Extension of Power Mean and Logarithms Mean

Yisa Anthonio; Abimbola Abolarinwa; Abdullai Abdurasid; Kamal Rauf; Michael Adeniyi; Adeyinka Ogunsanya; Christian Iluno

doi:doi:10.11648/j.sjams.20221002.12

Extension of Power Mean and Logarithms Mean

Yisa Anthonio, Abimbola Abolarinwa, Abdullai Abdurasid, Kamal Rauf, Michael Adeniyi, Adeyinka Ogunsanya, Christian Iluno

Published in Science Journal of Applied Mathematics and Statistics (Volume 10, Issue 2)

Received: 4 February 2022 Accepted: 7 March 2022 Published: 20 April 2022

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Abstract

Logarithms are indispensable in the revision of mathematics which are basic components tools in the theory of mathematical analysis. Logarithms have playing acute fundamental role in the study of the properties of power and arithmetic means as well as inequalities of Logarithms with their bound. This paper shows the properties of logarithms mean, power mean, arithmetic mean, Harmonic mean, geometric mean and later we use Minkowski’s inequality and Hölder’s inequality to establish the modified means. In the paper, we obtained the generalization of power mean, logarithms mean, arithmetic mean, Harmonic mean and geometric mean. The methodology adopted are Minkowski’s inequality and Hölder’s inequality to establish some means of order α of two distincts. These inequalities further generalize some existing results. This research work also demonstrated the importance of the Minkowski’s inequality and Hölder’s inequality over existing arithmetic mean, Harmonic mean and geometric mean and further extend the generalization to weighted logarithms mean. Hence, this article distinguished some present results on power mean, logarithms means and acquired more robust means by engaging modified Minkowski’s inequality and Hölder’s inequality with some ordinary theorems. The modified Minkowski’s inequality on power and logarithms mean further extends the generalized weighted logarithms mean of order α of two distincts.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 10, Issue 2)
DOI	10.11648/j.sjams.20221002.12
Page(s)	22-27
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), 2024. Published by Science Publishing Group

Keywords

Extension of Logarithms Mean, Power Mean, Arithmetic Mean, Geometric Mean, Harmonic Mean, Minkowski’s Inequality and Hölder’s Inequality

References

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[2]	E. B. Leach and M. C. Sholander: Extended Mean Values; Amer. Math. Monthly 85 (1978), 84-90.
[3]	E. B. Leach and M. C. Sholander: Corrections to "Extended Mean Values"; Amer. Math. Monthly 85 (8), (2018), 656-656, DOI: 10.1080/00029890.1978.11994664.
[4]	K. B. Stolarsky: Generalizations of the Logarithmic Mean; Amer. Math. Mag. 48 (1975), 87-92.
[5]	K. B. Stolarsky: The Power and Generalized Logarithmic Mean; Amer. Math. Monthly, to appear.
[6]	Tung-Po Lin: The Power Mean and the Logarithmic Mean; Amer. Math. Monthly 81 (1974), 879-883.
[7]	H. Alzer Uber reine einparametriqe familie von mittelwerten, steingsbes. Bayer. Akad wiss mat-natirin (1987).
[8]	H. Alzer Uber reine einparametriqe familie von mittelwerten II, steingsbes. Bayer. Akad wiss mat-natirin (1988).
[9]	B. C. Carlson: The Logarithmic Mean: Amer. Math. Monthly 79 (1972), 615-618.
[10]	Christopher Olutunde Imoru: The Power Mean and the Logarithmic Mean; Intrnat. J. Mah. Math. Sci. 5, (1982), 337-343.
[11]	C. E. M. Pecarce, J. Pečarič and V. simič: On Weighted Generalized Logarithms Means; Hou J. of mathematics. 24, (1998), 459-465.
[12]	S Chakraborty, A Short Note on the Versatile Power Mean, Resonance, Vol. 12, No. 9, pp. 76–79, (2007).
[13]	S. Furuichi and H. R. Moradi, Advances in mathematical inequalities, De Gruyter, 2020.
[14]	A. El Farissi, Z. Latreuch and B. Balaidi. Hadamard type inequalities for near convex functions, Gazeta Matematica Seria A, No. 1-2/2010.
[15]	Y. M. Chu and W. F. Xia. Inequalities for Generalized Logarithmic Means. J Inequal Appl 2009, 763252 (2010). https://doi.org/10.1155/2009/763252

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APA Style

Yisa Anthonio, Abimbola Abolarinwa, Abdullai Abdurasid, Kamal Rauf, Michael Adeniyi, et al. (2022). Extension of Power Mean and Logarithms Mean. Science Journal of Applied Mathematics and Statistics, 10(2), 22-27. https://doi.org/10.11648/j.sjams.20221002.12

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ACS Style

Yisa Anthonio; Abimbola Abolarinwa; Abdullai Abdurasid; Kamal Rauf; Michael Adeniyi, et al. Extension of Power Mean and Logarithms Mean. Sci. J. Appl. Math. Stat. 2022, 10(2), 22-27. doi: 10.11648/j.sjams.20221002.12

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AMA Style

Yisa Anthonio, Abimbola Abolarinwa, Abdullai Abdurasid, Kamal Rauf, Michael Adeniyi, et al. Extension of Power Mean and Logarithms Mean. Sci J Appl Math Stat. 2022;10(2):22-27. doi: 10.11648/j.sjams.20221002.12

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@article{10.11648/j.sjams.20221002.12,
  author = {Yisa Anthonio and Abimbola Abolarinwa and Abdullai Abdurasid and Kamal Rauf and Michael Adeniyi and Adeyinka Ogunsanya and Christian Iluno},
  title = {Extension of Power Mean and Logarithms Mean},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {10},
  number = {2},
  pages = {22-27},
  doi = {10.11648/j.sjams.20221002.12},
  url = {https://doi.org/10.11648/j.sjams.20221002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20221002.12},
  abstract = {Logarithms are indispensable in the revision of mathematics which are basic components tools in the theory of mathematical analysis. Logarithms have playing acute fundamental role in the study of the properties of power and arithmetic means as well as inequalities of Logarithms with their bound. This paper shows the properties of logarithms mean, power mean, arithmetic mean, Harmonic mean, geometric mean and later we use Minkowski’s inequality and Hölder’s inequality to establish the modified means. In the paper, we obtained the generalization of power mean, logarithms mean, arithmetic mean, Harmonic mean and geometric mean. The methodology adopted are Minkowski’s inequality and Hölder’s inequality to establish some means of order α of two distincts. These inequalities further generalize some existing results. This research work also demonstrated the importance of the Minkowski’s inequality and Hölder’s inequality over existing arithmetic mean, Harmonic mean and geometric mean and further extend the generalization to weighted logarithms mean. Hence, this article distinguished some present results on power mean, logarithms means and acquired more robust means by engaging modified Minkowski’s inequality and Hölder’s inequality with some ordinary theorems. The modified Minkowski’s inequality on power and logarithms mean further extends the generalized weighted logarithms mean of order α of two distincts.},
 year = {2022}
}

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TY  - JOUR
T1  - Extension of Power Mean and Logarithms Mean
AU  - Yisa Anthonio
AU  - Abimbola Abolarinwa
AU  - Abdullai Abdurasid
AU  - Kamal Rauf
AU  - Michael Adeniyi
AU  - Adeyinka Ogunsanya
AU  - Christian Iluno
Y1  - 2022/04/20
PY  - 2022
N1  - https://doi.org/10.11648/j.sjams.20221002.12
DO  - 10.11648/j.sjams.20221002.12
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 22
EP  - 27
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20221002.12
AB  - Logarithms are indispensable in the revision of mathematics which are basic components tools in the theory of mathematical analysis. Logarithms have playing acute fundamental role in the study of the properties of power and arithmetic means as well as inequalities of Logarithms with their bound. This paper shows the properties of logarithms mean, power mean, arithmetic mean, Harmonic mean, geometric mean and later we use Minkowski’s inequality and Hölder’s inequality to establish the modified means. In the paper, we obtained the generalization of power mean, logarithms mean, arithmetic mean, Harmonic mean and geometric mean. The methodology adopted are Minkowski’s inequality and Hölder’s inequality to establish some means of order α of two distincts. These inequalities further generalize some existing results. This research work also demonstrated the importance of the Minkowski’s inequality and Hölder’s inequality over existing arithmetic mean, Harmonic mean and geometric mean and further extend the generalization to weighted logarithms mean. Hence, this article distinguished some present results on power mean, logarithms means and acquired more robust means by engaging modified Minkowski’s inequality and Hölder’s inequality with some ordinary theorems. The modified Minkowski’s inequality on power and logarithms mean further extends the generalized weighted logarithms mean of order α of two distincts.
VL  - 10
IS  - 2
ER  -

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Author Information

Yisa Anthonio

Department of Pure and Applied Mathematics, Lagos State University of Science and Technology, Ikorodu, Nigeria
Abimbola Abolarinwa

Department of Mathematics, University of Lagos, Lagos, Nigeria
Abdullai Abdurasid

Department of Pure and Applied Mathematics, Lagos State University of Science and Technology, Ikorodu, Nigeria
Kamal Rauf

Department of Mathematics, University of Ilorin, Ilorin, Nigeria
Michael Adeniyi

Department of Pure and Applied Mathematics, Lagos State University of Science and Technology, Ikorodu, Nigeria
Adeyinka Ogunsanya

Department of Statistics, University of Ilorin, Ilorin, Nigeria
Christian Iluno

Department of Pure and Applied Mathematics, Lagos State University of Science and Technology, Ikorodu, Nigeria

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APA Style

Yisa Anthonio, Abimbola Abolarinwa, Abdullai Abdurasid, Kamal Rauf, Michael Adeniyi, et al. (2022). Extension of Power Mean and Logarithms Mean. Science Journal of Applied Mathematics and Statistics, 10(2), 22-27. https://doi.org/10.11648/j.sjams.20221002.12

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ACS Style

Yisa Anthonio; Abimbola Abolarinwa; Abdullai Abdurasid; Kamal Rauf; Michael Adeniyi, et al. Extension of Power Mean and Logarithms Mean. Sci. J. Appl. Math. Stat. 2022, 10(2), 22-27. doi: 10.11648/j.sjams.20221002.12

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AMA Style

Yisa Anthonio, Abimbola Abolarinwa, Abdullai Abdurasid, Kamal Rauf, Michael Adeniyi, et al. Extension of Power Mean and Logarithms Mean. Sci J Appl Math Stat. 2022;10(2):22-27. doi: 10.11648/j.sjams.20221002.12

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@article{10.11648/j.sjams.20221002.12,
  author = {Yisa Anthonio and Abimbola Abolarinwa and Abdullai Abdurasid and Kamal Rauf and Michael Adeniyi and Adeyinka Ogunsanya and Christian Iluno},
  title = {Extension of Power Mean and Logarithms Mean},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {10},
  number = {2},
  pages = {22-27},
  doi = {10.11648/j.sjams.20221002.12},
  url = {https://doi.org/10.11648/j.sjams.20221002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20221002.12},
  abstract = {Logarithms are indispensable in the revision of mathematics which are basic components tools in the theory of mathematical analysis. Logarithms have playing acute fundamental role in the study of the properties of power and arithmetic means as well as inequalities of Logarithms with their bound. This paper shows the properties of logarithms mean, power mean, arithmetic mean, Harmonic mean, geometric mean and later we use Minkowski’s inequality and Hölder’s inequality to establish the modified means. In the paper, we obtained the generalization of power mean, logarithms mean, arithmetic mean, Harmonic mean and geometric mean. The methodology adopted are Minkowski’s inequality and Hölder’s inequality to establish some means of order α of two distincts. These inequalities further generalize some existing results. This research work also demonstrated the importance of the Minkowski’s inequality and Hölder’s inequality over existing arithmetic mean, Harmonic mean and geometric mean and further extend the generalization to weighted logarithms mean. Hence, this article distinguished some present results on power mean, logarithms means and acquired more robust means by engaging modified Minkowski’s inequality and Hölder’s inequality with some ordinary theorems. The modified Minkowski’s inequality on power and logarithms mean further extends the generalized weighted logarithms mean of order α of two distincts.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Extension of Power Mean and Logarithms Mean
AU  - Yisa Anthonio
AU  - Abimbola Abolarinwa
AU  - Abdullai Abdurasid
AU  - Kamal Rauf
AU  - Michael Adeniyi
AU  - Adeyinka Ogunsanya
AU  - Christian Iluno
Y1  - 2022/04/20
PY  - 2022
N1  - https://doi.org/10.11648/j.sjams.20221002.12
DO  - 10.11648/j.sjams.20221002.12
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 22
EP  - 27
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20221002.12
AB  - Logarithms are indispensable in the revision of mathematics which are basic components tools in the theory of mathematical analysis. Logarithms have playing acute fundamental role in the study of the properties of power and arithmetic means as well as inequalities of Logarithms with their bound. This paper shows the properties of logarithms mean, power mean, arithmetic mean, Harmonic mean, geometric mean and later we use Minkowski’s inequality and Hölder’s inequality to establish the modified means. In the paper, we obtained the generalization of power mean, logarithms mean, arithmetic mean, Harmonic mean and geometric mean. The methodology adopted are Minkowski’s inequality and Hölder’s inequality to establish some means of order α of two distincts. These inequalities further generalize some existing results. This research work also demonstrated the importance of the Minkowski’s inequality and Hölder’s inequality over existing arithmetic mean, Harmonic mean and geometric mean and further extend the generalization to weighted logarithms mean. Hence, this article distinguished some present results on power mean, logarithms means and acquired more robust means by engaging modified Minkowski’s inequality and Hölder’s inequality with some ordinary theorems. The modified Minkowski’s inequality on power and logarithms mean further extends the generalized weighted logarithms mean of order α of two distincts.
VL  - 10
IS  - 2
ER  -

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