Inequalities between the arithmetic functions φ, ψ and σ. Part 2

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
Full paper (PDF, 161 Kb)

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

1. Atanassov, K. (2011). Note on φ, ψ and σ-functions. Part 3. Notes on Number Theory and Discrete Mathematics, 17 (3), 13–14.
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3. Mitrinovich, D. (1970). Analytic Inequalities. Springer Verlag, Berlin.
4. 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.
5. Sándor, J. (2018). Theory of means and their inequalities (online book)
6. Sándor, J. (2014). On certain inequalities for σ, φ, ψ and related functions. Notes on Number Theory and Discrete Mathematics, 20 (2), 52–60.