A basic logarithmic inequality, and the logarithmic mean

József Sándor
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 1, Pages 31–35
Full paper (PDF, 146 Kb)

Details

Authors and affiliations

József Sándor
Babeș-Bolyai University, Department of Mathematics
Str. Kogălniceanu nr. 1, 400084 Cluj-Napoca, Romania

Abstract

By using the basic logarithmic inequality ln x ≤ x − 1 we deduce integral inequalities, which particularly imply the inequalities G < L < A for the geometric, logarithmic, resp. arithmetic means.

Keywords

• Logaritmic function
• Logarithmic mean
• Means and their inequalities

AMS Classification

• 26D15
• 26D99

References

1. Carlson, B. C. (1966) Some inequalities for hypergeometric functions, Proc. Amer. Math.Soc.,17, 32–39
2. Carlson, B. C. (1972) The logarithmic mean, Amer. Math. Monthly, 79, 615–618.
3. Ostle, B., & Terwilliger, H. L. (1957) A comparison of two means, Proc. Montana Acad.Sci., 17, 69–70.
4. Sándor, J. (1990) On the identric and logarithmic means, Aeq. Math., 40, 261–270.
5. Lorenzen, G. (1994) Why means in two arguments are special, Elem. Math., 49, 32–37.