Continued Fraction Expansion of the Heinz Operator Mean

Kacem Belhroukia; Salah Salhi; Ali Kacha

doi:doi:10.11648/j.ajam.20200806.13

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Continued Fraction Expansion of the Heinz Operator Mean

Kacem Belhroukia, Salah Salhi, Ali Kacha

Published in American Journal of Applied Mathematics (Volume 8, Issue 6)

Received: 17 July 2020 Accepted: 9 September 2020 Published: 6 December 2020

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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.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 6)
DOI	10.11648/j.ajam.20200806.13
Page(s)	311-318
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Continued Fraction, Positive Definite Matrix, Heinz Operator Mean

References

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[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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APA Style

Kacem Belhroukia, Salah Salhi, Ali Kacha. (2020). Continued Fraction Expansion of the Heinz Operator Mean. American Journal of Applied Mathematics, 8(6), 311-318. https://doi.org/10.11648/j.ajam.20200806.13

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ACS Style

Kacem Belhroukia; Salah Salhi; Ali Kacha. Continued Fraction Expansion of the Heinz Operator Mean. Am. J. Appl. Math. 2020, 8(6), 311-318. doi: 10.11648/j.ajam.20200806.13

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AMA Style

Kacem Belhroukia, Salah Salhi, Ali Kacha. Continued Fraction Expansion of the Heinz Operator Mean. Am J Appl Math. 2020;8(6):311-318. doi: 10.11648/j.ajam.20200806.13

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@article{10.11648/j.ajam.20200806.13,
  author = {Kacem Belhroukia and Salah Salhi and Ali Kacha},
  title = {Continued Fraction Expansion of the Heinz Operator Mean},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {6},
  pages = {311-318},
  doi = {10.11648/j.ajam.20200806.13},
  url = {https://doi.org/10.11648/j.ajam.20200806.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200806.13},
  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.},
 year = {2020}
}

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TY  - JOUR
T1  - Continued Fraction Expansion of the Heinz Operator Mean
AU  - Kacem Belhroukia
AU  - Salah Salhi
AU  - Ali Kacha
Y1  - 2020/12/06
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200806.13
DO  - 10.11648/j.ajam.20200806.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 311
EP  - 318
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200806.13
AB  - 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.
VL  - 8
IS  - 6
ER  -

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Author Information

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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APA Style

Kacem Belhroukia, Salah Salhi, Ali Kacha. (2020). Continued Fraction Expansion of the Heinz Operator Mean. American Journal of Applied Mathematics, 8(6), 311-318. https://doi.org/10.11648/j.ajam.20200806.13

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ACS Style

Kacem Belhroukia; Salah Salhi; Ali Kacha. Continued Fraction Expansion of the Heinz Operator Mean. Am. J. Appl. Math. 2020, 8(6), 311-318. doi: 10.11648/j.ajam.20200806.13

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AMA Style

Kacem Belhroukia, Salah Salhi, Ali Kacha. Continued Fraction Expansion of the Heinz Operator Mean. Am J Appl Math. 2020;8(6):311-318. doi: 10.11648/j.ajam.20200806.13

Copy | Download

@article{10.11648/j.ajam.20200806.13,
  author = {Kacem Belhroukia and Salah Salhi and Ali Kacha},
  title = {Continued Fraction Expansion of the Heinz Operator Mean},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {6},
  pages = {311-318},
  doi = {10.11648/j.ajam.20200806.13},
  url = {https://doi.org/10.11648/j.ajam.20200806.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200806.13},
  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.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Continued Fraction Expansion of the Heinz Operator Mean
AU  - Kacem Belhroukia
AU  - Salah Salhi
AU  - Ali Kacha
Y1  - 2020/12/06
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200806.13
DO  - 10.11648/j.ajam.20200806.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 311
EP  - 318
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200806.13
AB  - 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.
VL  - 8
IS  - 6
ER  -

Copy | Download