\(^\ast \)Department of Computer Science, Technical University of Cluj-Napoca, Baritiu st. no. 28, Cluj-Napoca, Romania, e-mail: Iulia.Costin@cs.utcluj.ro.

\(^\S \)Department of Mathematics, Technical University of Cluj-Napoca, Baritiu st. no. 25, Cluj-Napoca, Romania, e-mail: Gheorghe.Toader@math.utcluj.ro.

1 Introduction

A mean is a function \(M:\mathbb {R}_{+}^{2}\rightarrow \mathbb {R}_{+},\) with the property

Each mean is reflexive, that is

This is also used as the definition of \(M(a,a)\).

A mean is symmetric if

it is homogeneous (of degree 1) if

We shall refer here to the following symmetric and homogeneous means:

- the power means \(\mathcal{A}_{p}\), defined by

- the geometric mean \(\mathcal{G}\), defined as \(\mathcal{G}(a,b)=\sqrt{ab},\) but verifying also the property

- the identric mean \(\mathcal{I}\) defined by

- the Gini mean \(\mathcal{S}\) defined by

- the first Seiffert mean \(\mathcal{P}\), defined in [ 9 ] by

- the second Seiffert mean \(\mathcal{T}\), defined in [ 10 ] by

- the Neuman-Sándor mean \(\mathcal{M}\), defined in [ 6 ] by

- the logarithmic mean \(\mathcal{L}\) defined by

As remarked B.C. Carlson in [ 1 ] , the logarithmic mean can be represented also by

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 \(M{\lt}N\) if \(M(a,b){\lt}N(a,b)\) for \(a\neq b.\) It is known that the family of power means is an increasing family of means, thus

The evaluation of a given mean \(M\) by power means assumes the determination of some real indices \(p\) and \(q\) such that \(\mathcal{A}_{p}{\lt}M{\lt}\mathcal{A}_{q}.\) The evaluation is optimal if \(p\) is the the greatest and \(q\) is the smallest index with this property. This means that \(M \) cannot be compared with \(\mathcal{A}_{r}\) if \(p{\lt}r{\lt}q.\)

Optimal evaluation were given for the logarithmic mean in [ 5 ]

for the identric mean in [ 8 ]

and for the first Seiffert mean in [ 3 ]

Following evaluations are also known:

proved in [ 4 ] ,

given in [ 10 ] ,

as it was shown in [ 6 ] and

as it is proved in [ 7 ] . In [ 2 ] it is proven that

and using some of the above results, it is obtained the following chain of inequalities

Here we retain another chain of inequalities

Our aim is to prove that \(\mathcal{A}_{4/3}\) can be put between \(\mathcal{M}\) and \(\mathcal{T}\) and \(\mathcal{A}_{2}\) can be replaced by \(\mathcal{A}_{5/3}\mathcal{.}\) 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.

we can assume that \(a{\gt}b\) and denote \(a/b=t^{3}{\gt}1.\) The inequality becomes

or

Denoting

we have to prove that \(f(t){\gt}0\) for \(t{\gt}1.\) As \(f(1)=0,\) we want to prove that \(f^{\prime }(t){\gt}0\) for \(t{\gt}1.\) We have

and so it is positive if

is positive. Or

so that the property is certainly true. The second inequality is a simple consequence of (1) because \(\mathcal{\mathcal{A}}_{4/3}{\lt}\mathcal{\mathcal{A}}_{3/2}\mathcal{.}\) For the last inequality

we can again consider \(\tfrac {a}{b}=t^{3}{\gt}1\) and we have to prove that

This is equivalent with the condition that the function

is positive for \(t{\gt}1.\) As \(h(1)=0\) and

we have \(h(t){\gt}0\) for \(t{\gt}1\) if \(h^{\prime }(t){\gt}0\) for \(t{\gt}1,\) thus if the function

is positive for \(t{\gt}1.\) Or this is obvious because