Schur Convexity and the Dual Simpson ’ s Formula

In this paper, we show that some functions related to the dual Simpson’s formula and BullenSimpson’s formula are Schur-convex provided that f is four-convex. These results should be compared to that of Simpson’s formula in Applied Math. Lett. (24) (2011), 1565-1568.


Introduction
Schur convexity is an important notion in the theory of convex functions, which were introduced by Schur in 1923 ( [1] [2]), its definition is stated in what follows.Let n R ≥ be denoted as, , , , ; and ( ) n R + ≥ be defined by, Then we recall (see, e.g., [3]- [5]) that a function f ( ) for all i, j {1, 2, , n} ∈  .Let f : R R I ⊆ → be a convex function defined on the interval I of real numbers and a, b I ∈ with a b < .The following inequality holds.This double inequality is called Hermite-Hadamard inequality for convex functions.Hermite-Hadamard inequality is improved though Schur convexity, c.f., [6]- [10].Among these paper, it is proven that if R I ∈ is an interval and f : I R → is continuous, then f is convex if and only if the mapping is an interval and f : I R → is continuous, then f is convex if and only if one of the following mappings which is called Simpson's formula, c.f. [14] and [15].For R I ∈ is an interval and f : In [15], the authors proved that if (4) : R f I → is continuous, then f is four-convex is equivalent to the mappings defined by ∫ is Schur-convex, this is an improvement of the Simpson's formula.
On the other hand, the dual Simpson's formula ( [14]) is stated as follows: if ( ) In [16], Bullen proved that, if f is four-convex, then the dual Simpson's quadrature formula is more accurate than Simpson's formula.That is, it holds that ( ) ( ) ( ) provided that f is four-convex.Now we can state our main results.In view of the dual Simpson's formula and the above Bullen-Simpson formula, we construct two mappings as follows: for b a ≠ , we set ( ) ( ) We shall show that if (4) : R f I → is continuous, then f is four-convex if and only if the mapping , S a b is Schur-convex.Obviously our results improve the dual-Simpson's formula and the Bullen- Simpson's formula, and hence complement the main result in [15].

Main Results
We now present our main theorem.( ) ( ) ( ) (e) For any a, b I ∈ with a b < , we have the dual Simpson inequality holds, i.e.: ( ) ( ) ( ) ( ) The equivalence of (a) (d) (g) was already proven in [15].Suppose that item (g) holds, then by the definition of the function , (by Simpson's formula (1.4) and four-convexity of f) hence, Here we denote ( ) ( ) Thus Hermite-Hadamard (1.2) holds for h(x) in a, 2 , this gives that ( ) , so by the criteria (1.1) 5 S is Schur-convex, item (b) is a consequence of item (g).Next we prove item (e) implies item (g).By item (e) and the dual Simpson's formula (1.6), we get , and a, b are arbitrary, it follows that f is four-convex.Now the equivalence of (b) (e) (g) is proven.We follow the same pattern to show the equivalence of (c) (f) (g).If item (c) holds, then , and a, b are arbitrary, item (g) follows again.It is only left to show that item (g) implies item (c).We give a lemma first.

Proof:
We only prove the first inequality.Denote that and that ( ) ( ) From the Hermite-Hadamard inequality for convex function ( ) g x , we see that ( ) Suppose that item (g) holds, by applying the lemma to f in , , , so by the criteria ( , S a b is Schur-convex, item (c) follows.
Remark 2.1.From Lemma 2.1, we add the two inequalities together to see that the following holds for fourconvex functions f: it is well-known, c.f., [14] or [15].Starting from this inequality (2.2), we deduce some properties for four-convex functions.As in the above, we define a pair of mappings 7 8 , S S by Proof: We observe that Here inequality (2.3) is due to inequality (2.2), and inequality (2.4) is a consequence of the Hermite-Hadamard inequality for convex function f ′′ , thus by the criteria (1.1) 8 S are Schur-convex on 2 I .Hence we get ( ) It is shown in [7] for a convex function g that the function is Schur-convex, specially we have ( )

9
, 0 S a b ≥ .We set g f′′ = , then it is convex, we see that RHS of inequa- lity (2.5) is non-negative, so by the criteria (1.1), 7 S is Schur-convex.Furthermore, we give a Schur-convexity theorem for the following mapping: Remark 2.2.For smooth four-convex functions, we see that both 8 S and 10 S are non-negative and Schur- convex functions, then the sum of 8 S and 10 S is also non-negative and Schur-convex function, especially it holds that ( ) function, and if I is an open interval and f : n I R → is symmetric and of class 1 C , then f is Schur-convex if and only if = case, which is no longer stated.) is Schur convex, and in this case, 1 (a, b)

6 ,
on I, then the following statements are equivalent: S a b is Schur-convex on 2 I .(d) For any a, b I ∈ with a b < , we have the Simpson inequality holds, i.e.: For any a, b I ∈ with a b < , we have the Bullen-Simpson inequality holds, i.e.: e., item (e) is valid if item (b) holds.
convex on I, then the following inequalities hold for any a, b I ∈ with b ≥ a:

6 ,
b ≥ a.The second inequality in the lemma is just the first inequality with b ≤ a, we omit its proof.The lemma is proven.Now we continue the proof of our main theorem.By the definition of ( ) S a b , we have

10 ,
≥ for convex function g f′′ = , as in the above, we can conclude that ( ) S a b are non- negative and Schur-convex.
For positive real numbers x, y, we denote the arithmetic mean, geometric mean, and logarithmic mean of x, y by A, G, L. Applying non-negativity of 7