Levinson-type inequalities via new Green functions and Montgomery identity

Abstract In this study, Levinson-type inequalities are generalized by using new Green functions and Montgomery identity for the class of k-convex functions (k ≥ 3). Čebyšev-, Grüss- and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, a new functional is introduced based on f {\mathfrak{f}} divergence and then some estimates for new functional are obtained. Some inequalities for Shannon entropies are obtained too.


Introduction and preliminaries
The theory of convex functions has encountered a fast advancement. This can be attributed to a few causes: first, applications of convex functions are directly involved in modern analysis; second, many important inequalities are results of applications of convex functions, and convex functions are closely related to inequalities (see [1]). Divided differences are found to be very helpful when we are dealing with functions having different degrees of smoothness. The following definition of divided difference is given in [1, p. 14].

kth-order divided difference
The kth-order divided difference of a function [ ] → ζ ζ : , 1 2 f at mutually distinct points x 0 ,…,x k ∈ [ζ 1 ,ζ 2 ] is defined recursively by The following definition of a real valued convex function is characterized by the kth-order divided difference (see [1, p. 15]).
If this inequality is reversed, then f is said to be k-concave. If the inequality is strict, then f is said to be a strictly k-convex (k-concave) function.
Note that 0-convex functions, 1-convex functions and 2-convex functions are non-negative functions, increasing functions and simply convex functions, respectively.
Functional form of (5) is defined as follows: Remark 1.1. It is essential to take note of that under the suppositions of Theorems A and B, if the function f is 3-convex, then ( (⋅)) ≥ J 0 i f for i = 1, 2.
In [6], (see also [3,  In [7], Mercer replaced the condition of symmetric distribution of points x ρ and y ρ with symmetric variances of points x ρ and y ρ .
In [8], Adeel et al. generalized Levinson's inequality for 3-convex functions; they also obtained some results for information theory. Similar results can be found in [9,10]. In [11], Witkowski showed the Levinson inequality with random variables. Furthermore, he showed that it is enough to assume that f is 3-convex and that assumption (7) can be weakened to inequality in a certain direction.
In [12], Pečarić et al. provided the probabilistic version of Levinson's inequality (2) under Mercer's assumption of equal variances for the family of 3-convex functions at a point. They showed that this is the largest family of continuous functions for which inequality (5) holds. The operator version of probabilistic Levinson's inequality was discussed in [13]. In [14], Pavic provided geometrical interpretation of different inequalities involving convex functions. Many other researchers generalized different inequalities for different classes of convex functions [15][16][17][18].
For our main results, we use the generalized Montgomery identity via Taylor's formula given in [19]. 1 2 I such that ζ 1 < ζ 2 . Then, the following identity holds: where , .
The error function ( ) e t can be represented in terms of the Green functions ( ) G t s , k , of the boundary value problem In [21], the following result holds.
, and let P F be its "two-point right focal" interpolating polynomial for ζ 1 ≤ a 1 < a 2 ≤ ζ 2 . Then, for k = 3 and p = 0, (12) becomes In [22], Mehmood et al. generalized Popoviciu-type inequalities via new Green functions and Montgomery identity. All generalizations that exist in the literature are only for one type of data points. But in this study, motivated by the above discussion Levinson-type inequalities are generalized via new Green functions and Montgomery identity involving two types of data points for higher order convex functions. Čebyšev-, Grüssand Ostrowski-type new bounds are also found for the functionals involving data points of two types.

Main results
Motivated by identity (6), we construct the following identities.

Generalization of Bullen-type inequalities for higher order convex functions
First, we define the following functional: , f is absolutely continuous. Then, we have the following new identities for t = 1, 2: , , and for t = 1, 2, G t (·,s) is defined in (14) and (16), respectively.
Using (25) in (24) and following the properties of ( (⋅)) J f , we get Execute Fubini's theorem in the last term to get (19) for t = 1, 2.
and if Proof. As function f is k-convex (k ≥ 3) and is k-times differentiable, so Then, we have Proof. It is clear that Green functions G t (·,s) defined in (14) and (16)  Next, we have a generalized form of Levinson-type inequality for 2n points given in [6] (see also [3]). □ Proof. Proof is similar to that of Theorem 2.3. □ In [7], Mercer made a significant improvement by replacing the condition x 1 + y 1 = … = x n + y n of symmetric distribution with the weaker one that the variances of the two sequences are equal.

Conclusion
This study is concerned with generalization of the Levinson-type inequalities (for real weights) for two types of data points implicating higher order convex functions. New Green functions and Montgomery identity are used for the class of k-convex functions, where k ≥ 3. As applications of new obtained results, Čebyšev-, Grüss-and Ostrowski-type new bounds are found. Moreover, the main results are applied to information theory via f-divergence and Shannon entropy. In future work, the main results can apply for other types of divergences and distances such as Rényi divergence, Rényi entropy and Zipf-Mandelbrot law.