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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 COSTINand 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,

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- 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,

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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.

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Denoting

f(t) = sinh−1 tt33−1+1−214 t3−1

t4+ 13 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)−2143t2(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+1t3−1

2 2 5 (t5+1)

3 5

is positive fort >1.Ash(1) = 0 and h0(t) = t63t+123t

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)

,

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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.

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