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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.
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참고문헌 (4)
참고문헌 신청
B. C. Carlson / 1972 / The logarithmic mean / Amer. Math. Monthly 79:615~618
Y. Lim / 2007 / The matrix golden mean and its applications to Riccati matrix equations / SIAM J. Matrix Anal. Appl 29(1):54~66
M. Livio / 2002 / The Golden Ratio / Broadway Books
P. Pesic / 2014 / Music and the Making of Modern Science / MIT Press
Y. Lim / 2007 / The matrix golden mean and its applications to Riccati matrix equations / SIAM J. Matrix Anal. Appl 29(1):54~66
M. Livio / 2002 / The Golden Ratio / Broadway Books
P. Pesic / 2014 / Music and the Making of Modern Science / MIT Press
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