On generalization of Levinson’s inequality involving averages of 3-convex functions

By using an integral arithmetic mean, a generalization of Levinson’s inequality given in (Pečarić et al. in Convex Functions, Partial Orderings, and Statistical Applications. Mathematics in Science and Engineering, vol. 187, 1992) and results from (Vukelić in Appl. Anal. Discrete Math. 14:670–684, 2020), we give extension of Wulbert’s result from (Wulbert in Math. Comput. Model. 37:1383–1391, 2003). Also, we obtain inequalities with divided differences for the functions of higher order.


Introduction
Suppose f is a continuous function defined on an interval I with a nonempty interior.Then, define x f (t) dt, x, y ∈ I, x = y, f (x), x = y ∈ I. (1.1) In a seminal work [14], Wulbert proved that the integral arithmetic mean F, defined in (1.1), exhibits convexity on the interval I 2 when the underlying function f is convex over the interval I.In a separate study [15], Zhang and Chu independently rediscovered this result without making any reference to Wulbert's findings.Their work revealed that the convexity of the integral arithmetic mean F hinges on the crucial condition that f must be convex on the interval I.
Since it will hold significant importance for our forthcoming analysis, let us take into consideration a real-valued function f defined on the interval [a, b].The divided difference of order n for the function f at distinct points x 0 , x 1 , . . ., x n ∈ [a, b] is defined recursively (as elucidated in [1,9]) in the following manner: The value f [x 0 , . . ., x n ] remains invariant regardless of the order in which the points x 0 , . . ., x n are arranged.
The definition can be further extended to accommodate scenarios where some (or all) of the points coincide.Provided that f (j-1) (x) exists, we establish the following notation: In the context of divided differences, the following holds: In conclusion, it is evident that the following property holds for divided differences: .
Under the condition that the function f has a continuous nth derivative on the interval [a, b], we can represent the divided difference f [x 0 , . . ., x n ] using integral notation (refer to [9, p. 15]) as i=0 u i .The notion of n-convexity is attributed to Popoviciu [10].For the present study, we adhere to the definition as presented by Karlin [6].
It is worth noting that Popoviciu's work demonstrated the fundamental result that any continuous n-convex function defined on the interval [a, b] can be represented as the uniform limit of a sequence of n-convex polynomials.Moreover, [7] provides an extensive collection of related results and essential inequalities attributed to Favard, Berwald, and Steffensen.
The proof of the Jensen inequality for divided differences can be found in [4]: is a convex function of the vector x = (x 0 , . . ., holds for all a i ≥ 0 such that m i=0 a i = 1.
In the context of future research, the notion of a generalized divided difference will hold relevance.Provided below is the definition for reference.
Consider a real-valued function f (x, y) defined on The divided difference of order (n, m) for the function f at distinct points x 0 , . . ., x n ∈ I and y 0 , . . ., y m ∈ J is defined as follows (see [9, p. 18]): where ω(x) = n i=0 (xx i ), w(y) = m j=0 (yy j ).Following the aforementioned definition, we can establish the concept of (n, m)convexity, which is as follows (see [9, p. 18]): If this inequality is strict, then f is said to be strictly (n, m)-convex.
In [11], Popoviciu presented and proved the following theorem: x n y m ≥ 0. (1.6) In this research, we build upon the generalization of Levinson's inequality, and thus, we begin by stating the fundamental Levinson's inequality as follows (see [8] and [12]): In [2], Bullen provided a proof for the generalization of Theorem 3: If f is strictly 3-convex there is equality in (1.9) if and only if ) holds for a continuous function f , (1.8) is satisfied by 2n-distinct points and It is shown in [9] that the condition (1.8) can be weakened, i.e., the following result holds: (1.12) Then (1.9) is valid.
The primary objective of this paper is to provide an extension of Wulbert's result, as presented in [14], for 3-convex functions.We will also consider relevant findings from [13].Moreover, we aim to establish an inequality involving divided differences by utilizing the generalization of Levinson's inequality given in [9].As a significant outcome, we will demonstrate the convexity of higher order for functions defined by divided differences.

Inequalities involving averages
Theorem 6 Let f be a real-valued 3-convex function on [a, b] and let F be defined in (1.1).Then for p i > 0 (1 and Proof Since the conditions from Theorem 5 are satisfied, by using inequality (1.9), we get Using the definition in (1.4), we get It is a known fact that if this property holds for all possible x, y, h > 0, then F is (3, 0)convex, as stated in [11].
If we put n = 2, Using the definition in (1.4), we get Continuing the previous arguments, since this property holds for all possible x, h 1 , y, h 2 > 0, we can deduce that F is (2, 1)-convex.
For similar results regarding Jensen's inequality involving averages of convex functions, refer to [3] and [5].Consequently,
Proof Since the conditions from Theorem 5 are satisfied, by using inequality (1.9), for the 3-convex function f (n) , we get As observed in the proof of Theorem 6, since this property holds for all possible y 0 , y 1 , y 2 , h > 0, we can conclude that G is (3, 0, 0)-convex.