Schur convexity of Bonferroni harmonic mean

In this paper, we research the Schur convexity, Schur geometric convexity and Schur harmonic convexity of the Bonferroni harmonic mean. Some inequalities identified with the Bonferroni harmonic mean are set up to represent the utilizations of the acquired outcomes.


Introduction
Arithmetic, Geometric and Harmonic means are three important means, which have been extensively used in the information aggregation [5,6,7,11,12,17,18,19,35,36]. For a collection of real numbers a i (i = 1, 2, … n), the Arithmetic mean (AM), the Geometric mean (GM) and the Harmonic mean (HM) are defined by: 1 3 respectively. The fundamental characteristic of arithmetic mean is that it focuses on the group opinions, while geometric mean gives more importance to the individual opinions and harmonic mean is the reciprocal of arithmetic mean, which is a conservative average to be used to provide for aggregation lying between the max and min operators and is widely used as a tool to aggregate central tendency data [30].
In the existing literature, the harmonic mean is generally considered as a fusion technique of numerical data information. However, in many situations the input arguments take the form of triangular fuzzy numbers because of time pressure, lack of knowledge, and peoples limited expertise related with problem domain. Therefore, "how to aggregate fuzzy data by using the harmonic mean?" is an interesting research topic and is worth paying attention too. So Xu [30] developed the fuzzy harmonic mean operators such as fuzzy weighted harmonic mean (FWHM) operator, fuzzy ordered weighted harmonic mean (FOWHM) operator and fuzzy hybrid harmonic mean (FHHM) operator and applied them to MAGDM. Wei [25] developed fuzzy induced ordered weighted harmonic mean (FIOWHM) operator and then based on the FWHM and FIOWHM operators, presented the approach to MAGDM. H. Sun and M. Sun [23] further applied the BM operator to fuzzy environment, introduced the fuzzy Bonferroni harmonic mean (FBHM) operator and the fuzzy ordered Bonferroni harmonic mean (FOBHM) operator and applied the FOBHM operator to multiple attribute decision making.
The Bonferroni mean operator was initially proposed by Bonferroni [2] and was also investigated intensively by Yager [32].
Beliakov et al. [1] further extended the BM operator by considering the correlations of any three aggregated arguments instead of any two.

Definition 1.2 [1]
Let p, q, r > 0, p + q + r ≠ 0 and let a i (i = 1, 2, … n) be a collection of non-negative numbers. If (1.2) GBM p,q,r a 1 , a 2 , … a n = 1 n(n − 1)(n − 2) Schur convexity of Bonferroni harmonic mean then GBM p,q,r is called the generalized Bonferroni mean (GBM) operator. In particular, if r = 0, then the GBM operator reduces to the BM operator. However, it is noted that both BM operator and the GBM operator do not consider the situation that i = j or j = k or i = k, and the weight vector of the aggregated arguments is not also considered. To overcome this drawback, Xia et al. [29]. defined the weighted version of the GBM operator.

Definition 1.3 [29]
Let p, q, r > 0, p + q + r ≠ 0 and let a i (i = 1, 2, … n) be a collection of non-negative numbers with the weight vector w = w 1 , w 2, … w n T such that If then GWBM p,q,r is called the generalized weighted Bonferroni mean (GWBM) operator. Some special cases can be obtained as the change of the parameters as follows: Case 1 If r = 0 then the GWBM operator reduces to the following: which is the weighted Bonferroni mean (WBM) operator.

Case 2
If q = 0 and r = 0 , then the GWBM operator reduces to the following: which is the generalized weighted averaging operator. Further in this case, let us look at the GWBM operator for some special cases of p.
1. If p = 1 , the GWBM operator reduces to the weighted averaging (WA) operator. 2. If p → 0 , then the GWBM operator reduces to the weighted geometric (WG) operator. 3. If p → +∞ , then the GWBM operator reduces to the max operator.
To aggregate the triangular fuzzy correlated information, based on the BM and weighted harmonic mean operators, H. Sun and M. Sun [23] developed the fuzzy Bonferroni harmonic mean operator. Since this operator considers the weight vector of the aggregated arguments, we redefine this operator as fuzzy weighted Bonferroni harmonic mean operator.
In particular, considering the triangular fuzzy numbers.
then the FWBHM operator (10) reduces to the uncertain weighted Bonferroni harmonic mean (UWBHM) operator as follows: Schur convexity of Bonferroni harmonic mean then the UWBHM operator reduces to the uncertain Bonferroni harmonic mean (UBHM) operator as follows: If a L i = a U i = a i for all, then the UBHM operator reduces to the weighted Bonferroni harmonic mean (WBHM) operator as follows: then the WBHM operator reduces to the Bonferroni harmonic mean (BHM) operator as follows: In recent years, the Schur convexity of functions relating to special means is a very significant research subject and has attracted the interest of many mathematicians. As supplements to the Schur convexity of functions, the Schur geometrically convex functions and Schur harmonically convex functions were investigated [8,21,26,27].
In [9]. the authors discussed the Schur convexity, Schur geometric convexity, Schur harmonic convexity and Schur m-power convexity of the geometric Bonferroni mean.
This motivated us to determine the Schur convexity, Schur geometric convexity, Schur harmonic convexity and Schur m-power convexity of the Bonferroni harmonic mean.
Our main results are as follows.

Theorem 1.1 For fixed non-negative real numbers p, q with
is Schur concave, Schur geometric convex and Schur harmonic convex on R n ++ ∶ = (0, +∞) n .

Preliminaries
We begin with recalling some basic concepts and notations in the theory of majorization. For more details, we refer the reader to [2,32].
x is said to be majorized by y (in symbols x ≺ y.), and y [1 ] ≥ … ≥ y [n ] are rearrangement of x and y and y in a descending order.

Let ⊆ R n , the function ∶ → R n is said to be schur convex function on if
x ≺ y on implies (x) ≤ (y) . is said to be a Schur concave function on , if and only if − is Schur convex function.

1.
⊆ R n is called geometrically convex set, if (x ∝ 1 y 1 , … , x n y n ) ∈ R n for any x and y ∈ , where , ∈ [0, 1] with + = 1. 2. Let ⊆ R n + the function ∶ → R + is said to be schur geometrically convex function on if (log x 1 , log x 2 … log x n )(log y 1 , log y 2 … log y n ) on implies (x) ≤ (y) . is said to be a Schur geometrically concave function on if and only if − is Schur geometrically convex function.

A set ⊆ R n . is said to be a harmonically convex set, if
for any x and y ∈ , and λ ∈ [0, 1]. holds for any x = x 1 , x 2 , x 3 … , x n ∈ 0 . Lemma 2.2 Let ⊆ R n be a symmetric geometrically convex set with non empty interior 0 . Let ∶ → R + be continuous on and differentiable on 0 . Then is Schur gemetrically convex (concave) function x = x 1 , x 2 , x 3 … , x n ∈ 0 if and only if is symmetric on and Lemma 2.3 Let ⊆ R n be symmetric harmonically convex set with non empty interior 0 Let ∶ → R + be continuous on . and differentiable on 0 Then . is Schur harmonically convex (concave) function x = x 1 , x 2 , x 3 … , x n ∈ Ω 0 . if and only if is symmetric on and.

The Bonferroni harmonic mean (BHM) is defined by
Taking the natural logarithm gives Which implies that Theorem 4.1 is proved.
A n (x) ≥ BHM p,q (x) Theorem 4.2 For fixed non-negative real numbers p, q with p + q ≠ 0 , and let c be a constant satisfying 0 ≤ c < A n (x), (X − c) = (x 1 − c, x 2 − c, … x n − c) then for arbitrary x ∈ R n ++ .
Proof By the majorization relationship given in Lemma (2.6), From Theorem (1.1) i.e., which implies that Theorem 4.2 is proved.

Conclusion
We prove the Bonferroni mean BHM p,q by introducing non-negative parameters p, q under the condition of Schur concave, Schur geometric convex and Schur harmonic convex on R n ++ . As an application of the Schur convexity, we establish two inequalities for generalized geometric Bonferroni mean BHM p,q . For details, we refer the interested reader to [15,16,24,28,33,34] and the references therein )BHM p,q (x)