OCTOGON MATHEMATICAL MAGAZINE Vol. 17, No.1, April 2009, pp 299-301
ISSN 1222-5657, ISBN 978-973-88255-5-0, www.hetfalu.ro/octogon
299
A note on inequalities for the logarithmic function
J´ozsef S´andor36
ABSTRACT.We show that the logarithmic inequalities from [1] and [2] are equivalent with known inequalities for means
Leta, b >0 and A=A(a, b) = a+b2 , G=G(a, b) =√ ab, L=L(a, b) = lnbb−−alna (a6=b), I =I(a, b) = 1e bb/aa1/(b−a)
(a6=b), L(a, a) =I(a, a) =a
be the well-known arithmetic, geometric,logarithmic, respectivelly identric means of argumentsa andb.
In papers [1], [2] certain logarithmic inequalities are offered.
However these inequalities are well-known, since they are in fact equivalent with certain inequalities for the above considered means.
The left side of Theorem 1 of [1] states that 3 x2−1
x2+ 4x+ 1 <lnx, x >1 (1)
Putx:=pa
b in (1) (as in Corollary 1.14), wherea >1.
Then becomes
L < 2G+A
3 (2)
whereL=L(a, b),etc. This is a known inequality of P´olya and Szeg˝o (see the References from [4], or [7]); and rediscovered by B.C. Carlson.
But inequality (2) implies also (1)! Puta=x2bin inequality (2). Then
36Received: 26.01.2009
2000Mathematics Subject Classification. 26D15, 26D99
Key words and phrases. Inequalities for means of two arguments.
300 J´ozsef S´andor
reducing with b, after some easy computations, we get (1).
The right side of Theorem 1 is
lnx < x3−1
(x+ 1)
33x(x2+ 1) , x >1 (3)
Lettingx=pa
b,where a > b; we get that (3) is equivalent with L > 3AG
2A+G (4)
(and notL > 2(2A+G)3AG ,as is stated in Corollary 1.14 of [1]).
As 2A+G3AG > G,inequality (4) is stronger than L > G, but weaker than the inequality
L > √3
G2A (5)
due to Leach and Scholander ([4]). This is exactly Theorem 2 of [1], attributed to W. Janous.
To show that √3
G2A > 2A+G3AG ,one has to verify the inequality 8A3−15A2G+ 6AG2+G3 >0,or dividing withG3,and letting
A
G =t: 8t3−15t2+ 6t+ 1>0 This is true, as
(t−1)2(8t+ 1)>0
We do not enter into all inequalities presented in [1], [2] but note that the identity
A
L −1 = ln I
G (6)
in page 983 of [2] is due to H.J. Seiffert ([5]). The proof which appears here has been discovered by the author in 1993 [5]
REFERENCES
[1] Bencze, M.,New inequalities for the function lnx (1), Octogon Mathematical Magazine, Vol. 16, No. 2, October 2008, pp. 965-980.
[2] Bencze, M.,New inequality for the function lnx and its applications (2), Octogon Mathematical Magazine, Vol. 16, No. 2, October 2008, pp. 981-983.
A note on inequalities for the logarithmic function 301 [3] Carlson, B.C., The logarithmic mean, Amer. Math. Monthly 79(1972), pp. 615-618.
[4] S´andor, J., On the identric and logarithmic means,Aequationes Math.
40(1990), pp. 261-270.
[5] S´andor, J., On certain identities for means, Studia Univ. Babes-Bolyai, Math., 18(1993), No.4., pp. 7-14.
[6] S´andor, J., Some simple integral inequalities, Octogon Mathematical Magazine, Vol. 16, No. 2, October 2008, pp. 925-933.
Babes-Bolyai University of Cluj and Miercurea Ciuc, Romania
