Weighted majorization inequalities for n-convex functions via extension of Montgomery identity using Green function

New identities and inequalities are given for weighted majorization theorem for n-convex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new generalizations of weighted majorization formulae are provided using Čebyšev type inequalities. Mean value theorems are also discussed for functional related to new results.


Definition 1.1
The nth order divided difference of a real-valued function f defined on an interval I of R at distinct points ξ i , ξ i+1 , . . . , ξ i+n for i ∈ N is defined recursively by The value ξ i , . . . , ξ i+n ; f is independent of the order of the points ξ i , ξ i+1 , . . . , ξ i+n . We can extend this definition by considering the cases in which two or more points coincide by taking respective limits.

Definition 1.3 A function f
: I → R is called convex of order n or n-convex if for all choices of (n + 1) distinct points ξ i , . . . , ξ i+n we have (n) f (ξ i ) ≥ 0.
when x ≺ y, x is said to be majorized by y.
This concept and notation of majorization were introduced by Hardy, Littlewood and Pólya [5]. Next, we state the well-known majorization theorem from [5]. The following weighted version of the majorization theorem was given by Fuchs in [4] (see also [11, p. 580] and [19, p. 323]). Proposition 1.6 Let p ∈ R m and let x, y be two nonincreasing real m-tuples such that Then, for any continuous convex function f : R → R, the following inequality holds Some integral inequalities in relation to majorization can be stated as follows. The following proposition is a consequence of Theorem 1 in [16] (see also [19, p. 328] hold, then for every continuous convex function f : R → R the following inequality holds then again inequality (1.7) holds.
In this article, we will state our results for nonincreasing x and y satisfying the assumption of Proposition 1.7, but they are still valid for nondecreasing x and y satisfying the above condition see for example [11, p. 584].
From [1], we have extracted here an extension of Montgomery identity via Taylor's formula which may be stated as follows. (1.9) In case n = 1, the sum n−2 k=0 · · · is empty, so identity (1.8) reduces to the famous Montgomery identity (see for instance [13]), where P (x, s) is the Peano kernel, defined by Finally, we define of Green function G : The function G is continuous and convex with respect to both t and s.
For any function f : we can obtain the following integral identity by simply using integration by parts (1.11) where the function G is defined as above in (1.10) (see also [24]). The aim of the present article is to introduce new weighted majorization theorems for convex functions via extension of Montgomery identity.

Majorization type identities and inequalities via extension of Montgomery identity
In this section, we state results related to weighted majorization identities and inequalities. First, we define notations in terms of positive linear functional: and by the notation where p i , ξ i , η i and f are defined in Proposition 1.6 and G is defined in (1.10). Also and by the notation ( p, ξ, η, G(·, s)) we would mean where p, ξ, η and f are as defined in Proposition 1.7 and G is defined in (1.10). Also "id" represents identity function, i.e., id(x) = x for all x.

Theorem 2.1
Let all the assumptions of Proposition 1.6 be valid. Also let f : we have the following identity: and G(·, s) is as defined in (1.10). We also have the following identity: where T n is defined in Proposition 1.9.
Proof Using (1.11) in (2.1) and using linearity of ( Differentiating (1.8) twice with respect to s, we get (2.8) Now using (2.8) in (2.7), we get and then applying Fubini's theorem for the last term to get (2.5).

Theorem 2.2 Let all the assumptions of Theorem 2.1 be valid with additional condition
where G is defined in (1.10) andT n in Theorem 2.1. Then for every n-convex function f : I → R, the following inequality holds: Proof Function f is n-convex, so we have f (n) ≥ 0. Using this fact and (2.10) in (2.5) we obtain required result.

Theorem 2.3 Let all the assumptions of Theorem
where G is as defined in (1.10) and T n as in Proposition 1.9. Then for every n-convex function f : I → R, the following inequality holds Proof Function f is n-convex, so we have f (n) ≥ 0. Using this fact and (2.12) in (2.6), we obtain required result.
Next, we state an important consequence.
Following is the integral version of our main results. Since proofs of these results are of similar nature we omit details.
whereT n is as in Theorem 2.1 and G is as defined in (1.10). Moreover, we also obtain the following identity: where T n is as in Proposition 1.9.

Theorem 2.6 Let all the assumptions of Theorem 2.5 hold with additional condition
where G is as defined in (1.10) andT n is defined in Theorem 2.1. Then, for every n-convex function f : I → R, the following inequality holds where G is as defined in (1.10) and T n is as in Proposition 1.9. Then for every n-convex function f : I → R the following inequality holds , ξ, η, G(·, s))ds , ξ, η, G(·, s) Here, we have another result.

Bounds for identities related to generalized linear inequalities
Let g, h : [a 0 , b 0 ] → R be two Lebesgue integrable functions. We consider theČebyšev functional defined by The following result can be found in [3]: Then we have the inequality The constant 1 √ 2 in (3.2) is the best possible.
Using aforementioned result, we are going to obtain generalizations of the result proved in previous section. Under the assumptions of Theorems 2.2, 2.3, 2.6 and 2.7, respectively, we define the following linear functionals Hence using these notations, we may defineČebyšev functional as follows (e.g., using ): where the remainder R 1 n ( f ; a 0 , b 0 ) satisfies the estimation Proof If we apply Proposition 3.1 for g → 1 and h → f (n) , then we obtain where R 1 n ( f ; a 0 , b 0 ) satisfies inequality (3.8). Now from identity (2.5), we obtain (3.7).
where the remainder R 2 n ( f ; a 0 , b 0 ) satisfies the estimation Finally, we obtain Ostrowski type inequalities for functions f whose n-th derivative belongs to L p spaces.
Proof Let us define functions and where Then, F 1 and F 2 are n-convex. Hence, k (F 1 ) ≥ 0 and k (F 2 ) ≥ 0 and If k ( f 0 ) = 0, then the statement obviously holds.
Applying Theorem 4.1 for function ω = k (h) f − k ( f )h, we get the following result.
assuming that both the denominators are non-zero. Remark 4.3 If the inverse of f (n) h (n) exists, then from the above mean value theorems we can give generalized means .
(4.6) Remark 4.4 Using the same method as in [8], we can construct new families of exponentially convex functions and Cauchy type means (see also [2]). Also, using the idea described in [8] we can obtain the results for n-convex functions at point.
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