^{1}

^{*}

^{2}

This research work considers the following inequalities:
λ
*A*(
a,
b) + (1-
λ)
*C*(
a,
b) ≤
C(
a,
b) ≤
μ
*A*(
a,
b) + (1-
μ)
*C*(
a,
b) and
*C*[
λ
*a* + (1-
λ)
*b*,
λ
*b* + (1-
λ)
*a*] ≤
C(
a,
b) ≤
*C*[
μ
*a* + (1-
μ)
*b*,
μ
*b* + (1-
μ)
*a*] with
. The researchers attempt to find an answer as to what are the best possible parameters
λ,
μ that (1.1) and (1.2) can be hold? The main tool is the optimization of some suitable functions that we seek to find out. By searching the best possible parameters such that (1.1) and (1.2) can be held. Firstly, we insert
*f*(
t) =
λ
*A*(
a,
b) + (1-
λ)
*C*(
a,
b) -
C(
a,
b) without the loss of generality. We assume that
a>
b and let
to determine the condition for
λ and
μ to become f (
t) ≤ 0. Secondly, we insert g(
t) =
μ
*A*(
a,
b) + (1-
μ)
*C*(
a,
b) -
C(
a,
b) without the loss of generality. We assume that
a>
b and let
to determine the condition for
λ and
μ to become
*g*(
t) ≥ 0.

For a , b > 0 with a ≠ b , the Centroidal mean C ¯ ( a , b ) , Harmonic mean A ( a , b ) and Contraharmonic mean C ( a , b ) are defined by:

C ¯ ( a , b ) = 2 ( a 2 + a b + b 2 ) 3 ( a + b ) , A ( a , b ) = a + b 2 ; C ( a , b ) = a 2 + b 2 a + b

respectively.

The main objective of this research work is to present optimization of the following inequalities:

λ A ( a , b ) + ( 1 − λ ) C ( a , b ) ≤ C ¯ ( a , b ) ≤ μ A ( a , b ) + ( 1 − μ ) C ( a , b ) (1.1)

and

C [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] ≤ C ¯ ( a , b ) ≤ C [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] (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 [

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. [

The basic function of mean value is to represent a given set of many values by some single value. In [

M p ( a , b ) = { ( a p + b 2 p ) 1 p ; p ≠ 0 a b ; p = 0

If we denote by

A ( a , b ) = 1 2 ( a + b ) , G ( a , b ) = a b and H ( a , b ) = 2 a b a + b ,

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 [

L ( a , b ) = { b − a log b − log a a ≠ b a a = b

I ( a , b ) = { 1 e ( b b a a ) 1 / ( b − a ) a ≠ b a a = b

Several authors investigated and developed relationship of optimal inequalities between the various means.

The well-known inequality that:

min { a , b } ≤ H ( a , b ) = M − 1 ( a , b ) ≤ G ( a , b ) = M 0 ( a , b ) ≤ L ( a , b ) ≤ I ( a , b ) ≤ A ( a , b ) = M 1 ( a , b ) ≤ max { a , b }

and all inequalities are strict for a ≠ b .

In [

Theorem (1) the double inequality: -

α 1 A ( a , b ) + ( 1 − α 1 ) H ( a , b ) ≤ L ( a , b ) ≤ β 1 A ( a , b ) + ( 1 − β 1 ) H ( a , b )

holds for all a , b > 0 if and only if α 1 ≤ 0 and β 1 ≥ 2 3 when proved that the parameters α 1 ≤ 0 and β 1 ≥ 2 3 cannot be improved.

Theorem (2) the double inequality: -

α 2 A ( a , b ) + ( 1 − α 2 ) H ( a , b ) ≤ L ( a , b ) ≤ β 2 A ( a , b ) + ( 1 − β 2 ) H ( a , b )

holds for all a , b > 0 if and only if α 2 ≤ 2 e and β 2 ≥ 5 6 when proved that the parameters α 2 ≤ 2 e and β 2 ≥ 5 6 cannot be improved.

Interestingly in [

M 0 ( a , b ) < L t ( a , b ) < M t l 3 ( a , b )

holds for all a , b > 0 with a ≠ b , and they found L 2 ( a , b ) the optimal lower generalized logarithmic means bound for the identric means I ( a , b ) for inequalities L 2 ( a , b ) < I ( a , b ) holds for all a, b are positive numbers with a ≠ b . Pursuing another line of investigation, in [

α L ( a , b ) + ( 1 − α ) T ( a , b ) ≤ N S ( a , b ) ≤ β L ( a , b ) + ( 1 − β ) T ( a , b )

holds for all a , b > 0 with a ≠ b is true if and only if α ≥ 1 4 and β ≤ 1 − π l [ 4 log ( 1 + 2 ) ] .

In [

The authors have to proven our main results several lemmas find the best possible parameters α i , β i / ( i = 1 , 2 , 3 , 4 ) such that the double inequalities

c α 1 ( a , b ) A 1 − α 1 ( a , b ) < R Q A ( a , b ) < c β 1 ( a , b ) A 1 − β 1 ( a , b )

c α 2 ( a , b ) A 1 − α 2 ( a , b ) < R Q A ( a , b ) < c β 2 ( a , b ) A 1 − β 2 ( a , b )

α 3 [ 1 3 C ( a , b ) + 2 3 A ( a , b ) ] + ( 1 − α 3 ) C 1 / 3 ( a , b ) A 2 / 3 ( a , b ) < R Q A ( a , b ) < β 3 [ 1 3 C ( a , b ) + 2 3 A ( a , b ) ] + ( 1 − β 3 ) C 1 / 3 ( a , b ) A 2 / 3 ( a , b ) ,

α 4 [ 1 6 C ( a , b ) + 5 6 A ( a , b ) ] + ( 1 − α 4 ) C 1 / 6 ( a , b ) A 5 / 6 ( a , b ) < R A Q ( a , b ) < β 4 [ 1 6 C ( a , b ) + 5 6 A ( a , b ) ] + ( 1 − β 4 ) C 1 / 6 ( a , b ) A 5 / 6 ( a , b )

holds for all a , b > 0 with a ≠ b .

In [

λ C ( a , b ) + ( 1 − λ ) A ( a , b ) ≤ M ( a , b ) ≤ μ C ( a , b ) + ( 1 − μ ) A ( a , b ) ,

with M ( a , b ) is the Neuman-S andor mean, hold for all holds for all a , b > 0 with a ≠ b if and only if λ ≤ 1 − log ( 1 + 2 ) log ( 1 + 2 ) and μ ≥ 1 6 . In [

C [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] ≤ T ( a , b ) ≤ C [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] ,

with T ( a , b ) = a − b 2 arctan ( a − b a + b ) , were proved to be valid for 1 2 < λ , μ < 1 and for all a , b < 0 with a ≠ b if and only if λ ≤ ( 1 + 4 − π π ) and μ ≥ 3 + 3 6 . Wen-Hui Li and Feng Qi [

λ Q ( a , b ) + ( 1 − λ ) M ( a , b ) ≤ C ¯ ( a , b ) ≤ μ Q ( a , b ) + ( 1 − μ ) M ( a , b ) ,

with Q ( a , b ) = a 2 + b 2 2 is the root-square mean, holds for all a , b > 0 with a ≠ b if and only if λ ≤ 1 2 and μ ≥ 3 − 4 ln ( 1 + 2 ) 3 [ 1 − 2 ln ( 1 + 2 ) ] = 0.7107 ⋯ .

For mor information on this topic, you can refer to the following references: [

Motivating by results mentioned above, we naturally ask a question: what are the best possible parameters λ , μ 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 a > 0 , b > 0 with a b > 1 then,

1) if λ ∈ ( 2 3 , + ∞ ) and μ ∈ ( − ∞ , 2 3 ) then, the double inequality (1.1) holds.

2) if λ ∈ ( 3 − 3 6 , 3 + 3 6 ) and μ ∈ ( − ∞ , − 6 ] ∪ [ 0, 3 − 3 6 ) ∪ ( 3 + 3 6 , + ∞ ) then the double inequality (1.2) holds.

Proof. 1): Assuming a > 0 , b > 0 with a b > 1

λ ( a + b 2 ) + ( 1 − λ ) ( a 2 + b 2 a + b ) ≤ 2 ( a 2 + a b + b 2 ) 3 ( a + b ) ≤ μ ( a + b 2 ) + ( 1 − μ ) ( a 2 + b 2 a + b )

Set t = a b > 1 . Then, we obtain

λ ( b ( t + 1 ) 2 ) + ( 1 − λ ) ( b ( t 2 + 1 ) t + 1 ) ≤ 2 b ( t 2 + t + 1 ) 3 ( t + 1 ) ≤ μ ( b ( t + 1 ) 2 ) + ( 1 − μ ) ( b ( t 2 + 1 ) t + 1 )

We start by showing that

λ ( b ( t + 1 ) 2 ) + ( 1 − λ ) ( b ( t 2 + 1 ) t + 1 ) − 2 b ( t 2 + t + 1 ) 3 ( t + 1 ) ≤ 0,

⇔ λ 3 b ( t + 1 ) 2 6 ( t + 1 ) + ( 1 − λ ) 6 b ( t 2 + 1 ) 6 ( t + 1 ) − 4 b ( t 2 + t + 1 ) 6 ( t + 1 ) ≤ 0

Because t > 0 therefore the study amounts to proving that

λ 3 b ( t + 1 ) 2 + ( 1 − λ ) 6 b ( t 2 + 1 ) − 4 b ( t 2 + t + 1 ) ≤ 0.

Let

f ( t ) = 3 λ b ( t + 1 ) 2 + 6 ( 1 − λ ) b ( t 2 + 1 ) − 4 b ( t 2 + t + 1 )

We have to prove that the function f is negative under certain conditions on the parameter λ , a.e: f ( t ) ≤ 0 . So

f ( t ) = 3 λ b ( t + 1 ) 2 + 6 ( 1 − λ ) b ( t 2 + 1 ) − 4 b ( t 2 + t + 1 ) ≤ 0

Because f ( 1 ) = 0 , it will suffice to show that f is decreasing for all t > 1 , which amounts to studying the sign of the derivative f ′ of f. We have:

f ′ ( t ) = 6 λ b ( t + 1 ) + 12 ( 1 − λ ) b t − 4 b ( 2 t + 1 )

Because f ′ ( 1 ) = 0 , it will suffice to show that f ′ is decreasing for all t > 1 , which amounts to studying the sign of the derivative f ″ of f ′ . We have:

f ″ ( t ) = 2 b ( 2 − 3 λ ) < 0 ⇔ λ > 2 3

so that f ′ is decreasing for t > 1 and therefore, we obtain that f ( t ) < 0 because f ( 1 ) = 0 .

Finally in this part for a > 0 , b > 0 with a b > 1 , we obtain that

λ ( b ( t + 1 ) 2 ) + ( 1 − λ ) ( b ( t 2 + 1 ) t + 1 ) ≤ 2 b ( t 2 + t + 1 ) 3 ( t + 1 ) , for all λ > 2 3 .

To show the second inequality in this first case, we proceed by similar calculations. This is done by considering the function g defined by

g ( t ) = g ( t ) = 3 μ b ( t + 1 ) 2 + 6 ( 1 − μ ) b ( t 2 + 1 ) − 4 b ( t 2 + t + 1 ) .

So, after all the calculations, we get that for a > 0 , b > 0 with a b > 1 , that g ( t ) ≥ 0 , for all μ < 2 3 . a.e:

2 ( a 2 + a b + b 2 ) 3 ( a + b ) ≤ μ ( a + b 2 ) + ( 1 − μ ) ( a 2 + b 2 a + b )

2): Assuming a > 0 , b > 0 with a b > 1 and with similar calculations and by the same idea we obtain that for all

λ ∈ ( 3 − 3 6 , 3 + 3 6 ) and μ ∈ ( − ∞ , − 6 ] ∪ [ 0, 3 − 3 6 ) ∪ ( 3 + 3 6 , + ∞ )

then the double inequality

C [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] ≤ C ¯ ( a , b ) ≤ C [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] ,

holds.

Conclusion 1. In our work, we studied the following double inequalities: respectively (1.1) and (1.2)

λ A ( a , b ) + ( 1 − λ ) C ( a , b ) ≤ C ¯ ( a , b ) ≤ μ A ( a , b ) + ( 1 − μ ) C ( a , b )

and

C [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] ≤ C ¯ ( a , b ) ≤ C [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ]

by searching the best possible parameters such that (1.1) and (1.2) can be hold.

Firstly, we have inserted

f ( t ) = λ A ( a , b ) + ( 1 − λ ) C ( a , b ) − C ¯ ( a , b )

and

g ( t ) = μ A ( a , b ) + ( 1 − μ ) C ( a , b ) − C ¯ ( a , b )

Without loss of generality, we have assumed that a > b and let t = a b > 1 to determine the condition for λ and μ to become f ( t ) ≤ 0 and g ( t ) ≥ 0 .

Secondly, we have inserted Without loss of generality, we assume that a > b and let

And finally, we got that:

1) if

2) if

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.

The authors declare no conflicts of interest regarding the publication of this paper.

El Mokhtar Ould El Mokhtar, M. and Alharbi, H. (2020) On Two Double Inequalities (Optimal Bounds and Sharps Bounds) for Centroidal Mean in Terms of Contraharmonic and Arithmetic Means. Journal of Applied Mathematics and Physics, 8, 1039-1046. https://doi.org/10.4236/jamp.2020.86081