", SIAM J. Matrix Anal. Appl, 29(1), pp.54~66 (2007)
- M. Livio, "The Golden Ratio" (2002), Broadway Books
- P. Pesic, "Music and the Making of Modern Science" (2014), MIT Press