New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity

The inequality of Popoviciu, which was improved by Vasić and Stanković (Math. Balk. 6:281-288, 1976), is generalized by using new identities involving new Green’s functions. New generalizations of an improved Popoviciu inequality are obtained by using generalized Montgomery identity along with new Green’s functions. As an application, we formulate the monotonicity of linear functionals constructed from the generalized identities, utilizing the recent theory of inequalities for n-convex functions at a point. New upper bounds of Grüss and Ostrowski type are also computed.


Introduction
Higher order convexity was introduced by Popoviciu, who defined it under the context of divided differences of a function (see Ch., []). Inequalities of higher order convex functions are very important and many physicists used them while dealing with higher dimensions. It is interesting to note that results for convex functions may not be true for convex functions of higher order. There are remarkable changes in the results, which force to think about the existence of such results. Butt  In order to obtain our main results in the present paper, we use the generalized Montgomery identity via Taylor's formula given in paper [].
Theorem  Let n ∈ N, ψ : I → R be such that ψ (n-) is absolutely continuous, I ⊂ R is an open interval, δ  , δ  ∈ I, δ  < δ  . Then the following identity holds: In the case n = , the sum n- l= · · · is empty, so identity () reduces to the well-known Montgomery identity (see, for instance, []) where P(z, ξ ) is the Peano kernel defined by Remark  It is important to note that R n ≥  for even n, as The complete reference concerning the Abel-Gontscharoff polynomial and theorem for 'two-point right focal' problem is given in []. As a special choice for n = , the Abel-Gontscharoff polynomial for 'two-point right focal' interpolating polynomial can be given as where G , (z, w) is the Green's function for 'two-point right focal problem' given as In the next section, we present our main results by introducing some new types of Green's functions.

Main results
We start this section by our nice observation about Abel-Gontscharoff identity () and the related Green's function for 'two-point right focal problem' . Therefore, keeping in view the Abel-Gontscharoff Green's function for 'two-point right focal problem' , we would like to introduce some new types of Green's functions G k : The graphical representations of G k , k = , , , , are depicted in Figure  which shows that all four Green's functions are continuous and symmetric. Moreover, all functions are convex with respect to both the variables z and w. These new Green's functions enable us to introduce some new identities, stated in the form of the following lemma.
Proof We can give the proofs of the above identities by following the same integrating scheme. Therefore we would like to give the proof of () only. As Now by simplifying terms, we will get our identity ().
Remark  Lemma  gives another proof of the special case of Abel-Gontscharoff identity (). G  and G  are new Green's functions, but the results are not so simple as in other two cases.
The inequality of Popoviciu, which was improved by Vasić and Stanković [], is generalized by using the above new Green's functions. In Theorem  we have that q i (i = , . . . , s) are positive real numbers. Now we give the generalization of that result for real values of q i (i = , . . . , s) with s i= q i =  using the new Green's functions G k , k = , , , , as defined in Lemma .
Then the following statements are equivalent: Proof (i) ⇒ (ii): Let (i) be valid. Fix k = , , , . Then as the functions for all k G k (·, w) (w ∈ [δ  , δ  ]) are also continuous and convex, it follows that for these functions () also holds for each fix k, i.e., () is valid.
is valid for each k = , , , , then it follows that for every convex function Here we can eliminate the differentiability condition due to the fact that it is possible to approximate uniformly a continuous convex function by convex polynomials (see [], p.).
Analogous to the above proof, we can give the proof of the last part of our theorem.
Next we formulate generalized identities with the help of identities defined in Lemma  and Fink's identity.
Theorem  Let all the assumptions of Theorem  be valid with n > , and let m, s ∈ N, with R n (·, v) and G k (·, w), (k = , , , ) be the same as defined in () and Lemma , respectively. Then we have the following new identities for k = , , , : Differentiating () twice with respect to the first variable, we have By executing Fubini's theorem in the last term, we have () respectively for k = , , , .
Next, using formula () on the function ψ , replacing n by n - (n ≥ ) and rearranging the indices, we have Similarly, using () in () and employing Fubini's theorem, we get () respectively for k = , , , .
As an application of the above obtained identities, the next theorem gives artistic generalization of Popoviciu-type inequalities for n-convex functions involving new Green's functions.
Theorem  Let all the assumptions of Theorem  be satisfied and n ≥ . Also let ψ be an n-convex function such that ψ (n-) is absolutely continuous. Then we have the following two results: If for k = , , , , then Proof Fix k = , , , . Since ψ (n-) is absolutely continuous on [δ  , δ  ], ψ (n) exists almost everywhere. As ψ is n-convex, so ψ (n) (z) ≥  for all z ∈ [δ  , δ  ] (see [], p.). Hence we can apply Theorem  to obtain () and () respectively.

Remark 
OR () be satisfied and Then we have POP z, q; ψ(z) ≥ .

()
Proof It is clear from Figure  that Green's function G k (z, w) is convex for all k = , , , , and the weights are assumed to be positive. Therefore, applying Theorem  and taking into account Remark , we can obtain POP[z, q; G k (z, w)] ≥  for all k = , , , .

Applications to (n + 1)-convex functions at a point
In the present section we give related results for the class of (n + )-convex functions at a point introduced in [].

A function is (n + )-convex on an interval if and only if it is (n + )-convex at every point of the interval (see []). Pečarić, Praljak and Witkowski in [] studied necessary and sufficient conditions on two linear functionals : C([δ  , c]) → R and : C([c, δ  ]) → R so that the inequality (ψ) ≤ (ψ) holds for every function ψ that is (n+)-convex at point c.
In the present section we give inequalities of such type for the particular linear functionals obtained from the inequalities in the previous section. Let σ i denote the monomials σ i (z) = z i , i ∈ N  . For the rest of this section, k (ψ) and k (ψ) for fixed k = , , ,  will denote the linear functionals obtained as the difference of the L.H.S. and R.H.S. of inequality (), applied to the intervals [δ  , c] and [c, δ  ], respectively, i.e., for It is important to notify that by introducing new linear functionals for k = , , , , k (ψ) and k (ψ), identity () applied to the respective intervals [δ  , c] and [c, δ  ] takes the shape: Now we are ready to state the following theorem for inequalities involving (n + )-convex function at a point.
Proof Using Definition , construct the function (z) = ψ(z) -Z c n! σ n in such a way that the function is n-concave on [δ  , c] and n-convex on [c, δ  ]. Fix k = , , , , and applying Theorem  to on the interval [δ  , c], we have Analogously applying Theorem  to on the interval [c, δ  ], we get Moreover, identities () and () applied to the function σ n gives Therefore assumption () is equivalent to So, from () and (), we obtain the desired result.
Remark  In the proof of Theorem , we have shown that for k = , , , , More importantly, inequality () still holds if we replace assumption () with the weaker assumption that is Z c ( k (σ n )k (σ n )) ≥ .
We conclude this section by adding the following remark.
Remark  Similar results can also be given by constructing linear functionals from inequality () involving new Green's functions G k for k = , , , .

New upper bounds of Grüss and Ostrowski type for generalized identities
In the present section we use Čebyšev's functional defined for Lebesgue integrable func- to construct some new upper bounds.
The following inequalities of Grüss type were given in [].
Then the inequality holds with  √  being the best possible constant.
and F  : [δ  , δ  ] → R be a monotonic nondecreasing function. Then the inequality holds with the best possible constant   .
In the sequel, we consider the above theorems to construct new estimations of generalized identities proved earlier.
In what follows, we let for k = , , , , First we express some Ostrowski-type inequalities affiliated with our generalized Popoviciu's inequality.
The constants on the R.H.S. of () and () are sharp for  < r ≤ ∞ and best possible for r = .
Proof Fix k = , , , . Rearrange identity () in such a way that Employing the classical Holder's inequality to R.H.S. of () yields (). The proof for sharpness is similar to Theorem . in [] (see also []). The proof of () is similar to that of (), but we utilize identity () instead of ().
Next we give some upper bounds of Grüss type.
where k (·) is defined by (). Theorem  enables us to define Cauchy means for (k = , , , ), in fact , means that ξ is the mean of δ  , δ  for given functions ψ and μ. We conclude our paper with the following remark.

Conclusions
By integration techniques new Green's functions are constructed, which are convex symmetric and continuous. Graphical representation of these new Green's functions is also included. These new Green's functions are then used to extend the inequality of Popoviciu given by Vasić and Stanković from nonnegative to real weights. Generalized identities are obtained using generalized Montgomery identity and new Green's functions which further establish the extension of Popoviciu inequality from a convex function to higher order convex functions along with real weights. The obtained results are then applied to establish the monotonicity of the linear functionals constructed from generalized inequalities.
New upper bounds are obtained using the Čebyšev functional. A new way is introduced to construct new n-exponential and logarithmic convex functions, which are then further used to give the Stolarsky means.