József Sándor

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 22, 2016, Number 4, Pages 20–24

**Full paper (PDF, 139 Kb)**

## Details

### Authors and affiliations

József Sándor

*Babeș-Bolyai University
Cluj-Napoca, Romania
*

### Abstract

We show how a logarithmic inequality from the book [1] is connected to means, and we offer new proofs, as well as refinements. We show that Karamata’s [2] and Leach–Sholander’s [3] inequality are in fact equivalent.

### Keywords

- Logarithmic function
- Logarithmic mean
- Leach–Sholander inequality

### AMS Classification

- 26D15
- 26D99

### References

- Hardy, K., & Williams, K. S. (1997) The green book of mathematics, Dover Publ., USA.
- Karamata, J. (1960) Sur quelque problemes poses par Ramanujan, J. Indian Math. Soc., 24, 343–365.
- Leach, E. B., & Sholander, M. C. (1983) Extended mean values II., J. Math. Anal. Appl., 92(2), 207–223.
- Sándor, J. (1988) Some integral inequalities, Elem. Math., 43, 177–180.
- Sándor, J. (1990) On the identric and logarithmic means, Aequationes Math., 40, 261–270.
- Sándor, J. (1991) A note on some inequalities for means, Arch. Math. (Basel), 56(5), 471–473.
- Sándor, J. (1996) On certain inequalities for menas II, J. Math. Anal. Appl., 199(2), 629–635.
- Sándor, J. (2001) On certain inequalities for means III, Arch. Math. (Basel), 76, 34–40.
- Sándor, J. (2002) On certain conjectures by Russo, Smarandache Notions J., 13(1–3), 21–22.
- Sándor, J. (2003) On the Leach–Sholander and Karamata theorems, Octogon Math. Mag., 11(2), 542–544.
- Sándor, J. (2012) On Huygens’ inequalities and the theory of means, Intern. J. Math. Math. Sci., Vol. 2012, Article ID 97490, 9 pages.
- Sándor, J. (2012) On a logarithmic inequality, Bull. Intern. Math. Virt. Inst. (Banja Luka), 2, 219–221.
- Sándor, J. (2013) New refinements of two inequalities for means, J. Math. Ineq., 7(2), 251–254.
- Sándor, J. (2014) On two new means of two variables, Notes Numb. Th. Discr. Math., 20(1), 1–9.
- Sándor, J. (2015) A basic logarithmic inequality and the logarithmic mean, Notes Number Th. Discr. Math., 21(1), 31–35.
- Sándor, J. (2015) A note on log–convexity of power means, Ann. Math. Inf., 45, 107–110.
- Sándor, J. (2016) A note on the logarithmic mean, Amer. Math. Monthly, 123(1), 112.
- Sándor, J. (2016) Applications of the Cauch–Bouniakowsky inequality in the theory of means, Adv. Stud. Contemp. Math., 26(2), 237–254.
- Sándor, J. (2016) Series expansions related to the logarithmic mean, Notes Number Th. Discr. Math., 22(2), 54–57.

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## Cite this paper

Sándor, J. (2016). On certain logarithmic inequalities. *Notes on Number Theory and Discrete Mathematics*, 22(4), 20-24.