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Article

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

Department of Mathematics, Josip Juraj Strossmayer University of Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, Croatia
Academic Editor: Janusz Brzdęk
Mathematics 2021, 9(15), 1720; https://doi.org/10.3390/math9151720
Received: 28 June 2021 / Revised: 17 July 2021 / Accepted: 20 July 2021 / Published: 22 July 2021
(This article belongs to the Special Issue Mathematical Inequalities with Applications)

Abstract

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.
Keywords: Hermite–Hadamard–Fejér inequalities; weighted three-point formulae; higher-order convex functions; w-harmonic sequences of functions; harmonic sequences of polynomials Hermite–Hadamard–Fejér inequalities; weighted three-point formulae; higher-order convex functions; w-harmonic sequences of functions; harmonic sequences of polynomials

1. 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)).
Theorem 1
(The Hermite–Hadamard–Fejér inequalities). Let h : [ a , b ] R be a convex function. Then
h a + b 2 a b u ( x ) d x a b u ( x ) h ( x ) d x 1 2 h ( a ) + 1 2 h ( b ) a b u ( x ) d x ,
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.
If u 1 , then we are talking about the Hermite–Hadamard inequalities.
Hermite–Hadamard and Hermite–Hadamard–Fejér-type inequalities have many applications in mathematical analysis, numerical analysis, probability and related fields. Their generalizations, refinements and improvements have been an important topic of research (see [1,2,3,4,5,6,7,8,9,10,11,12,13], and the references listed therein). In the past few years, Hermite–Hadamard–Fejér-type inequalities for superquadratic functions [2], GA-convex functions [7], quasi-convex functions [11] and convex functions [13] have been largely investigated in the literature.
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 j k } k = 1 , , n by:
w j 1 ( t ) = w ( t ) , t [ x j 1 , x j ] , w j k ( t ) = w j , k 1 ( t ) , t [ x j 1 , x j ] , k = 2 , 3 , , n .
Further, the function W n , w is defined as follows:
W n , w ( t , σ ) = w 1 n ( t ) , t [ a , x 1 ] , w 2 n ( t ) , t ( x 1 , x 2 ] , . . . w m n ( t ) , t ( x m 1 , b ] .
The following theorem gives a general integral identity (see [14]).
Theorem 2.
Let f : [ a , b ] R be such that f ( n ) is piecewise continuous on [ a , b ] . Then, the following holds:
a b w ( t ) f ( t ) d t = k = 1 n ( 1 ) k 1 w m k ( b ) f ( k 1 ) ( b ) + j = 1 m 1 w j k ( x j ) w j + 1 , k ( x j ) f ( k 1 ) ( x j ) w 1 k ( a ) f ( k 1 ) ( a ) + ( 1 ) n a b W n , w ( t , σ ) f ( n ) ( t ) d t .
In [15], the authors proved the following Fejér-type inequalities by using identity (4).
Theorem 3.
Let f : [ a , b ] R be ( n + 2 ) -convex on [ a , b ] and f ( n ) piecewise continuous on [ a , b ] . Further, let us suppose that the function W n , w , defined in (3), is nonnegative and symmetric about a + b 2 (i.e., W n , w ( t , σ ) = W n , w ( a + b t , σ ) ). Then
U n ( σ ) · f ( n ) a + b 2 ( 1 ) n a b w ( t ) f ( t ) d t k = 1 n ( 1 ) k 1 w m k ( b ) f ( k 1 ) ( b ) + j = 1 m 1 w j k ( x j ) w j + 1 , k ( x j ) f ( k 1 ) ( x j ) w 1 k ( a ) f ( k 1 ) ( a ) U n ( σ ) · 1 2 f ( n ) ( a ) + 1 2 f ( n ) ( b ) ,
where
U n ( σ ) = ( 1 ) n n ! a b w ( t ) · t n d t ( 1 ) n k = 1 n ( 1 ) k 1 ( n k + 1 ) ! · w m k ( b ) b n k + 1 + j = 1 m 1 w j k ( x j ) w j + 1 , k ( x j ) x j n k + 1 w 1 k ( a ) a n k + 1 .
If W n , w ( t , σ ) 0 or f is an ( n + 2 ) -concave function on [ a , b ] , then the inequalities in (5) hold with reversed inequality signs.
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
f [ x i ] = f ( x i ) , ( i = 0 , , n )
and
f [ x 0 , , x n ] = f [ x 1 , , x n ] f [ x 0 , , x n 1 ] x n x 0 .
The value f [ x 0 , , x n ] is independent of the order of points x 0 , , x n .
Definition 2.
A function f : [ a , b ] R is said to be n-convex on [ a , b ] , 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
f [ x 0 , , x n ] 0 .
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
B ( x , y ) = 0 1 s x 1 ( 1 s ) y 1 d s ,
Γ denotes the gamma function defined as:
Γ ( x ) = 2 0 s 2 x 1 e s 2 d s ,
and
F α , β , γ ; z = 1 B ( β , γ β ) 0 1 t β 1 ( 1 t ) γ β 1 ( 1 z t ) α d t
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.

2. 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].
In [16] (p. 54), the authors proved the following theorem.
Theorem 4.
Let w : [ a , b ] R be an integrable function, x [ a , a + b 2 ) , and let L j , x j = 0 , 1 , , n , n N , be a sequence of harmonic polynomials such that d e g L j , x j 1 and L 0 , x 0 . Further, let us suppose that { w j k } k = 1 , . . , n are w-harmonic sequences of functions on [ x j 1 , x j ] , for j = 1 , 2 , 3 , 4 , defined by the following relations:
w 1 k ( t ) = 1 ( k 1 ) ! a t t s k 1 w ( s ) d s , t [ a , x ] ,
w 2 k ( t ) = 1 ( k 1 ) ! x t t s k 1 w ( s ) d s + L k , x ( t ) , t x , a + b 2 ,
w 3 k ( t ) = 1 ( k 1 ) ! t a + b x t s k 1 w ( s ) d s + ( 1 ) k L k , x ( a + b t ) , t a + b 2 , a + b x ,
w 4 k ( t ) = 1 ( k 1 ) ! t b t s k 1 w ( s ) d s , t ( a + b x , b ] .
If f : [ a , b ] R is such that f ( n ) is piecewise continuous on a , b , then we have
a b w ( t ) f ( t ) d t = k = 1 n A k ( x ) f ( k 1 ) ( x ) + ( 1 ) k 1 f ( k 1 ) ( a + b x ) + k = 1 n B k ( x ) f ( k 1 ) a + b 2 + ( 1 ) n a b W n , w ( t , x ) f ( n ) ( t ) d t ,
where
A k ( x ) = ( 1 ) k 1 1 ( k 1 ) ! a x x s k 1 w ( s ) d s L k , x ( x ) , k 1 ,
B k ( x ) = 2 1 ( k 1 ) ! x a + b 2 a + b 2 s k 1 w ( s ) d s + L k , x a + b 2 , f o r   o d d k 1 ,
and
B k ( x ) = 0 , f o r   e v e n k 1 ,
such that
W n , w ( t , x ) = w 1 n ( t ) , t [ a , x ] , w 2 n ( t ) , t x , a + b 2 , w 3 n ( t ) , t a + b 2 , a + b x , w 4 n ( t ) , t ( a + b x , b ] .
Remark 1.
If we assume w ( t ) = w ( a + b t ) , for each t a , b , then the following symmetry conditions hold for k = 1 , , n :
w 1 k ( t ) = ( 1 ) k w 4 k ( a + b t ) , f o r t [ a , x ] ,
and
w 2 k ( t ) = ( 1 ) k w 3 k ( a + b t ) , f o r t x , a + b 2 .
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).
Theorem 5.
Let w : [ a , b ] R be an integrable function such that w ( t ) = w ( a + b t ) , for each t a , b and x [ a , a + b 2 ) . Let the function W 2 n , w , defined by (10), be nonnegative. If f : [ a , b ] R is ( 2 n + 2 ) -convex on a , b and f ( 2 n ) is piecewise continuous on a , b , then
U n , w ( x ) · f ( 2 n ) a + b 2 a b w ( t ) f ( t ) d t k = 1 2 n A k ( x ) f ( k 1 ) ( x ) + ( 1 ) k 1 f ( k 1 ) ( a + b x ) k = 1 , k o d d 2 n B k ( x ) f ( k 1 ) a + b 2 U n , w ( x ) · 1 2 f ( 2 n ) ( a ) + 1 2 f ( 2 n ) ( b ) ,
where
U n , w ( x ) = 1 ( 2 n ) ! a b w ( t ) · t 2 n d t k = 1 2 n A k ( x ) x 2 n k + 1 + ( 1 ) k 1 ( a + b x ) 2 n k + 1 ( 2 n k + 1 ) ! k = 1 , k o d d 2 n B k ( x ) ( a + b ) 2 n k + 1 2 2 n k + 1 ( 2 n k + 1 ) ! ,
and A k and B k are defined as in Theorem 4. If W 2 n , w ( t , x ) 0 or f is a ( 2 n + 2 ) -concave function, then inequalities (11) hold with reversed inequality signs.
Proof. 
Let us observe that the function f is ( 2 n + 2 ) -convex. Hence, f ( 2 n ) is a convex function. It follows from Remark 1 that the function W 2 n , w is symmetric about a + b 2 , i.e., W 2 n , w ( t , x ) = W 2 n , w ( a + b t , x ) . Thus, inequalities (11) follow directly from Theorem 1, replacing a nonnegative and symmetric function u by a nonnegative and symmetric function W 2 n , w , and a convex function h by a convex function f ( 2 n ) , and then using identity (7) in a b W 2 n , w ( t , x ) f ( 2 n ) ( t ) d t .
Identity (7) yields U n , w ( x ) by substituting n with 2 n and putting f ( t ) = t 2 n ( 2 n ) ! . Then, f ( 2 n ) ( t ) = 1 and f ( k 1 ) ( t ) = 1 ( 2 n k + 1 ) ! · t 2 n k + 1 . On the other hand, if W 2 n , w t , x is nonpositive, then W 2 n , w t , x is nonnegative, from where there follow reversed signs in (11).
Further, let us assume that f is a ( 2 n + 2 ) -concave function. Hence, the function f ( 2 n ) is convex. Reversed signs in (11) are obtained by putting f ( 2 n ) and the nonnegative function W 2 n , 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 1 k , w 2 k , w 3 k and w 4 k .
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:
L 0 , x ( t ) = 0 , f o r t x , a + b 2 , L 1 , x ( x ) = a x w ( s ) d s 2 ( a + b 2 x ) 2 a b s 2 a + b 2 2 w ( s ) d s , L j , x ( x ) = 1 ( j 1 ) ! a x x s j 1 w ( s ) d s , j = 2 , 3 , 4 , 5 , 6 , L j , x ( t ) = k = 1 6 j L k , x ( x ) ( t x ) j k ( j k ) ! , for t x , a + b 2 , j = 1 , , n .
Therefore, we have
A 1 ( x ) = 2 ( a + b 2 x ) 2 a b s 2 a + b 2 2 w ( s ) d s ,
B 1 ( x ) = a b w ( s ) d s 2 A 1 ( x ) ,
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 weighted integral formula:
a b w ( t ) f ( t ) d t = A 1 ( x ) f ( x ) + f ( a + b x ) + a b w ( s ) d s 2 A 1 ( x ) f a + b 2 + T n , w ( x ) + ( 1 ) n a b W n , w ( t , x ) f ( n ) ( t ) d t ,
where
T n , w ( x ) = k = 7 n A k ( x ) f ( k 1 ) ( x ) + ( 1 ) k 1 f ( k 1 ) ( a + b x ) + k = 5 , o d d k n B k ( x ) f ( k 1 ) a + b 2 .
Now, applying results from Theorem 5 to identity (15), we get the following results.
Corollary 1.
Let w : [ a , b ] R be an integrable function such that w ( t ) = w ( a + b t ) , for each t a , b and let x [ a , a + b 2 ) . Let the function W 2 n , w , defined by (10), be nonnegative and let L j , x be defined by (13). If f : [ a , b ] R is ( 2 n + 2 ) -convex on a , b and f ( 2 n ) is piecewise continuous on a , b , then
U n , w ( x ) · f ( 2 n ) a + b 2 a b w ( t ) f ( t ) d t A 1 ( x ) f ( x ) + f ( a + b x ) a b w ( s ) d s 2 A 1 ( x ) f a + b 2 T 2 n , w ( x ) U n , w ( x ) · 1 2 f ( 2 n ) ( a ) + 1 2 f ( 2 n ) ( b ) ,
where
U n , w ( x ) = 1 ( 2 n ) ! a b w ( t ) · t 2 n d t A 1 ( x ) x 2 n + ( a + b x ) 2 n ( 2 n ) ! a b w ( s ) d s 2 A 1 ( x ) ( a + b ) 2 n 2 2 n ( 2 n ) ! k = 7 2 n A k ( x ) x 2 n k + 1 + ( 1 ) k 1 ( a + b x ) 2 n k + 1 ( 2 n k + 1 ) ! k = 5 , k o d d 2 n B k ( x ) ( a + b ) 2 n k + 1 2 2 n k + 1 ( 2 n k + 1 ) ! .
If W 2 n , w ( t , x ) 0 or f is a ( 2 n + 2 ) -concave function, then inequalities (17) hold with reversed inequality signs.
Proof. 
The proof follows from Theorem 5 for the special choice of the polynomials L j , x .  □
Remark 3.
If we assume B 5 ( x ) = 0 , then we get
x = a + b 2 a b s a + b 2 4 w ( s ) d s a b s 2 a + b 2 2 w ( s ) d s .
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.

3. 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.

3.1. 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
W n G L ( t , x ) = w 1 n ( t ) = ( t a ) n n ! , t [ a , x ] , w 2 n ( t ) = ( t x ) n n ! + L n , x ( t ) , t x , a + b 2 , w 3 n ( t ) = ( t a b + x ) n n ! + ( 1 ) n L n , x ( a + b t ) , t a + b 2 , a + b x , w 4 n ( t ) = ( t b ) n n ! , t ( a + b x , b ] ,
and
A k G L ( x ) = ( 1 ) k 1 x a k k ! L k , x ( x ) , for k 1 ,
B k G L ( x ) = 2 a + b 2 x k k ! + L k , x a + b 2 , for   odd k 1 ,
and
B k G L ( x ) = 0 , for   even k > 1 .
Corollary 2.
Let w 2 , 2 n ( t ) 0 , for all t x , a + b 2 and for n N . If f : [ a , b ] R is a ( 2 n + 2 ) -convex function and f ( 2 n ) is piecewise continuous on a , b , then
U n G L ( x ) · f ( 2 n ) a + b 2 a b f ( t ) d t k = 1 2 n A k G L ( x ) f ( k 1 ) ( x ) + ( 1 ) k 1 f ( k 1 ) ( a + b x ) k = 1 , k o d d 2 n B k G L ( x ) f ( k 1 ) a + b 2 U n G L ( x ) · 1 2 f ( 2 n ) ( a ) + 1 2 f ( 2 n ) ( b ) ,
where
U n G L ( x ) = b 2 n + 1 a 2 n + 1 ( 2 n + 1 ) ! k = 1 2 n A k G L ( x ) x 2 n k + 1 + ( 1 ) k 1 ( a + b x ) 2 n k + 1 ( 2 n k + 1 ) ! k = 1 , k o d d 2 n B k G L ( x ) ( a + b ) 2 n k + 1 2 2 n k + 1 ( 2 n k + 1 ) ! .
If f is a ( 2 n + 2 ) -concave function, then inequalities (20) hold with reversed inequality signs.
Proof. 
A special case of Theorem 5 for w ( t ) = 1 , t a , b , and a nonnegative function W 2 n G L defined by (19).  □
If we assume that the polynomials L j , x ( t ) are such that
L 0 , x ( t ) = 0 , for t x , a + b 2 , L 1 , x ( x ) = x a ( b a ) 3 6 ( a + b 2 x ) 2 , L j , x ( x ) = ( x a ) j j ! , j = 2 , 3 , 4 , 5 , 6 , L j , x ( t ) = k = 1 6 j L k , x ( x ) ( t x ) j k ( j k ) ! , for t x , a + b 2 , j = 1 , , n ,
we get A 1 G L ( x ) = ( b a ) 3 6 ( a + b 2 x ) 2 , A k G L ( x ) = 0 , for k = 2 , 3 , 4 , 5 , 6 , B 1 G L ( x ) = b a 2 A 1 G L ( x ) and B 3 G L ( x ) = 0 . Thus, we obtain the following non-weighted three-point quadrature formulae:
a b f ( t ) d t = ( b a ) 3 6 ( a + b 2 x ) 2 f ( x ) + f ( a + b x ) + b a ( b a ) 3 3 ( a + b 2 x ) 2 f a + b 2 + T n G L ( x ) + ( 1 ) n a b W n G L ( t , x ) f ( n ) ( t ) d t ,
where
T n G L ( x ) = k = 7 n A k G L ( x ) f ( k 1 ) ( x ) + ( 1 ) k 1 f ( k 1 ) ( a + b x ) + k = 5 , o d d k n B k G L ( x ) f ( k 1 ) a + b 2 .
In particular, according to Remark 3, for [ a , b ] = [ 1 , 1 ] and x = 15 5 , we get B 5 G L ( 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.
If the assumptions of Corollary 1 hold for w ( t ) = 1 , t [ 1 , 1 ] , and if f : [ 1 , 1 ] R is a ( 2 n + 2 ) -convex function, we derive:
U n G L 15 5 · f ( 2 n ) 0 1 1 f ( t ) d t 1 9 5 f 15 5 + 8 f ( 0 ) + 5 f 15 5 T 2 n G L 15 5 U n G L 15 5 · 1 2 f ( 2 n ) ( 1 ) + 1 2 f ( 2 n ) ( 1 ) ,
where
U n G L 15 5 = 2 · 5 n 1 2 ( 2 n + 1 ) · 3 n 2 5 n 1 ( 2 n + 1 ) ! k = 7 2 n A k G L 15 5 ( 15 ) 2 n k + 1 + ( 1 ) k 1 ( 15 ) 2 n k + 1 5 2 n k + 1 ( 2 n k + 1 ) ! .
In a special case, for n = 3 , we get
1 15 , 750 · f ( 6 ) 0 1 1 f ( t ) d t 1 9 5 f 15 5 + 8 f ( 0 ) + 5 f 15 5 1 15 , 750 · 1 2 f ( 6 ) ( 1 ) + 1 2 f ( 6 ) ( 1 ) .

3.2. Gauss–Chebyshev Three-Point Quadrature Formula of the First Kind

Let us assume that w ( t ) = 1 1 t 2 , t ( 1 , 1 ) and x 1 , 0 .
From Theorem 4, there follow:
W n , w G C 1 ( t , x ) = w 1 n ( t ) = 1 ( n 1 ) ! 1 t ( t s ) n 1 1 s 2 d s , t [ 1 , x ] , w 2 n ( t ) = 1 ( n 1 ) ! x t ( t s ) n 1 1 s 2 d s + L n , x ( t ) , t ( x , 0 ] , w 3 n ( t ) = 1 ( n 1 ) ! t x ( t s ) n 1 1 s 2 d s + ( 1 ) n L n , x ( t ) , t ( 0 , x ] , w 4 n ( t ) = 1 ( n 1 ) ! t 1 ( t s ) n 1 1 s 2 d s , t ( x , 1 ] ,
A k G C 1 ( x ) = ( 1 ) k 1 x + 1 k 1 / 2 π 2 Γ 1 2 + k F 1 2 , 1 2 , 1 2 + k , x + 1 2 L k , x ( x ) , k 1 ,
and
B k G C 1 ( x ) = 2 ( 1 ) k 1 ( k 1 ) ! x 0 s k 1 1 s 2 d s + L k , x ( 0 ) , for   odd k 1 ,
and
B k G C 1 ( x ) = 0 , for   even k > 1 .
Corollary 3.
Let w 2 , 2 n ( t ) 0 , for all t ( x , 0 ] and for n N . If f : [ 1 , 1 ] R is a ( 2 n + 2 ) -convex function and f ( 2 n ) is piecewise continuous on 1 , 1 , then
U n G C 1 ( x ) · f ( 2 n ) 0 1 1 f ( t ) 1 t 2 d t k = 1 2 n A k G C 1 ( x ) f ( k 1 ) ( x ) + ( 1 ) k 1 f ( k 1 ) ( x ) k = 1 , k o d d 2 n B k G C 1 ( x ) f ( k 1 ) 0 U n G C 1 ( x ) · 1 2 f ( 2 n ) ( 1 ) + 1 2 f ( 2 n ) ( 1 ) ,
where
U n G C 1 ( x ) = 1 ( 2 n ) ! B 1 2 , 1 2 + n k = 1 2 n A k G C 1 ( x ) x 2 n k + 1 + ( 1 ) k 1 ( x ) 2 n k + 1 ( 2 n k + 1 ) ! .
If f is a ( 2 n + 2 ) -concave function, then inequalities (28) hold with reversed inequality signs.
Proof. 
A special case of Theorem 5 for w ( t ) = 1 1 t 2 , t 1 , 1 , and a nonnegative function W 2 n , w G C 1 defined by (27).  □
If we assume that the polynomials L j , x ( t ) are such that
L 0 , x ( t ) = 0 , for t x , 0 , L 1 , x ( x ) = arcsin x + π 2 π 4 x 2 , L j , x ( x ) = x + 1 j 1 / 2 π 2 Γ 1 2 + j F 1 2 , 1 2 , 1 2 + j , x + 1 2 , j = 2 , 3 , 4 , 5 , 6 , L j , x ( t ) = k = 1 6 j L k , x ( x ) ( t x ) j k ( j k ) ! , for t x , 0 , j = 1 , , n ,
we calculate A 1 G C 1 ( x ) = π 4 x 2 , A k G C 1 ( x ) = 0 , for k = 2 , 3 , 4 , 5 , 6 , B 1 G C 1 ( x ) = π π 2 x 2 and B 3 G C 1 ( x ) = 0 .
Now, we derive
1 1 f ( t ) 1 t 2 d t = π 4 x 2 f ( x ) + π π 2 x 2 f 0 + π 4 x 2 f ( x ) + T n , w G C 1 ( x ) + ( 1 ) n 1 1 W n , w G C 1 ( t , x ) f ( n ) ( t ) d t ,
where
T n , w G C 1 ( x ) = k = 7 n A k G C 1 ( x ) f ( k 1 ) ( x ) + ( 1 ) k 1 f ( k 1 ) ( x ) + k = 5 , o d d k n B k G C 1 ( x ) f ( k 1 ) 0 .
In particular, there follows a generalization of the Gauss–Chebyshev three-point quadrature formula of the first kind for x = 3 2 . Now, we derive Hermite–Hadamard-type estimates for the Gauss–Chebyshev three-point quadrature formula of the first kind.
If the assumptions of Corollary 1 hold for w ( t ) = 1 1 t 2 , t ( 1 , 1 ) , and if f : [ 1 , 1 ] R is a ( 2 n + 2 ) -convex function, we get
U n G C 1 3 2 · f ( 2 n ) 0 1 1 f ( t ) 1 t 2 d t π 3 f 3 2 + f ( 0 ) + f 3 2 T 2 n , w G C 1 3 2 U n G C 1 3 2 · 1 2 f ( 2 n ) ( 1 ) + 1 2 f ( 2 n ) ( 1 ) ,
where
U n G C 1 3 2 = 1 ( 2 n ) ! B 1 2 , 1 2 + n π · 3 n 1 2 2 n 1 ( 2 n ) ! k = 7 2 n A k G C 1 3 2 ( 3 ) 2 n k + 1 + ( 1 ) k 1 ( 3 ) 2 n k + 1 2 2 n k + 1 ( 2 n k + 1 ) ! .
In a special case, for n = 3 , we obtain
π 23 , 040 · f ( 6 ) 0 1 1 f ( t ) 1 t 2 d t π 3 f 3 2 + f ( 0 ) + f 3 2 π 23 , 040 · 1 2 f ( 6 ) ( 1 ) + 1 2 f ( 6 ) ( 1 ) .

3.3. Gauss–Chebyshev Three-Point Quadrature Formula of the Second Kind

Assuming w ( t ) = 1 t 2 , t [ 1 , 1 ] and x 1 , 0 and using Theorem 4, we obtain
W n , w G C 2 ( t , x ) = w 1 n ( t ) = 1 ( n 1 ) ! 1 t ( t s ) n 1 1 s 2 d s , t [ 1 , x ] , w 2 n ( t ) = 1 ( n 1 ) ! x t ( t s ) n 1 1 s 2 d s + L n , x ( t ) , t ( x , 0 ] , w 3 n ( t ) = 1 ( n 1 ) ! t x ( t s ) n 1 1 s 2 d s + ( 1 ) n L n , x ( t ) , t ( 0 , x ] , w 4 n ( t ) = 1 ( n 1 ) ! t 1 ( t s ) n 1 1 s 2 d s , t ( x , 1 ] ,
A k G C 2 ( x ) = ( 1 ) k 1 x + 1 k + 1 / 2 2 π Γ 3 2 + k F 1 2 , 3 2 , 3 2 + k , x + 1 2 L k , x ( x ) , k 1 ,
B k G C 2 ( x ) = 2 ( 1 ) k 1 ( k 1 ) ! x 0 s k 1 1 s 2 d s + L k , x ( 0 ) , , for   odd k 1 ,
and
B k G C 2 ( x ) = 0 , for   even k > 1 .
Corollary 4.
Let w 2 , 2 n ( t ) 0 , for all t ( x , 0 ] and for n N . If f : [ 1 , 1 ] R is a ( 2 n + 2 ) -convex function and f ( 2 n ) is piecewise continuous on 1 , 1 , then
U n G C 2 ( x ) · f ( 2 n ) 0 1 1 f ( t ) 1 t 2 d t k = 1 2 n A k G C 2 ( x ) f ( k 1 ) ( x ) + ( 1 ) k 1 f ( k 1 ) ( x ) k = 1 , k o d d 2 n B k G C 2 ( x ) f ( k 1 ) 0 U n G C 2 ( x ) · 1 2 f ( 2 n ) ( 1 ) + 1 2 f ( 2 n ) ( 1 ) ,
where
U n G C 2 ( x ) = 1 ( 2 n ) ! B 3 2 , 1 2 + n k = 1 2 n A k G C 2 ( x ) x 2 n k + 1 + ( 1 ) k 1 ( x ) 2 n k + 1 ( 2 n k + 1 ) ! .
If f is a ( 2 n + 2 ) -concave function, then inequalities (35) hold with reversed inequality signs.
Proof. 
A special case of Theorem 5 for w ( t ) = 1 t 2 , t 1 , 1 , and a nonnegative function W 2 n , w G C 2 defined by (34).  □
If the polynomials L j , x ( t ) are such that
L 0 , x ( t ) = 0 , for t x , 0 , L 1 , x ( x ) = 1 2 arcsin x + π 2 π 8 x 2 + x 1 x 2 2 , L j , x ( x ) = x + 1 j + 1 / 2 2 π Γ 3 2 + j F 1 2 , 3 2 , 3 2 + j , x + 1 2 , j = 2 , 3 , 4 , 5 , 6 , L j , x ( t ) = k = 1 6 j L k , x ( x ) ( t x ) j k ( j k ) ! , for t x , 0 , j = 1 , , n ,
we have A 1 G C 2 ( x ) = x 1 x 2 4 π 16 x 2 , A k G C 2 ( x ) = 0 , for k = 2 , 3 , 4 , 5 , 6 , B 1 G C 2 ( x ) = π 2 x 1 x 2 2 + π 8 x 2 and B 3 G C 2 ( x ) = 0 , so we obtain
1 1 f ( t ) 1 t 2 d t = A 1 G C 2 ( x ) f ( x ) + f ( x ) + B 1 G C 2 ( x ) f 0 + T n , w G C 2 ( x ) + ( 1 ) n 1 1 W n , w G C 2 ( t , x ) f ( n ) ( t ) d t ,
where
T n , w G C 2 ( x ) = k = 7 n A k G C 2 ( x ) f ( k 1 ) ( x ) + ( 1 ) k 1 f ( k 1 ) ( x ) + k = 5 , o d d k n B k G C 2 ( x ) f ( k 1 ) 0 .
In particular, a generalization of the Gauss–Chebyshev three-point quadrature formula of the second kind follows for x = 2 2 . Now, we derive Hermite–Hadamard-type estimates for the Gauss–Chebyshev three-point quadrature formula of the second kind.
Applying Corollary 1 to w ( t ) = 1 t 2 , t [ 1 , 1 ] , x = 2 2 , and a ( 2 n + 2 ) -convex function f, we obtain
U n G C 2 2 2 · f ( 2 n ) 0 1 1 f ( t ) 1 t 2 d t π 8 f 2 2 + 2 f ( 0 ) + f 2 2 T 2 n , w G C 2 2 2 U n G C 2 2 2 · 1 2 f ( 2 n ) ( 1 ) + 1 2 f ( 2 n ) ( 1 ) ,
where
U n G C 2 2 2 = 1 ( 2 n ) ! B 3 2 , 1 2 + n π 2 n + 2 ( 2 n ) ! k = 7 2 n A k G C 2 2 2 ( 2 ) 2 n k + 1 + ( 1 ) k 1 ( 2 ) 2 n k + 1 2 2 n k + 1 ( 2 n k + 1 ) ! .
As a special case, for n = 3 , we obtain
π 92 , 160 · f ( 6 ) 0 1 1 f ( t ) 1 t 2 d t π 8 f 2 2 + 2 f ( 0 ) + f 2 2 π 92 , 160 · 1 2 f ( 6 ) ( 1 ) + 1 2 f ( 6 ) ( 1 ) .

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Conflicts of Interest

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