1. Introduction
The main objective of this research work is to present optimization of inequality in the one-parameter, arithmetic and harmonic means as follows:
(1.1)
with
and
Our motivation for this study is to find out such inequality that arises in the search for determination of a point of reference about which some function of variants would be minimum or maximum. Since very early times, people have been interested in the problem of choosing the best single quantity, which could summarize the whole information contained in a number of observations (measurements). Moreover, the theory of means has its roots in the work of the Pythagorean who introduced the harmonic, geometric, and arithmetic means. The strong relations and introduction of the theory of means with the theories of inequalities, function equations, probability and statistics add greatly to its importance. This single element is usually called a means or averages. The term “means” or “average” (middle value) has for a long time been used in all branches of human activity.
The basic function of mean value is to represent a given set of many values by some single value. In [1], the authors for the first time introduced power means defining the meaning of the term “representation” as determination of appointing of reference about which some function of variants would be minimum. More recently the means were the subject of research, study, and essential areas in several applications such as physics, economics, electrostatics, heat conduction, medicine, and even in meteorology. It can be observed the power mean
(see as [2]).
If we denote by
the arithmetic means, geometric means and harmonic means of two positive numbers a and b, respectively. In addition, the logarithmic and identric means of two positive real numbers a and b were defined by [3]
Several authors investigated and developed relationship of optimal inequalities between the various means.
The well-known inequality that:
and all inequalities are strict for
.
In [3], researchers studied what are the best possible parameters
and
by two theorems:
Theorem (1) the double inequality:
holds for all
if and only if
and
when proved that the parameters
and
cannot be improved.
Theorem (2) the double inequality:
holds for all
if and only if
and
when proved that the parameters
and
cannot be improved; holds for all
with
, and they found
the optimal lower generalized logarithmic means bound for the identric means
for inequalities
; holds for all a, b are positive numbers with
. Pursuing another line of investigation, in [4] the authors showed the sharp upper and lower bounds for the Neuman-Sandor
[5] in terms of the linear convex combination of the logarithmic means
, and second Seiffert means
[6] of two positive numbers a and b, respectively for the double inequalities
holds for all
with
is true if and only if
and
.
In [7], HZ Xu et al. have improvements and refinements, for they found several sharp upper and lower bounds for the Sandor-Yang means
and
. In terms of combinations of the arithmetic means
, there is [8]; and in terms of the contraharmonic mean
there is [9].
2. Main Results
Our main results are set in the following theorem:
Theorem 1. Assume
then, there exist
reals such that
1) If
,
and
with
then, the double inequality (1.1) holds.
2) If
,
and
with
(
small) then the double inequality (1.1) holds.
Proof. 1) Assuming
with
We have
Set
. Then, we obtain
and
Because
and
therefore the study amounts to proving that
Let
We have to prove that the function f is negative under certain conditions on the parameters
and p, a.e:
. So
Because
, it will suffice to show that f is decreasing for all
, which amounts to studying the sign of the derivative
of
. We have:
Because
, it will suffice to show that
is decreasing for all
, which amounts to studying the sign of the derivative
of
. We have:
There exixt
with
such that
for all
so it will suffice to show that
is decreasing for all
, which amounts to studying the sign of the derivative
of
. We have:
and we get
and since
we obtain that
so, we will have
By the same process we find that
then that
and
.
Finally in this part there exixt
with
such that for all
we have:
i.e.:
To show the second inequality in this first case, we proceed by similar calculations. This is done by considering the function g defined by
So, after all the calculations, we get that for
and
:
. a.e:
2) With similar calculations and by the same idea we obtain that if
,
and
with
and
(
small) then,
Conclusion 1. In our work, we studied the following double inequality
by searching the best possible parameters such that (1.1) can be held.
Firstly, we have inserted
Without loss of generality, we have assumed that
and let
for 1) and
(t small) for 2) to determine the condition for
and
to become
.
Secondly, have inserted
Without loss of generality, we assume that
and let
for 1) and
(t small) for 2) to determine the condition for
and
to become
.
And finally, we got that:
1) if
,
and
with
then, the double inequality (1.1) holds;
2) if
,
and
with
(
small) then the double inequality (1.1) holds.