Rev. Anal. Num´er. Th´eor. Approx., vol. 41 (2012) no. 2, pp. 125–129 ictp.acad.ro/jnaat
A SEPARATION OF SOME SEIFFERT-TYPE MEANS BY POWER MEANS
IULIA COSTIN∗and GHEORGHE TOADER†
Abstract. Consider the identric meanI, the logarithmic meanL,two trigono- metric means defined by H. J. Seiffert and denoted by P and T, and the hy- perbolic meanMdefined by E. Neuman and J. S´andor.There are a number of known inequalities between these means and some power meansAp.We add to these inequalities some new results obtaining the following chain of inequalities
A0<L<A1/3<P<A2/3<I<A3/3<M<A4/3<T <A5/3.
MSC 2000. 26E60.
Keywords. Seiffert type means; power means; logarithmic mean; identric mean; inequalities of means.
1. INTRODUCTION
A meanis a functionM :R2+ →R+,with the property min(a, b)≤M(a, b)≤max(a, b), ∀a, b >0.
Each mean is reflexive, that is
M(a, a) =a, ∀a >0.
This is also used as the definition of M(a, a).
A mean is symmetricif
M(b, a) =M(a, b), ∀a, b >0;
it is homogeneous (of degree 1) if
M(ta, tb) =t·M(a, b), ∀a, b, t >0.
We shall refer here to the following symmetric and homogeneous means:
- the power means Ap, defined by Ap(a, b) =ap+bp
2
1p
, p6= 0;
∗Department of Computer Science, Technical University of Cluj-Napoca, Baritiu st. no.
28, Cluj-Napoca, Romania, e-mail: [email protected].
†Department of Mathematics, Technical University of Cluj-Napoca, Baritiu st. no. 25,
- the geometric mean G, defined as G(a, b) = √
ab, but verifying also the property
p→0limAp(a, b) =A0(a, b) =G(a, b);
- the identric meanI defined by I(a, b) = 1e aa
bb
a−b1
, a6=b;
- the Gini meanS defined by
S(a, b) =
aabba+b1
; - the first Seiffert mean P, defined in [9] by
P(a, b) = a−b
2 sin−1a−b a+b
, a6=b;
- the second Seiffert mean T, defined in [10] by T(a, b) = a−b
2 tan−1a−b a+b
, a6=b;
- the Neuman-S´andor meanM, defined in [6] by M(a, b) = a−b
2 sinh−1a−b a+b
, a6=b;
- the logarithmic meanL defined by
L(a, b) = lna−ba−lnb, a6=b.
As remarked B.C. Carlson in [1], the logarithmic mean can be represented also by
L(a, b) = a−b
2 tanh−1a−b a+b
, a6=b, thus the last four means are very similar.
Being rather complicated, these means were evaluated by simpler means, first of all by power means. For two means M and N we write M < N if M(a, b)< N(a, b) for a6=b.It is known that the family of power means is an increasing family of means, thus
Ap <Aq ifp < q.
The evaluationof a given meanM by power means assumes the determina- tion of some real indices p and q such that Ap < M <Aq. The evaluation is optimalifpis the the greatest andq is the smallest index with this property.
This means that M cannot be compared with Ar ifp < r < q.
Optimal evaluation were given for the logarithmic mean in [5]
A0 <L<A1/3, for the identric mean in [8]
A2/3 <I <Aln 2,
and for the first Seiffert mean in [3]
Aln 2/lnπ <P <A2/3. Following evaluations are also known:
A1/3 <P <A2/3, proved in [4],
A1<T <A2, given in [10],
A1 <M<T, as it was shown in [6] and
S >A2 as it is proved in [7]. In [2] it is proven that
(1) M<A3/2 <T
and using some of the above results, it is obtained the following chain of inequalities
A0 <L<A1/2 <P <A1<M<A3/2 <T <A2. Here we retain another chain of inequalities
(2) A0 <L<A1/3<P <A2/3 <I <A1 <M<T <A2 <S.
Our aim is to prove that A4/3 can be put between M and T and A2 can be replaced by A5/3.We obtain so another nice separation of these means by
“equidistant” power means.
2. MAIN RESULTS
We add to the inequalities (2) the next results.
Theorem 1. The following inequalities
M<A4/3<T <A5/3 hold.
Proof. As the means are symmetric and homogenous, for the first inequality
a−b 2 sinh−1a−b
a+b
<
a4/3+b4/3 2
34
, a6=b,
we can assume thata > b and denote a/b=t3 >1.The inequality becomes
t3−1 2 sinh−1t3−1
t3+1
<
t4+1 2
34
, t >1, or
2 3 4(t3−1)
3 <sinh−1tt33−1+1, t >1.
Denoting
f(t) = sinh−1 tt33−1+1−2−14 t3−1
t4+ 1−3 4
we have to prove thatf(t)>0 for t >1.As f(1) = 0,we want to prove that f0(t)>0 for t >1.We have
f0(t) = 6t2
(t3+1)√
2(t6+1)−2−143t2(t+1)
(t4+1) 7 4
=
3t2
"
2 3 4(t4+1)
7
4−(t+1)(t3+1)√t6+1
#
2 1
4 (t3+1)√
t6+1(t4+1) 7 4
and so it is positive if g(t) =
234 t4+ 174 4
−h
(t+ 1) t3+ 1 p
t6+ 1i4
is positive. Or
g(t) = (t−1)4(7t24+ 24t23+ 48t22+ 68t21+ 112t20 + 184t19+ 264t18+ 296t17+ 344t16+ 428t15 + 512t14+ 488t13+ 466t12+ 488t11+ 512t10 + 428t9+ 344t8+ 296t7+ 184t5+ 112t4 + 68t3+ 48t2+ 24t+ 7)
so that the property is certainly true. The second inequality is a simple con- sequence of (1) becauseA4/3 <A3/2.For the last inequality
a−b 2 tan−1a−b
a+b
<
a5/3+b5/3 2
35
, a6=b,
we can again consider ab =t3>1 and we have to prove that
t3−1 2 tan−1t3−1
t3+1
<
t5+1 2
35
, t >1.
This is equivalent with the condition that the function h(t) = tan−1tt33−1+1 − t3−1
2 2 5 (t5+1)
3 5
is positive fort >1.Ash(1) = 0 and h0(t) = t63t+12 − 3t
2(t2+1)
2 2 5 (t5+1)
8 5
=
3t2
"
2 2 5(t5+1)
8
5−(t2+1)(t6+1)
#
2 2 5 (t5+1)
8 5 (t6+1)
,
we have h(t)>0 fort >1 if h0(t)>0 for t >1,thus if the function k(t) =
225 t5+ 185 5
−
t2+ 1
t6+ 15
is positive fort >1.Or this is obvious because
k(t) = (t−1)4(185t28+ 200t27+ 221t26+ 365t24 + 410t22+ 520t19+ 580t18+ 520t17+ 430t16 + 400t15+ 410t14+ 440t13+ 365t12+ 284t11 + 221t10+ 200t9+ 185t8+ 140t7+ 90t6 + 60t5+ 45t4+ 25t2+ 40t3+ 12t+ 3).
Remark 2. For the factorization of the polynomialsg and kwe have used
the computer algebra Maple.
Remark 3. It is an open problem for us to find a mean N, related to the above mentioned means, with the property that
A5/3 < N <A2.
For instance, the mean S,which is similar to I, is not convenient as follows
from (2).
Corollary 4. For each x∈(0,1)we have the following evaluations 1< x
sinh−1x <A4/3(1−x,1 +x)< tanx−1x <A5/3(1−x,1 +x).
REFERENCES
[1] B.C. Carlson,The logarithmic mean, Amer. Math. Monthly,79(1972), pp. 615–618.
[2] I. Costin and G. Toader, A nice evaluation of some Seiffert type means by power means, Int. J. Math. Math. Sc., 2012, Article ID 430692, 6 pages, doi:10.1155/2012/430692.
[3] P.A. H¨ast¨o, Optimal inequalities between Seiffert’s means and power means, Math.
Inequal. Appl.,7(2004) no. 1, pp. 47–53.
[4] A.A. Jagers,Solution of problem 887, Niew Arch. Wisk. (Ser. 4),12(1994), pp. 230–
231.
[5] T.P. Lin, The power and the logarithmic mean, Amer. Math. Monthly, 81 (1974), pp. 879–883.
[6] E. NeumanandJ. S´andor,On the Schwab-Borchardt mean, Math. Panon.,14(2003) no. 2, pp. 253–266.
[7] E. NeumanandJ. S´andor,Comparison inequalities for certain bivariate means, Appl.
Anal. Discrete Math.,3(2009), pp. 46–51.
[8] A.O. Pittenger,Inequalities between arithmetic and logarithmic means, Univ. Beograd.
Publ. Elektrotehn. Fak. Ser. Mat. Fiz.,678-715(1980), pp. 15–18.
[9] H.-J. Seiffert,Problem 887, Niew Arch. Wisk. (Ser. 4),11(1993), pp. 176–176.
[10] H.-J. Seiffert,Aufgabeβ16, Die Wurzel,29(1995), pp. 221–222.
Received by the editors: June 12, 2012.