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Continued Fraction Expansion of the Heinz Operator Mean
American Journal of Applied Mathematics
Volume 8, Issue 6, December 2020, Pages: 311-318
Received: Jul. 17, 2020; Accepted: Sep. 9, 2020; Published: Dec. 6, 2020
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Authors
Kacem Belhroukia, Department of Mathematics, Science Faculty, Ibn Toufail University, Kenitra, Morocco
Salah Salhi, Department of Mathematics, Regional Center for Education and Training Professions, Errachidia, Morocco
Ali Kacha, Department of Mathematics, Science Faculty, Ibn Toufail University, Kenitra, Morocco
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Abstract
We recall that means arise in various contexts and contribute to solving many scientific problems. The aim of the present paper is to give a continued fraction expansion of the Heinz operator mean for two positive definite matrices. We note that the direct calculation of the Heinz operator mean proves difficult by the appearance of rational exponents of matrices. The main motivation of this work is to overcome these difficulties and to present a practical and efficient method for this calculation. We use the matrix continued fraction algorithm. At the end of our paper, we deduce a continued fraction representation of the symmetric operator entropy.
Keywords
Continued Fraction, Positive Definite Matrix, Heinz Operator Mean
To cite this article
Kacem Belhroukia, Salah Salhi, Ali Kacha, Continued Fraction Expansion of the Heinz Operator Mean, American Journal of Applied Mathematics. Vol. 8, No. 6, 2020, pp. 311-318. doi: 10.11648/j.ajam.20200806.13
Copyright
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
T. ANDO, Topics on operators inequalities, Ruyuku Univ. Lecteure Note Series, 1 (1978).
[2]
M. ALAKHRASS, M. SABABHEH, Matrix mixed mean inequalities. Results Math. 74, no 1 (2019), Art 213.
[3]
A. A. BURQAN, Comparisons of Heinz Operator Parameters. Malaysian Journal of Science, Vol. 38, S1, (2019), 33-42
[4]
S. S. DRAGOMIR, Some inequalities for Heinz operator mean, Mathematica Moravica, vol. 24, no 1 (2020), 71- 82.
[5]
T. H. DINH, R. DUMITRU, J. A. FRANCO, THe matrix power mean and interpolations. Adv. Oper. Theory 3 (2018), 647-654.
[6]
B. W. HELTON, Logarithms of matrices, Proc. Amer. Math. Scoc., 19 (1968), 733-738.
[7]
F. R. GANTMACHER, The Theory of matrices, Vol. I. Chelsa. New York, Elsevier Science Publiscers, (1992).
[8]
Y. KAPIL, C. CONDE, M. S. MOSLEHIAN, M. SABABHEH and M. SING. Norm inequalities related to the Heron and Heinz means. Mediterr. J. Math. 14, (2017), Art. 213.
[9]
A. N. KHOVANSKI, The applications of continued fractions and their Generalisation to problemes in approximation theory, (1963), Noordhoff, Groningen, The Netherlands.
[10]
L. LORENTZEN, H. WADELAND, Continued fractions with applications, Elsevier Science Publishers, 1992.
[11]
G. J. MURPHY, C∗-Algebras and operators theory, Chapter 2, (1990), Academic press, INC Harcourt Brace Jovanovich, publishers.
[12]
N. NEGOESCU, Convergence theorems on noncommutative continued fractions, Rev. Anal. Num´ e. Th´ erie Approx., 5 (1977), 165-180.
[13]
G. NETLLER, On trenscendental numbers whose sum, difference, quotient and product are transcendental numbers, Math. Student 41, No. 4 (1973), 339-348.
[14]
M. RAISSOULI, A. KACHA, Convergence for matrix continued fractions, Linear Algebra Appl., 320 (2000), 115-129.
[15]
M. RAISSOULI, A. KACHA, S. SALHI, The Arabian Journal for Science and Engineering, Volume 31, Number 1 A (2006), 1-15.
[16]
M. SABABHEH, Convexity and matrix means. Linear Algebra Appl. 506 (2016), 588-602.
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