Let I ⊂ ℝ 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 → ℝ .

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 → ℝ 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 → ℝ is a continuous strictly monotone function.

Given the continuous functions φ , ψ : I → ℝ 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 → ℝ 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 → ℝ 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 ⊂ ℝ , φ , ψ : I → ( 0, + ∞ ) are continuous functions and φ is strictly increasing, ψ is strictly decreasing.

Lemma 1. Let I ⊂ ℝ 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 ⊂ ℝ 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 ⊂ ℝ 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 ∈ ℝ ,   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 ∈ ℝ ,   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 ⊂ ℝ 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 ∈ ℝ ,   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 ∈ ℝ ,   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 ∈ ℝ .

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] .

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

The author declares no conflicts of interest regarding the publication of 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. https://doi.org/10.4236/jamp.2018.612206