Invariance of Weighted Bajraktarević Mean with Respect to the Beckenbach-Gini means

Under some conditions on the functions φ and ψ defined on I, the weighted Bajraktarević mean is given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , , 1 , : , , , 1 x y B x y x y I x y φ ψ λ μ λφ λ φ φ ψ μψ μ ψ −   + −   = ∈       + −     where [ ] , 0,1 λ μ ∈ . In this paper, we study the invariance of the weighted Bajraktarević mean with respect to Beckenbach-Gini means.

A more general mean is the class of the weighted quasi-arithmetic means, which is defined by where : I ϕ →  is a continuous strictly monotone function, and the constant ( ) A Lagrangian mean is defined by Given the continuous functions , : and ϕ ψ is one-to-one, the Bajraktarević mean of generators ϕ and ψ [1] is defined by is a strict mean, and it is a generalization of quasi-arithmetic mean.Note that if where the mean [ ] B ψ is called Beckenbach-Gini mean of a generator ψ [2].

Quotient mean [ ]
, 2 where the functions ϕ and ψ are continuous, positive, and of different type of strict monotonicity in I [3].For ( ) ( ) ( ) x y xy ϕ ψ = =  , where  is geometric mean.Now we define the weighted Bajraktarević mean as follows: . Without any loss of generality, we can assume that ϕ is strictly increasing and ψ is strictly decreasing.
where , ,    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 Then Matkowski solved the above equation under assumptions that ( ) ( ) x ψ 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 where I ⊂  , ( ) , : 0, I ϕ ψ → +∞ are continuous functions and ϕ is strictly increasing, ψ is strictly decreasing.

Main Result
If the function , . 2 Proof.By the definition of [ ] B ϕ , we have , , B x y x x y x x y y x y Lemma 2. Let I ⊂  be an interval and Differentiating the above equation with respect to x, we get that Then, letting y x and Lemma 1 we obtain Thus we can get that (2.3) holds.
Theorem 1.Let I ⊂  be an interval and that the functions ( ) , : 0, I ϕ ψ → +∞ is twice differentiable, ϕ strictly increasing, ψ strictly decreasing and ϕ ψ is one-to-one.Then if the weighted Bajraktarević is invariant with respect to the mean-type mapping , that is (1.6) holds, then there exist , , , x a x b x I is invariant with respect to the mean-type . Then the equality (2.4) is satisfied.Differentiating two times (2.4) with respect to x, we (1 ) From Formula (2.5), after simple calculations, we have Solving this equation we obtain, for some , , 0, 0  Then assuming , ϕ ψ are three times differentiable, we can find the result for this case in [21].

Supporting
Let I ⊂  be an open interval.A two-variable function 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., arithmetic mean, generated by the function ϕ , is defined by , strictly monotone function : I ϕ →  .
 are continuous, positive, and of different type of strict monotonicity and ϕ ψ is one-to-one.Note that if 1 2

Let
invariant with respect to the mean-type mappings ( ) , M N , shortly, ( )

[
+∞ is differentiable, ϕ strictly increasing, ψ strictly decreasing and ϕ ψ is one-to-one.If [ ] the definition of the mean[ ]

1 .
Let I ⊂  be an interval and [ ]