On Two Double Inequalities (Optimal Bounds and Sharps Bounds) for Centroidal Mean in Terms of Contraharmonic and Arithmetic Means ()
1. Introduction
For
with
, the Centroidal mean
, Harmonic mean
and Contraharmonic mean
are defined by:
respectively.
The main objective of this research work is to present optimization of the following inequalities:
(1.1)
and
(1.2)
Recently, both mean values have been the subject of intensive research. In particular, many remarkable inequalities and properties for these means can be found in the literature [1] [2].
This work finds 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. Peter et al. [3] introduced seven other means and gave the well-known elegant geometric proof of the celebrated inequalities among 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 average. 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 [4], the authors were the first time introduced power means defined the meaning of the term “representation” as determination of appoint of reference about which some function of variants would be minimum. More recently the means were the subject of research and study whereas essential areas in several applications such as: physics, economics, electrostatics, heat conduction, medicine and even in meteorology. It can be observed that the power mean
of order p can be rewritten as (see as [5])
If we denote by
the arithmetic, geometric 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 defined by [6]
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 [7], 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.
Interestingly in [5] B. Long et al., proved that the following results:
and
are the best possible lower and upper power bounds for the generalized logarithmic mean
for any fixed
the double inequalities
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 [8] the authors showed the sharp upper and lower bounds for the Neuman-sandor
. [9] in terms of the liner convex combination of the logarithmic means
, and second seiffert means
[10] of two positive numbers a and b, respectively for the double inequalities
holds for all
with
is true if and only if
and
.
In [11] have improvements and refinements by HZ Xu et al., for they found several sharp upper and lower bounds for the Sandor-yang means
and
. [12] [13] in terms of combinations of the arithmetic means
and the contra-harmonic mean
. [4] [14].
The authors have to proven our main results several lemmas find the best possible parameters
such that the double inequalities
holds for all
with
.
In [15], Neuman proved that the double inequalities
with
is the Neuman-S andor mean, hold for all holds for all
with
if and only if
and
. In [2] Shen, the inequalities sharps bounds for Seiffert mean in terms of Contraharmonic mean
with
, were proved to be valid for
and for all
with
if and only if
and
. Wen-Hui Li and Feng Qi [16], proved that the double inequality
with
is the root-square mean, holds for all
with
if and only if
and
.
For mor information on this topic, you can refer to the following references: [17] [18] [19].
2. Main Results
Motivating by results mentioned above, we naturally ask a question: what are the best possible parameters
that (1.1) and (1.2) can be hold?
The aim of this paper is to answer this question. The solution to this question may be stated as the following Theorem:
Theorem 1. Assuming
with
then,
1) if
and
then, the double inequality (1.1) holds.
2) if
and
then the double inequality (1.2) holds.
Proof. 1): Assuming
with
Set
. Then, we obtain
We start by showing that
Because
therefore the study amounts to proving that
Let
We have to prove that the function f is negative under certain conditions on the parameter
, 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 f. 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:
so that
is decreasing for
and therefore, we obtain that
because
.
Finally in this part for
with
, we obtain that
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
with
, that
, for all
. a.e:
2): Assuming
with
and with similar calculations and by the same idea we obtain that for all
then the double inequality
holds.
Conclusion 1. In our work, we studied the following double inequalities: respectively (1.1) and (1.2)
and
by searching the best possible parameters such that (1.1) and (1.2) can be hold.
Firstly, we have inserted
and
Without loss of generality, we have assumed that
and let
to determine the condition for
and
to become
and
.
Secondly, we have inserted Without loss of generality, we assume that
and let to determine the condition for and to become
And finally, we got that:
1) if and then, the double inequality (1.1) holds.
2) if and then the double inequality (1.2) holds.
Acknowledgements
The authors gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the material support for this research WORK under the number (1063) during the academic year 1441AH/2020AD.