On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function

In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the ‘two-point right focal’ problem. We use Fink’s identity and a new Abel-Gontscharoff-type Green’s function for a ‘two-point right focal’ to generalize the refinement of Jensen’s inequality given in (Horváth and Pečarić in Math. Inequal. Appl. 14: 777-791, 2011) from convex function to higher order convex function. Also we formulate the monotonicity of the linear functional obtained from these identities using the recent theory of inequalities for n-convex function at a point. Further we give the bounds for the identities related to the generalization of the refinement of Jensen’s inequality using inequalities for the Cebyšev functional. Some results relating to the Grüss and Ostrowski-type inequalities are constructed.


Introduction and preliminary results
Divided difference is a helpful tool when we are dealing with functions that have different degrees of smoothness. In [], p., the divided difference is given as follows.
Definition  Let g be a real valued function defined on [α, β]. For r +  distinct points u  , u  , . . . , u r ∈ [α, β], the rth order divided difference is defined recursively by This is equivalent to [u  , u  , . . . , u r ; g] = r j= g(u j ) w (u j ) , where w(u) = r j= (uu j ).
We can include the case when some or all points are the same. In this case [u, u, . . . , u l-times where f (l-) is supposed to exist. The r-convex function is characterized by the rth order divided difference as follows (see [], p.).

Definition  A function
Introduce the sets I l ⊂ {, . . . , n} l (m - ≥ l ≥ ) inductively by Obviously the set I  = {, . . . , n} by (H  ) and this ensures that For m ≥ l ≥ , and, for any (j  , . . . , j l- ) ∈ I l- , let With the help of these sets they define the functions η I m ,l : They define some special expressions for  ≤ l ≤ m, as follows: and prove the following theorem.
Theorem . Assume (H  ), and let f : . , x n ∈ I and p  , . . . , p n are positive real numbers such that n i= p i = , then In [], A. M. Fink gave the following result.
continuous then the following identity holds: where The complete reference about Abel-Gontscharoff polynomial and theorem for 'two- The Abel-Gontscharoff polynomial for 'two-point right focal' interpolating polynomial for n =  can be given as where In the next section, we will present our main results by introducing some new types of Green functions defined as which enables us to introduce some new Abel-Gontscharoff-type identities, stated in the following lemma.
Proof The proofs of these identities requires some simple integration scheme, therefore we just give the proof of () only as follows: Simplifying we get the result (). We define the following functionals by taking the differences of refinement of Jensen's inequality given in (): (i) For k = , ,  we have the following identities: Proof (i) Using Abel-Gontsharoff-type identities (), (), () in i (f ), i = , , and using properties of i (f ), we get From identity (), we get Using () and () and applying Fubini's theorem we get the result () for k = , , .
for k = , , . If f is an m-convex function, then (i) For k = , , , the following holds:

Bounds for identities related to generalization of refinement of Jensen's inequality
For two Lebesgue integrable functions f  , f  : [α  , α  ] → R, we consider the Čebyšev functional where the integrals are assumed to exist. In [], the following theorems are given.
The constant  √  in () is the best possible.
The constant   in () is the best possible. Now we consider Theorem . and Theorem . to generalize results given in previous section. Let us first denote for ζ ∈ [α  , α  ]