Hermite–Hadamard–Fejér-Type Inequalities and Weighted Three-Point Quadrature Formulae

: The goal of this paper is to derive Hermite–Hadamard–Fejér-type inequalities for higher-order convex functions and a general three-point integral formula involving harmonic sequences of polynomials and w -harmonic sequences of functions. In special cases, Hermite–Hadamard–Fejér-type estimates are derived for various classical quadrature formulae such as the Gauss–Legendre three-point quadrature formula and the Gauss–Chebyshev three-point quadrature formula of the ﬁrst and of the second kind. formulae; higher-order convex functions; w -harmonic sequences of functions; harmonic sequences of polynomials


Introduction
The Hermite-Hadamard inequalities and their weighted versions, the so-called Hermite-Hadamard-Fejér inequalities, are the most well-known inequalities related to the integral mean of a convex function (see [1] (p. 138)).
where u : [a, b] → R is nonnegative, integrable and symmetric about a+b 2 . If h is a concave function, then the inequalities in (1) are reversed.
The importance and significance of our paper are reflected in the way in which we prove new Hermite-Hadamard-Fejér-type inequalities for higher-order convex functions and the general weighted three-point quadrature formula by using inequality (1), and a weighted version of the integral identity expressed by w-harmonic sequences of functions.
For this purpose, let us introduce the notations and terminology used in relation to w-harmonic sequences of functions (see [14]).
Let us consider a subdivision σ = {a = x 0 < x 1 < · · · < x m = b} of the segment [a, b], m ∈ N. Let w : [a, b] → R be an arbitrary integrable function. For each segment [x j−1 , x j ], j = 1, . . . , m, we define w-harmonic sequences of functions {w jk } k=1,...,n by: Further, the function W n,w is defined as follows: The following theorem gives a general integral identity (see [14]).
Further, let us recall the definition of the divided difference and the definition of an n-convex function (see [1] (p. 15)).
Definition 1. Let f be a real-valued function defined on the segment [a, b]. The divided difference of order n of the function f at distinct points x 0 , . . . , x n ∈ [a, b] is defined recursively by The value f [x 0 , . . . , x n ] is independent of the order of points x 0 , . . . , x n .
, n ≥ 0, if, for all choices of (n + 1) distinct points x 0 , . . . , x n ∈ [a, b], the n-th order divided difference in f satisfies From the previous definitions, the following property holds: if f is an (n + 2)-convex function, then there exists the n-th order derivative f (n) , which is a convex function (see, e.g., [1] (pp. 16, 293)).
The paper is organized as follows. After this introduction, in Section 2, we establish Hermite-Hadamard-Fejér-type inequalities for weighted three-point quadrature formulae by using the integral identity with w-harmonic sequences of functions, the properties of harmonic sequences of polynomials and the properties of n-convex functions. Since we deal with three-point quadrature formulae that contain values of the function in nodes x, a+b 2 and a + b − x and values of higher-ordered derivatives in inner nodes, the level of exactness of these quadrature formulae is retained. In Section 3, we derive Hermite-Hadamard-Fejér-type estimates for a generalization of the Gauss-Legendre three-point quadrature formula, and a generalization of the Gauss-Chebyshev three-point quadrature formula of the first and of the second kind.
Throughout the paper, the symbol B denotes the beta function defined by Γ denotes the gamma function defined as: is a hypergeometric function with γ > β > 0, z < 1.
In the paper, we assume that all considered integrals exist and that they are finite.

Hermite-Hadamard-Fejér-Type Inequalities for Three-Point Quadrature Formulae
In this section, we establish Hermite-Hadamard-Fejér-type inequalities for the weighted three-point formula using a weighted version of the integral identity expressed by w-harmonic sequences of functions that are given in Theorem 2 and the method that originated in [15].
Using Theorems 1 and 4, the properties of both n-convex functions and w-harmonic sequences of functions, and the method that originated in [15], in the next theorem, we derive new Hermite-Hadamard-Fejér-type inequalities for the weighted three-point quadrature Formula (7).
for each t ∈ [a, b] and x ∈ [a, a+b 2 ). Let the function W 2n,w , defined by (10), be nonnegative.
and A k and B k are defined as in Theorem 4. If W 2n,w (t, x) ≤ 0 or f is a (2n + 2)-concave function, then inequalities (11) hold with reversed inequality signs.
Further, let us assume that f is a (2n + 2)-concave function. Hence, the function − f (2n) is convex. Reversed signs in (11) are obtained by putting − f (2n) and the nonnegative function W 2n,w (t, x) in (1). This completes the proof.

Remark 2.
The value of U n,w (x) can be obtained from Theorem 3 by taking an appropriate subdivision of the segment [a, b] and applying the properties of functions w 1k , w 2k , w 3k and w 4k .
To get a maximum degree of exactness of quadrature Formula (7) for fixed x ∈ a, a+b 2 , we consider a sequence of harmonic polynomials {L j,x } j=0,1,...,n defined as follows: Therefore, we have A k (x) = 0, for k = 2, 3, 4, 5, 6 and B k (x) = 0, for k = 2, 3, 4. Finally, from identity (7), for x ∈ a, a+b 2 , we obtain the following three-point where Now, applying results from Theorem 5 to identity (15), we get the following results.
and let x ∈ [a, a+b 2 ). Let the function W 2n,w , defined by (10), be nonnegative and let L j,x be defined by (13).
Proof. The proof follows from Theorem 5 for the special choice of the polynomials L j,x . Therefore, for such a choice of x, we obtain the quadrature formula with three nodes, which is accurate for the polynomials of degree at most 5, and the approximation formula includes derivatives of order 6 and more.

Special Cases
Considering some special cases of the weight function w, in our results given in the previous section, we obtain estimates for the Gauss-Legendre three-point quadrature formula and for the Gauss-Chebyshev three-point quadrature formula of the first and of the second kind.

Gauss-Legendre Three-Point Quadrature Formula
Let us assume that w(t) = 1, t ∈ [a, b] and x ∈ a, a+b 2 . Now, from Theorem 4, we calculate and and B GL k (x) = 0, for even k > 1.

Proof.
A special case of Theorem 5 for w(t) = 1, t ∈ [a, b], and a nonnegative function W GL 2n defined by (19).
If we assume that the polynomials L j,x (t) are such that 3 (x) = 0. Thus, we obtain the following non-weighted three-point quadrature formulae: where In particular, according to Remark 3, for [a, b] = [−1, 1] and x = − √ 15 5 , we get B GL 5 (x) = 0, and there follows a generalization of the Gauss-Legendre three-point formula. Now, we derive Hermite-Hadamard-Fejér-type estimates for this generalization of the Gauss-Legendre three-point formula.