Let be an open interval. A two-variable function is called a mean on the interval I if
holds. If for all , these inequalities are strict, M is called strict. Obviously, if M is a mean, then M is reflexive, i.e., for all .
A quasi-arithmetic mean, generated by the function , is defined by
for a continuous, strictly monotone function .
A more general mean is the class of the weighted quasi-arithmetic means, which is defined by
where is a continuous strictly monotone function, and the constant .
A Lagrangian mean is defined by
where is a continuous strictly monotone function.
Given the continuous functions satisfy for and is one-to-one, the Bajraktarević mean of generators and  is defined by
is a strict mean, and it is a generalization of quasi-arithmetic mean. Note that if , we have
where the mean is called Beckenbach-Gini mean of a generator  .
Quotient mean is defined by
where the functions and are continuous, positive, and of different type of strict monotonicity in I  . For , we have , where is geometric mean.
Now we define the weighted Bajraktarević mean as follows:
where , are continuous, positive, and of different type of strict monotonicity and is one-to-one. Note that if , . If , the weighted Bajraktarević mean becomes quotient mean, that is . Without any loss of generality, we can assume that is strictly increasing and is strictly decreasing.
Let be means. A mean is called invariant with respect to the mean-type mappings , shortly, -invariant  , if
The simplest example when the invariance equation holds is the well-known identity
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   presented in 1914, in which he gave analytic solutions for the invariance equation
Then Matkowski solved the above equation under assumptions that and are twice continuously differentiable  . These regularity assumptions were weaken step-by-step by Daróczy, Maksa and Páles in   . Finally, without any regularity assumptions, the problem was solved by Daróczy and Páles in  .
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  . Daróczy, Hajdu, Jarczyk and Matkowski studied the invariance equation involving three weighted quasi-arithmetic means    . Matkowski solved the invariance equation involving the arithmetic mean in class of Lagrangian mean-type mappings  . In  , 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  . All pairs of Stolarsky’s means for which the geometric mean is invariant were determined in  . Zhang and Xu considered the invariance of the geometric mean with respect to generalized quasi-arithmetic means in  and some invariance of the quotient mean with respect to Makó-Páles means in  . Recently, Jarczyk provided a review on the invariance of means  .
Matkowski studied the invariance of the quotient mean with respect to weighted quasi-arithmetic mean type mapping  . He also studied the invariance of the Bajraktarević means with respect to quasi-arthmetic means in  and the invariance of the Bajraktarević means with respect to the Beckenbach-Gini means in  . 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 , are continuous functions and is strictly increasing, is strictly decreasing.
2. Main Result
Lemma 1. Let be an interval. Suppose that the function is differentiable, then we have
If the function is twice differentiable, then we have
Proof. By the definition of , we have
then let , we can get that .
Also we have
letting , we can get (2.2).
Lemma 2. Let be an interval and . Suppose that the functions is differentiable, strictly increasing, strictly decreasing and is one-to-one. If is invariant with respect to the mean-type mapping i.e., the Equation (1.6) holds, then there exists a positive number c such that
Proof. By the definition of the mean and (1.6) we have
whence, for all
Differentiating the above equation with respect to x, we get that
Then, letting , since and Lemma 1 we obtain
Thus we can get that (2.3) holds.
Theorem 1. Let be an interval and . Suppose that the functions is twice differentiable, strictly increasing, strictly decreasing and is one-to-one. Then if the weighted Bajraktarević mean is invariant with respect to the mean-type mapping , that is (1.6) holds, then there exist , such that
Proof. Assume that is invariant with respect to the mean-type mapping . Then the equality (2.4) is satisfied. Differentiating two times (2.4) with respect to x, we get
Letting and dividing , since Lemma 1, we get that
From Formula (2.5), after simple calculations, we have
Substituting them into Equation (2.7), we get that
Solving this equation we obtain, for some
Since Lemma 2, we can get that where and .
Corollary 1. Let be an interval and . Suppose that the functions is twice differentiable, strictly increasing, strictly decreasing and is one-to-one. Then the following conditions are equivalent:
1) is invariant with respect to the mean-type mapping , i.e.,
2) there exist , such that
3) there exist such that
for all .
Remark 1. For the case , since (2.5) and is strictly increasing, is strictly decreasing, we have . Then the Equation (2.7) becomes
Then assuming are three times differentiable, we can find the result for this case in  .
Funded by Longshan academic talent research supporting program of SWUST (17LZXY12) and Doctoral fund of SWUST (18zx7166, 15zx7142).
 Matkowski, J. (2011) Quotient Mean, Its Invariance with Respect to a Quasi-Arithmetic Mean-Type Mapping, and Some Applications. Aequationes Mathematicae, 82, 247-253.
 Matkowski, J. (2005) Lagrangian Mean-Type Mappings for Which the Arithmetic mean Is Invariant. Journal of Mathematical Analysis and Applications, 309, 15-24.
 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.
 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.
 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.
 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.