Three versions of an inequality

Krassimir Atanassov
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 639–645
DOI: 10.7546/nntdm.2025.31.3.639-645
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Authors and affiliations

Krassimir 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

Abstract

Three versions of the inequality

    \[\sum_{i=1}^{n-1} \frac{1}{a_i(a_{i+1} + 1)} \geq \frac{n}{1 + \prod\limits_{i=1}^n a_i} - \frac{1}{a_{n}(a_1 + 1)}\]

are formulated and proved, where a_1, \dots, a_n > 0 are real numbers. An open problem is formulated.

Keywords

  • Arithmetic inequality
  • Combinatorics

2020 Mathematics Subject Classification

  • 11A99

References

  1. Cvetkovski, Z. (2012). Inequalities: Theorems, Techniques and Selected Problems. Springer, Heidelberg.
  2. Hardy, G., Littlewood, J., & Pólya, G. (1988). Inequalities. (2nd ed.). Cambridge University Press, Cambridge.
  3. Mitrinović, D. (1970). Analytic Inequalities. Springer, New York.
  4. Wikipedia contributors. (2025). List of inequalities. In: Wikipedia, The Free Encyclopedia. Retrieved September 17, 2025, from https://en.wikipedia.org/w/index.php?title=List_of_inequalities&oldid=1285635451 (Last edited April 14, 2025).

Manuscript history

  • Received: 30 January 2024
  • Revised: 17 September 2025
  • Accepted: 22 September 2025
  • Online First: 25 September 2025

Copyright information

Ⓒ 2025 by the Author.
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).

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Cite this paper

Atanassov, K. (2025). Three versions of an inequality. Notes on Number Theory and Discrete Mathematics, 31(3), 639-645, DOI: 10.7546/nntdm.2025.31.3.639-645.

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