Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THE k-GOLDEN MEAN OF TWO POSITIVE NUMBERS AND ITS APPLICATIONS

  • Choi, Jin Ho (Department of Mathematics Kyungpook National University) ;
  • Kim, Young Ho (Department of Mathematics Kyungpook National University)
  • Received : 2018.04.19
  • Accepted : 2018.07.05
  • Published : 2019.03.31
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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 [2] for which the matrix golden mean.

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E1BMAX_2019_v56n2_521_f0001.png 이미지

FIGURE 1. A square inscribed in a semi-ellipse and k-golden rectangle

E1BMAX_2019_v56n2_521_f0002.png 이미지

FIGURE 2. Approximation of k-golden spirals

References

  1. B. C. Carlson, The logarithmic mean, Amer. Math. Monthly 79 (1972), 615-618. https://doi.org/10.1080/00029890.1972.11993095
  2. Y. Lim, The matrix golden mean and its applications to Riccati matrix equations, SIAM J. Matrix Anal. Appl. 29 (2006/07), no. 1, 54-66. https://doi.org/10.1137/050645026
  3. M. Livio, The Golden Ratio, Broadway Books, New York, 2002.
  4. P. Pesic, Music and the Making of Modern Science, London, MIT Press, 2014.