József Sándor and Karol Gryszka
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 361–369
DOI: 10.7546/nntdm.2025.31.2.361-369
Full paper (PDF, 193 Kb)
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Authors and affiliations
József Sándor 
Department of Mathematics, Babeș-Bolyai University
400084 Cluj-Napoca, Romania
Karol Gryszka
Institute of Mathematics, University of the National Education Commission, Krakow
30-084 Krakow, Poland
Abstract
We obtain generalizations of certain results from [2] and [4]. The unitary variants are also considered. Some new arithmetic functions and their inequalities are also considered.
Keywords
- Arithmetic functions
- Inequalities
2020 Mathematics Subject Classification
- 11A25
- 26D15
References
- Dimitrov, S. (2023). Lower bounds on expressions dependent on functions φ(n), ψ(n), and σ(n). Notes on Number Theory and Discrete Mathematics, 29(4), 713–716.
- Dimitrov, S. (2024). Lower bounds on expressions dependent on functions φ(n), ψ(n), and σ(n), II. Notes on Number Theory and Discrete Mathematics, 30(3), 547–556.
- Sándor, J. (2014). On certain inequalities for σ, φ, ψ and related functions. Notes on Number Theory and Discrete Mathematics, 20(2), 52–60.
- Sándor, J. (2024). On certain inequalities for σ, φ, ψ and related functions, II. Notes on Number Theory and Discrete Mathematics, 20(3), 575–579.
- Sándor, J., & Cristici, B. (2004). Handbook of Number Theory II, Springer.
Manuscript history
- Received: 10 December 2024
- Revised: 3 June 2025
- Accepted: 4 June 2025
- Online First: 9 June 2025
Copyright information
Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Related papers
- Sándor, J. (2024). On certain inequalities for σ, φ, ψ and related functions, II. Notes on Number Theory and Discrete Mathematics, 20(3), 575–579.
- Dimitrov, S. (2024). Lower bounds on expressions dependent on functions φ(n), ψ(n), and σ(n), II. Notes on Number Theory and Discrete Mathematics, 30(3), 547–556.
- Dimitrov, S. (2023). Lower bounds on expressions dependent on functions φ(n), ψ(n), and σ(n). Notes on Number Theory and Discrete Mathematics, 29(4), 713–716.
- Sándor, J. (2014). On certain inequalities for σ, φ, ψ and related functions. Notes on Number Theory and Discrete Mathematics, 20(2), 52–60.
Cite this paper
Sándor, J., & Gryszka, K. (2025). On certain inequalities for φ, ψ, σ and related functions, III. Notes on Number Theory and Discrete Mathematics, 31(2), 361-369, DOI: 10.7546/nntdm.2025.31.2.361-369.
