On certain logarithmic inequalities

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
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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

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  8. Sándor, J. (2001) On certain inequalities for means III, Arch. Math. (Basel), 76, 34–40.
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  10. Sándor, J. (2003) On the Leach–Sholander and Karamata theorems, Octogon Math. Mag., 11(2), 542–544.
  11. Sándor, J. (2012) On Huygens’ inequalities and the theory of means, Intern. J. Math. Math. Sci., Vol. 2012, Article ID 97490, 9 pages.
  12. Sándor, J. (2012) On a logarithmic inequality, Bull. Intern. Math. Virt. Inst. (Banja Luka), 2, 219–221.
  13. Sándor, J. (2013) New refinements of two inequalities for means, J. Math. Ineq., 7(2), 251–254.
  14. Sándor, J. (2014) On two new means of two variables, Notes Numb. Th. Discr. Math., 20(1), 1–9.
  15. Sándor, J. (2015) A basic logarithmic inequality and the logarithmic mean, Notes Number Th. Discr. Math., 21(1), 31–35.
  16. Sándor, J. (2015) A note on log–convexity of power means, Ann. Math. Inf., 45, 107–110.
  17. Sándor, J. (2016) A note on the logarithmic mean, Amer. Math. Monthly, 123(1), 112.
  18. Sándor, J. (2016) Applications of the Cauch–Bouniakowsky inequality in the theory of means, Adv. Stud. Contemp. Math., 26(2), 237–254.
  19. 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

    APA

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

    Chicago

    Sándor, József. “On Certain Logarithmic Inequalities.” Notes on Number Theory and Discrete Mathematics 22, no. 4 (2016): 20-24.

    MLA

    Sándor, József. “On Certain Logarithmic Inequalities.” Notes on Number Theory and Discrete Mathematics 22.4 (2016): 20-24. Print.

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