József Sándor and Krassimir Atanassov

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 2, Pages 30—35

DOI: 10.7546/nntdm.2019.25.2.30-35

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## Details

### Authors and affiliations

József Sándor

*Department of Mathematics, Babeș–Bolyai University
Str. Kogalniceanu 1, 400084 Cluj-Napoca, Romania
*

Krassimir T. Atanassov

*Department of Bioinformatics and Mathematical Modelling
Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences,
Acad. G. Bonchev Str., Bl. 105, Sofia-1113, Bulgaria*

*Intelligent Systems Laboratory, Prof. Asen Zlatarov University
Bourgas-8000, Bulgaria*

### Abstract

We prove inequalities related to *φ*(*n*)^{φ(n)}.*ψ*(*n*)^{ψ(n)} or *φ*(*n*)^{ψ(n)}.*ψ*(*n*)^{φ}^{(n)} and related powers, where *φ* and *ψ* denote the Euler, resp. Dedekind arithmetic functions. More general theorem for the arithmetical functions *f*, *g* and *h* is formulated and proved.

### Keywords

- Arithmetic function
- Inequality

### 2010 Mathematics Subject Classification

- 11A25
- 26D15.

### References

- Atanassov, K. (2011). Note on
*φ*,*ψ*and*σ*-functions. Part 3, Notes on Number Theory and Discrete Mathematics, 17 (3), 13–14. - Kannan, V. & Srikanth, R. (2013). Note on
*φ*and*ψ*functions, Notes on Number Theory and Discrete Mathematics, 19 (1), 19–21. - Mitrinovich, D. (1970). Analytic Inequalities. Springer Verlag, Berlin.
- Atanassov, K., & Sándor, J. (2019). Inequalities between the arithmetic functions
*φ*,*ψ*and*σ*. Part 1. Notes on Number Theory and Discrete Mathematics, 25 (1), 50–53. - Sándor, J. (2018). Theory of means and their inequalities (online book)
- Sándor, J. (2014). On certain inequalities for
*σ*,*φ*,*ψ*and related functions. Notes on Number Theory and Discrete Mathematics, 20 (2), 52–60.

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## Cite this paper

APASándor, J. & Atanassov, K. (2019). Inequalities between the Arithmetic Functions *φ*, *ψ* and *σ*. Part 2. Notes on Number Theory and Discrete Mathematics, 25(2), 30-35, doi: 10.7546/nntdm.2019.25.2.30-35.

Sándor, J. and Atanassov K. “Inequalities between the Arithmetic Functions *φ*, *ψ* and *σ*. Part 2.” Notes on Number Theory and Discrete Mathematics 25, no. 2 (2019): 30-35, doi: 10.7546/nntdm.2019.25.2.30-35.

Sándor, J. and Atanassov K. “Inequalities between the Arithmetic Functions *φ*, *ψ* and *σ*. Part 2.” Notes on Number Theory and Discrete Mathematics 25.2 (2019): 30-35. Print, doi: 10.7546/nntdm.2019.25.2.30-35.