Difference Mappings Associated with Nonsymmetric Monotone Types of Fejér’s Inequality

Two mappings Lw and Pw, in connection with Fejér’s inequality, are considered for the convex and nonsymmetric monotone functions. Some basic properties and results along with some refinements for Fejér’s inequality according to these new settings are obtained. As applications, some special means type inequalities are given.


Introduction
In 1906, L. Fejér [1] proved the following integral inequalities known in the literature as Fejér's inequality: where f : [a, b] → R is convex and g : [a, b] → R + = [0, +∞) is integrable and symmetric to x = a+b 2 g(x) = g(a + b − x), ∀x ∈ [a, b] . If in (1) we consider g ≡ 1, we recapture the classic Hermite-Hadamard inequality [2,3]: In [4], two difference mappings L and P associated with Hermite-Hadamard's inequality have been introduced as follows: Some properties for L and P, refinements for Hermite-Hadamard's inequality and some applications were raised in [4] as well: Theorem 1 (Theorem 1 in [4]). Let f : I ⊂ R → R be a convex mapping on the interval I and let a < b be fixed in I • . Then, we have the following: (i) The mapping L is nonnegative, monotonically nondecreasing, and convex on [a, b] (ii) The following refinement of Hadamard's inequality holds: for each y ∈ [a, b]. (iii) The following inequality holds: for every t, s ∈ [a, b] and each α ∈ [0, 1].
Theorem 2 (Theorem 2 in [4]). Let f : I ⊂ R → R be a convex mapping on the interval I and let a < b be fixed in I • . Then, we have the following: (i) The mapping P is nonnegative and monotonically nondecreasing on [a, b].
(ii) The following inequality holds: (iii) The following refinement of Hadamard's inequality holds: The main results obtained in [4] (Theorems 1 and 2) are based on the facts that if f : [a, b] → R is convex, then for all x, y ∈ [a, b] with x = y we have (see, [5,6]): where f + (y) is the right-derivative of f at y. Motivated by the above concepts, inequalities and results, we introduce two difference mappings, L w and P w , related to Fejér's inequality: In the case that w ≡ 1, the mappings L w and P w reduce to L and P, respectively.
In this paper we obtain some properties for L w and P w that imply some refinements for Fejér's inequality in the case that w is a nonsymmetric monotone function. Also, our results generalize Theorems 1 and 2 from Hermite-Hadamard's type to Fejér's type. Furthermore as applications, we find some numerical and special means type inequalities.
To obtain our respective results, we need the modified version of Theorem 5 in [7] which includes the left and right part of Fejér's inequality in the monotone nonsymmetric case.
Theorem 3. Let f : I ⊂ R → R be a convex function on the interval I and differentiable on I • . Consider a, b ∈ I • with a < b such that w : [a, b] → R is a nonnegative, integrable and monotone function. Then The main point in Theorem 3 (1) (w (x) ≤ 0), is that we have (2) for any x, y ∈ [a, b] with f (x) ≤ f (y) without the need for w to be symmetric with respect to x+y 2 . Also similar properties hold for other parts of the above theorem.
It is clear that f is convex and w is nonsymmetric and decreasing. If we consider 0 < x ≤ y, then from the fact that (y − x) 2 ≥ 0 we obtain that This inequality implies that 2 It follows that 2 shows that f and w satisfy (3) on [x, y], where w is not symmetric. Also, we can see that f and w satisfy (2).

Main Results
The first result of this section is about some properties of the mapping L w where the function w is nonincreasing.
(iii) The following refinement of (2) holds: for any y ∈ [a, b] with f (a) ≤ f (y). (iv) If f is nondecreasing, then the following inequality holds: Furthermore when | f | is convex on [a, b], then: Proof. (i) We need only the inequality for all t ∈ [a, b]. This happens according to Theorem 3 (1).
(ii) Without loss of generality for a ≤ y < x < b consider the following identity: Dividing with "x − y" and then letting x → y we obtain that Also from the convexity of f we have which, along with the fact that w is nonincreasing, implies that So from (9) and (10) we get x y w(s)ds.
On the other hand from (8) and Theorem 3 (1), we have x y w(s)ds, and, along with (11), we obtain that This implies the convexity of L w (t).
For the fact that L is monotonically nondecreasing, from convexity of f on [a, b] we have for all y ∈ [a, b] and so for any x > y.
(iii) Since L w is monotonically nondecreasing we have 0 ≤ L w (y) ≤ L w (b), for all y ∈ [a, b] and so Also, by the use of Theorem 3 (1) we get (12) and (13), we have the result.
(iv) Since L w is convex, then from the fact that for any u, v ∈ [a, b] and each t ∈ [0, 1], we have the result.
(v) The following identity was obtained in [8]: for any t ∈ [a, b] where Since w is nonincreasing, then we obtain Now by the use of (15) in (14) we get for any t ∈ [a, b]. Using the change of variable x = sa + (1 − s)t and some calculations imply that , then from (16) and by the use of the change of variable x = sa + (1 − s)t we get The following result is including some properties of the mapping P w in the case that w is nondecreasing.
Theorem 5. Let f : I ⊂ R → R be a convex function on the interval I and differentiable on I • . Consider a, b ∈ I • with a < b such that w : [a, b] → R is a nonnegative and continuous function with w (x) ≥ 0 for all a ≤ x ≤ b. Then Furthermore when | f | is convex on [a, b], then: (iv) The following inequality holds: provided that f (a) ≤ f a+t 2 for all t ∈ [a, b]. (v) If for any x < y we have f (x) ≤ f ( x+y 2 ), then the following refinement of (3) holds: Proof. (i) It follows from Theorem 3 (2).
(ii) Suppose that a ≤ x < y < b. So from Theorem 3 (2) and the facts that w is nondecreasing and f is convex, we get Since P w is nondecreasing we have P w (t) ≤ P w (b) for all t ∈ [a, b], i. e.
Then we have the right side of (20).

Remark 2.
(i) By the use of Theorem 3 (2) (w is nonincreasing on [a, b]) in the proof of Theorem 5, we can obtain some different properties for P w with new corresponding results. The details are omitted.
(ii) Theorem 5 gives a generalization of Theorem 2, along with some new results.

Applications
The following means for real numbers a, b ∈ R are well known: A(a, b) = a + b 2 arithmetic mean, L n (a, b) = b n+1 − a n+1 (n + 1)(b − a) 1 n generalized log−mean, n ∈ R, a < b.
The following result holds between the two above special means: Theorem 6. For any a, b ∈ R with 0 < a < b and n ∈ N we have