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 JAMP  Vol.6 No.12 , December 2018
Invariance of Weighted Bajraktarević Mean with Respect to the Beckenbach-Gini means
Abstract: Under some conditions on the functions and defined on I, the weighted Bajraktarević mean is given by where . In this paper, we study the invariance of the weighted Bajraktarević mean with respect to Beckenbach-Gini means.

1. Introduction

Let I be an open interval. A two-variable function M : I 2 I is called a mean on the interval I if

min { x , y } M ( x , y ) max { x , y } , x , y I

holds. If for all x , y I , x y , these inequalities are strict, M is called strict. Obviously, if M is a mean, then M is reflexive, i.e., M ( x , x ) = x for all x I .

A quasi-arithmetic mean, generated by the function φ , is defined by

M ( x , y ) = A φ ( x , y ) : = φ 1 ( φ ( x ) + φ ( y ) 2 ) , x , y I ,

for a continuous, strictly monotone function φ : I .

A more general mean is the class of the weighted quasi-arithmetic means, which is defined by

M ( x , y ) = A φ , λ ( x , y ) : = φ 1 ( λ φ ( x ) + ( 1 λ ) φ ( y ) ) , x , y I ,

where φ : I is a continuous strictly monotone function, and the constant λ ( 0,1 ) .

A Lagrangian mean is defined by

M ( x , y ) = L φ ( x , y ) : = { φ 1 ( 1 y x x y φ ( t ) d t ) , if x y , x , if x = y , x , y I ,

where φ : I is a continuous strictly monotone function.

Given the continuous functions φ , ψ : I satisfy ψ ( x ) 0 for x I and φ ψ is one-to-one, the Bajraktarević mean of generators φ and ψ [1] is defined by

M ( x , y ) = B [ φ , ψ ] : = ( φ ψ ) 1 ( φ ( x ) + φ ( y ) ψ ( x ) + ψ ( y ) ) , x , y I . (1.1)

B [ φ , ψ ] is a strict mean, and it is a generalization of quasi-arithmetic mean. Note that if φ ( x ) ψ ( x ) = x , x I , we have

B [ φ , ψ ] = B [ ψ ] : = x ψ ( x ) + y ψ ( y ) ψ ( x ) + ψ ( y ) , x , y I , (1.2)

where the mean B [ ψ ] is called Beckenbach-Gini mean of a generator ψ [2] .

Quotient mean Q [ φ , ψ ] : I 2 is defined by

Q [ φ , ψ ] ( x , y ) : = ( φ ψ ) 1 ( φ ( x ) ψ ( y ) ) , x , y I , (1.3)

where the functions φ and ψ are continuous, positive, and of different type of strict monotonicity in I [3] . For I = ( 0 , ) , φ ( x ) = x , ψ ( x ) = 1 x , we have Q [ φ , ψ ] ( x , y ) = x y = G , where G is geometric mean.

Now we define the weighted Bajraktarević mean as follows:

M ( x , y ) = B λ , μ [ φ , ψ ] : = ( φ ψ ) 1 ( λ φ ( x ) + ( 1 λ ) φ ( y ) μ ψ ( x ) + ( 1 μ ) ψ ( y ) ) , x , y I , (1.4)

where λ , μ [ 0,1 ] , φ , ψ : I are continuous, positive, and of different type of strict monotonicity and φ ψ is one-to-one. Note that if λ = μ = 1 2 , B λ , μ [ φ , ψ ] = B [ φ , ψ ] . If λ = 1 , μ = 0 , the weighted Bajraktarević mean becomes quotient mean, that is B λ , μ [ φ , ψ ] = Q [ φ , ψ ] ( x , y ) . Without any loss of generality, we can assume that φ is strictly increasing and ψ is strictly decreasing.

Let M , N : I 2 I be means. A mean K : I 2 I is called invariant with respect to the mean-type mappings ( M , N ) , shortly, ( M , N ) -invariant [4] , if

K ( M ( x , y ) , N ( x , y ) ) = K ( x , y ) , x , y I .

The simplest example when the invariance equation holds is the well-known identity

G ( A ( x , y ) , H ( x , y ) ) = G ( x , y ) , x , y > 0 ,

where A , H , G denote the arithmetic, harmonic and geometric means, respectively.

The invariance of the arithmetic mean with respect to various quasi-arithmetic means has been extensively investigated. Firstly we came upon the work of Sutô [5] [6] presented in 1914, in which he gave analytic solutions for the invariance equation

A φ ( x , y ) + A ψ ( x , y ) = x + y , x , y I . (1.5)

Then Matkowski solved the above equation under assumptions that φ ( x ) and ψ ( x ) are twice continuously differentiable [4] . These regularity assumptions were weaken step-by-step by Daróczy, Maksa and Páles in [7] [8] . Finally, without any regularity assumptions, the problem was solved by Daróczy and Páles in [9] .

Also, the form of Equation (1.5) was generalized by many authors. Concretely, Burai considered the invariance of the arithmetic mean with respect to weighted quasi-arithmetic means in [10] . Daróczy, Hajdu, Jarczyk and Matkowski studied the invariance equation involving three weighted quasi-arithmetic means [11] [12] [13] . Matkowski solved the invariance equation involving the arithmetic mean in class of Lagrangian mean-type mappings [14] . In [15] , Makó and Páles investigated the invariance of the arithmetic mean with respect to generalized quasi-arithmetic means. The invariance of the geometric mean in class of Lagrangian mean-type mappings has been studied by Głazowska and Matkowski in [16] . All pairs of Stolarsky’s means for which the geometric mean is invariant were determined in [17] . Zhang and Xu considered the invariance of the geometric mean with respect to generalized quasi-arithmetic means in [18] and some invariance of the quotient mean with respect to Makó-Páles means in [19] . Recently, Jarczyk provided a review on the invariance of means [20] .

Matkowski studied the invariance of the quotient mean with respect to weighted quasi-arithmetic mean type mapping [3] . He also studied the invariance of the Bajraktarević means with respect to quasi-arthmetic means in [21] and the invariance of the Bajraktarević means with respect to the Beckenbach-Gini means in [22] . Motivated by the above mentioned works, in this paper, we study the invariance of the weighted Bajraktarević mean with respect to the Beckenbach-Gini means, i.e., solve the functional equation

B λ , μ [ φ , ψ ] ( B [ φ ] ( x , y ) , B [ ψ ] ( x , y ) ) = B λ , μ [ φ , ψ ] ( x , y ) , x , y I , (1.6)

where I , φ , ψ : I ( 0, + ) are continuous functions and φ is strictly increasing, ψ is strictly decreasing.

2. Main Result

Lemma 1. Let I be an interval. Suppose that the function φ : I ( 0, + ) is differentiable, then we have

B [ φ ] ( x , x ) x = 1 2 . (2.1)

If the function φ : I ( 0, + ) is twice differentiable, then we have

2 B [ φ ] ( x , x ) x 2 = φ ( x ) 2 φ ( x ) . (2.2)

Proof. By the definition of B [ φ ] , we have

B [ φ ] ( x , y ) x = φ 2 ( x ) + φ ( x ) φ ( y ) + x φ ( x ) φ ( y ) y φ ( x ) φ ( y ) ( φ ( x ) + φ ( y ) ) 2 ,

then let y = x , we can get that B [ φ ] ( x , x ) x = 1 2 .

Also we have

2 B [ φ ] ( x , y ) x 2 = 2 φ ( x ) φ ( y ) + x φ ( x ) φ ( y ) y φ ( x ) φ ( y ) ( φ ( x ) + φ ( y ) ) 2 2 φ ( x ) ( x φ ( x ) φ ( y ) y φ ( x ) φ ( y ) ) ( φ ( x ) + φ ( y ) ) 3

letting y = x , we can get (2.2).

Lemma 2. Let I be an interval and λ , μ [ 0,1 ] , λ 1 2 , μ 1 2 . Suppose that the functions φ , ψ : I ( 0, + ) is differentiable, φ strictly increasing, ψ strictly decreasing and φ ψ is one-to-one. If B λ , μ [ φ , ψ ] is invariant with respect to the mean-type mapping ( B [ φ ] , B [ ψ ] ) i.e., the Equation (1.6) holds, then there exists a positive number c such that

ψ ( x ) = c φ ( x ) 1 2 λ 1 2 μ , x I . (2.3)

Proof. By the definition of the mean B λ , μ [ φ , ψ ] and (1.6) we have

( φ ψ ) 1 ( λ φ ( B [ φ ] ( x , y ) ) + ( 1 λ ) φ ( B [ ψ ] ( x , y ) ) μ ψ ( B [ φ ] ( x , y ) ) + ( 1 μ ) ψ ( B [ ψ ] ( x , y ) ) ) = ( φ ψ ) 1 ( λ φ ( x ) + ( 1 λ ) φ ( y ) μ ψ ( x ) + ( 1 μ ) ψ ( y ) ) , x , y I ,

whence, for all x , y I

( λ φ ( B [ φ ] ( x , y ) ) + ( 1 λ ) φ ( B [ ψ ] ( x , y ) ) ) ( μ ψ ( x ) + ( 1 μ ) ψ ( y ) ) = ( μ ψ ( B [ φ ] ( x , y ) ) + ( 1 μ ) ψ ( B [ ψ ] ( x , y ) ) ) ( λ φ ( x ) + ( 1 λ ) φ ( y ) ) (2.4)

Differentiating the above equation with respect to x, we get that

( λ φ ( B [ φ ] ) B [ φ ] x + ( 1 λ ) φ ( B [ ψ ] ) B [ ψ ] x ) ( μ ψ ( x ) + ( 1 μ ) ψ ( y ) ) + ( λ φ ( B [ φ ] ) + ( 1 λ ) φ ( B [ ψ ] ) ) μ ψ (x)

= ( μ ψ ( B [ φ ] ) B [ φ ] x + ( 1 μ ) ψ ( B [ ψ ] ) B [ ψ ] x ) ( λ φ ( x ) + ( 1 λ ) φ ( y ) ) + ( μ ψ ( B [ φ ] ) + ( 1 μ ) ψ ( B [ ψ ] ) ) λ φ (x)

Then, letting y = x , since B [ φ ] ( x , x ) = B [ ψ ] ( x , x ) = x and Lemma 1 we obtain

( 1 2 λ ) φ ( x ) ψ ( x ) = ( 1 2 μ ) φ ( x ) ψ ( x ) , x I , (2.5)

that is,

ψ ( x ) ψ ( x ) = 1 2 λ 1 2 μ φ ( x ) φ ( x ) . (2.6)

Thus we can get that (2.3) holds.

Theorem 1. Let I be an interval and λ , μ [ 0,1 ] , λ 1 2 , μ 1 2 . Suppose that the functions φ , ψ : I ( 0, + ) is twice differentiable, φ strictly increasing, ψ strictly decreasing and φ ψ is one-to-one. Then if the weighted Bajraktarević mean B λ , μ [ φ , ψ ] is invariant with respect to the mean-type mapping ( B [ φ ] , B [ ψ ] ) , that is (1.6) holds, then there exist a , b , p , q , p , q 0 , a , b > 0 , such that

φ ( x ) = a e p x , ψ ( x ) = b e q x , x I ;

where q = 1 2 λ 1 2 μ p .

Proof. Assume that B λ , μ [ φ , ψ ] is invariant with respect to the mean-type mapping ( B [ φ ] , B [ ψ ] ) . Then the equality (2.4) is satisfied. Differentiating two times (2.4) with respect to x, we get

( λ φ ( B [ φ ] ) ( B [ φ ] x ) 2 + ( 1 λ ) φ ( B [ ψ ] ) ( B [ ψ ] x ) 2 + λ φ ( B [ φ ] ) 2 B [ φ ] x 2 + ( 1 λ ) φ ( B [ ψ ] ) 2 B [ ψ ] x 2 ) ( μ ψ ( x ) + ( 1 μ ) ψ ( y ) ) + 2 ( λ φ ( B [ φ ] ) B [ φ ] x + ( 1 λ ) φ ( B [ ψ ] ) B [ ψ ] x ) μ ψ ( x ) + ( λ φ ( B [ φ ] ) + ( 1 λ ) φ ( B [ ψ ] ) ) μ ψ (x)

= ( μ ψ ( B [ φ ] ) ( B [ φ ] x ) 2 + ( 1 μ ) ψ ( B [ ψ ] ) ( B [ ψ ] x ) 2 + μ ψ ( B [ φ ] ) 2 B [ φ ] x 2 + ( 1 μ ) ψ ( B [ ψ ] ) 2 B [ ψ ] x 2 ) ( λ φ ( x ) + ( 1 λ ) φ ( y ) ) + 2 ( μ ψ ( B [ φ ] ) B [ φ ] x + ( 1 μ ) ψ ( B [ ψ ] ) B [ ψ ] x ) λ φ ( x ) + ( μ ψ ( B [ φ ] ) + ( 1 μ ) ψ ( B [ ψ ] ) ) λ φ (x)

Letting y = x and dividing φ ( x ) ψ ( x ) , since Lemma 1, we get that

( 1 4 λ ) φ ( x ) φ ( x ) ( 1 4 μ ) ψ ( x ) ψ ( x ) + ( 1 2 3 2 λ + 1 2 μ ) φ ( x ) φ ( x ) ψ ( x ) ψ ( x ) + λ 2 ( φ ( x ) φ ( x ) ) 2 1 μ 2 ( ψ ( x ) ψ ( x ) ) 2 = 0. (2.7)

From Formula (2.5), after simple calculations, we have

ψ ( x ) ψ ( x ) = 1 2 λ 1 2 μ φ ( x ) ψ ( x ) , ψ ( x ) ψ ( x ) = 1 2 λ 1 2 μ φ ( x ) φ ( x ) + 1 2 λ 1 2 μ ( 1 2 λ 1 2 μ 1 ) ( φ ( x ) φ ( x ) ) 2 .

Substituting them into Equation (2.7), we get that

φ ( x ) φ ( x ) ( φ ( x ) φ ( x ) ) 2 = 0 ,

that is

( φ ( x ) φ ( x ) ) = 0.

Solving this equation we obtain, for some a , p , p 0 , a > 0

φ ( x ) = a e p x . (2.8)

Since Lemma 2, we can get that ψ ( x ) = b e q x where q = 1 2 λ 1 2 μ p and b = c a > 0 .

Corollary 1. Let I be an interval and λ , μ [ 0 , 1 ] , λ 0 , μ 1 2 , λ + μ = 1 . Suppose that the functions φ , ψ : I ( 0, + ) is twice differentiable, φ strictly increasing, ψ strictly decreasing and φ ψ is one-to-one. Then the following conditions are equivalent:

1) B λ , μ [ φ , ψ ] is invariant with respect to the mean-type mapping ( B [ φ ] , B [ ψ ] ) , i.e.,

B λ , μ [ φ , ψ ] ( B [ φ ] , B [ ψ ] ) = B λ , μ [ φ , ψ ] ;

2) there exist a , b , p , p 0 , a , b > 0 , such that

φ ( x ) = a e p x , ψ ( x ) = b e p x , x I ;

3) there exist p , p 0 such that

B λ , μ [ φ , ψ ] ( x , y ) = x + y 2 , B [ φ ] ( x , y ) = x e p x + y e p y e p x + e p y , B [ ψ ] = x e p x + y e p y e p x + e p y

for all x , y .

Remark 1. For the case ( 1 2 λ ) ( 1 2 μ ) = 0 , since (2.5) and φ is strictly increasing, ψ is strictly decreasing, we have λ = μ = 1 2 . Then the Equation (2.7) becomes

φ ( x ) φ ( x ) ( φ ( x ) φ ( x ) ) 2 = ψ ( x ) ψ ( x ) ( ψ ( x ) ψ ( x ) ) 2 , x , y I . (2.9)

Then assuming φ , ψ are three times differentiable, we can find the result for this case in [21] .

Supporting

Funded by Longshan academic talent research supporting program of SWUST (17LZXY12) and Doctoral fund of SWUST (18zx7166, 15zx7142).

Cite this paper: Zhang, Q. (2018) Invariance of Weighted Bajraktarević Mean with Respect to the Beckenbach-Gini means. Journal of Applied Mathematics and Physics, 6, 2453-2460. doi: 10.4236/jamp.2018.612206.
References

[1]   Bajraktarevic, M. (1958) Sur uneéquationfonctionelle aux valeursmoyennes. 13, 243-248.

[2]   Bullen, P.S., Mitrinovic, D.S. and Vasic, P.M. (1988) Means and Their Inequalities, Mathematics and Its Applications, D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo.

[3]   Matkowski, J. (2011) Quotient Mean, Its Invariance with Respect to a Quasi-Arithmetic Mean-Type Mapping, and Some Applications. Aequationes Mathematicae, 82, 247-253.
https://doi.org/10.1007/s00010-011-0088-8

[4]   Matkowski, J. (1999) Invariant and Complementary Quasi-Arithmetic Means. Aequationes Mathematicae, 57, 87-107.
https://doi.org/10.1007/s000100050072

[5]   Sutô, O. (1914) Studies on Some Functional Equations I. Tohoku Mathematical Journal, 6, 1-15.

[6]   Sutô, O. (1914) Studies on some Functional Equations II. Tohoku Mathematical Journal, 6, 82-101.

[7]   Daróczy, Z., Maksa, G. and Páles, Z. (2000) Extension Theorems for the Matkowski-Sutô Problem. Demonstratio Mathematica, 33, 547-556.
https://doi.org/10.1515/dema-2000-0311

[8]   Daróczy, Z. and Páles, Z. (2001) On Means That Are Both Quasi-Arithmetic and Conjugate Arithmetic. Acta Mathematica Hungarica, 90, 271-282.
https://doi.org/10.1023/A:1010641702978

[9]   Daróczy, Z. and Páles, Z. (2002) Gauss-Composition of Means and the Solution of the Matkowski-Sutô Problem. Publicationes Mathematicae, 61, 157-218.

[10]   Burai, P. (2007) A Matkowski-Sutô Type Equation. Publicationes Mathematicae, 70, 233-247.

[11]   Daróczy, Z. and Hajdu, G. (2005) On Linear Combinations of Weighted Quasi-Arithmetic Means. Aequationes Mathematicae, 69, 58-67.
https://doi.org/10.1007/s00010-004-2746-6

[12]   Jarczyk, J. (2007) Invariance of Weighted Quasi-Arithmetic Means with Continuous Generators. Publicationes Mathematicae, 71, 279-294.

[13]   Jarczyk, J. and Matkowski, J. (2006) Invariance in the Class of Weighted Quasi-Arithmetic Means. Annales Polonici Mathematici, 88, 39-51.
https://doi.org/10.4064/ap88-1-3

[14]   Matkowski, J. (2005) Lagrangian Mean-Type Mappings for Which the Arithmetic mean Is Invariant. Journal of Mathematical Analysis and Applications, 309, 15-24.
https://doi.org/10.1016/j.jmaa.2004.10.033

[15]   Makó, Z. and Páles, Z. (2009) The Invariance of the Arithmetic Mean with Respect to Generalized Quasi-Arithmetic Means. Journal of Mathematical Analysis and Applications, 353, 8-23.
https://doi.org/10.1016/j.jmaa.2008.11.071

[16]   GLazowska, D. and Matkowski, J. (2007) An Invariance of the Geometric Mean with Respect to Lagrangian Means. Journal of Mathematical Analysis and Applications, 331, 1187-1199.
https://doi.org/10.1016/j.jmaa.2006.09.005

[17]   BLasińska-Lesk, J., GLazowska, D. and Matkowski, J. (2003) An Invariance of the Geometric Mean with Respect to Stolarsky Mean-Type Mappings. Results in Mathematics, 43, 42-55.
https://doi.org/10.1007/BF03322720

[18]   Zhang, Q. and Xu, B. (2011) An Invariance of Geometric Mean with Respect to Generalized Quasi-Arithemtic Means. Journal of Mathematical Analysis and Applications, 379, 65-74.
https://doi.org/10.1016/j.jmaa.2010.12.025

[19]   Zhang, Q. and Xu, B. (2017) On Some Invariance of the Quotient Mean with Respect to Makó-Páles Means. Aequationes Mathematicae, 91, 1147-1156.
https://doi.org/10.1007/s00010-017-0502-y

[20]   Jarczyk, J. and Jarczyk, W. (2018) Invariance of Means. Aequationes Mathematicae, 92, 801-872.
https://doi.org/10.1007/s00010-018-0564-5

[21]   Matkowski, J. (2012) Invariance of Bajraktarevic Mean with Respect to Quasi-Arithmetic Means. Publicationes Mathematicae Debrecen, 80, 441-455.
https://doi.org/10.5486/PMD.2012.5151

[22]   Matkowski, J. (2013) Invariance of the Bajraktarevic Means with Respect to the Beckenbach-Gini Means. Mathematica Slovaca, 63, 493-502.
https://doi.org/10.2478/s12175-013-0111-8

 
 
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