Sharps Bounds for Power Mean in Terms of Contraharmonic Mean

In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters , α β that (1.1) can be held? The main tool is the optimization of some suita-ble functions that we seek to find out. Without loss of generality, we have assumed that a b > and let 1 a t b = > for 1) and a b < , 1 a t b =  (t small) for 2) to determine the condition for α and β to become ( ) 0 f t ≤ and ( ) 0 g t ≥ .

ble functions that we seek to find out. Without loss of generality, we have assumed that a b > and let 1 a t b = > for 1) and a b < , for 2) to determine the condition for α and β to become ( ) 0 f t ≤ and

Introduction
The main objective of this research work is to present optimization of inequality in the one-parameter, arithmetic and harmonic means as follows:  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 ( ) , p M a b (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 a b ≠ .
In [3], researchers studied what are the best possible parameters 1 2 1 , , α α β and 2 β by two theorems: Theorem (1) the double inequality: Journal of Applied Mathematics and Physics In

Main Results
Our main results are set in the following theorem: 1 0 p t + > therefore the study amounts to proving We have to prove that the function f is negative under certain conditions on the parameters , α β and p, a.e: ( ) 0 = , it will suffice to show that f is decreasing for all 1 t > , which amounts to studying the sign of the derivative f ′ of f . We have: 1 0 f ′ = , it will suffice to show that f ′ is decreasing for all 1 t > , which amounts to studying the sign of the derivative f ′′ of f ′ . We have: 2) With similar calculations and by the same idea we obtain that if Without loss of generality, we have assumed that a b > and let 1 a t b = > for 1) and , 1 a a b t b < =  (t small) for 2) to determine the condition for α

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.