The main objective of this research work is to present optimization of inequality in the one-parameter, arithmetic and harmonic means as follows:
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 , 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 ).
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 
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 , 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  the authors showed the sharp upper and lower bounds for the Neuman-Sandor  in terms of the linear convex combination of the logarithmic means , and second Seiffert means  of two positive numbers a and b, respectively for the double inequalities
holds for all with is true if and only if and .
In , 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 ; and in terms of the contraharmonic mean there is .
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
Set . Then, we obtain
Because and therefore the study amounts to proving that
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:
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:
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.
 Chen, J.-J., Lei, J.-J. and Long, B.-Y. (2017) Optimal Bounds for Neuman-Sandor Means in Term of the Convex Combination of the Logarithmic and the Second Seiffert Means. Journal of Inequalities and Applications, No. 1, 251.
 Seiffert, S., Kaselowesky, J., Jungk, A. and Claassen, N. (1995) Observed and Calculated Potassium Uptake by Maize as Affected by Soil Water Content and Bulk Density. Agronomy Journal, 87, 1070-1077.
 Xu, H.-Z., Chu, Y.-M. and Qian, W.-M. (2018) Sharp Bounds for the Sandor-Yang Means in Terms of Arithmetic and Contra-Harmonic Means. Journal of Inequalities and Applications, No. 1, 127.