Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function

Mohamed Jleli; Bessem Samet

doi:10.1515/math-2023-0178

Article Open Access

Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function

Mohamed Jleli and Bessem Samet

Published/Copyright: March 12, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

We establish weighted Hermite-Hadamard-type inequalities for some classes of differentiable functions without assuming any symmetry property on the weight function. Next, we apply our obtained results to the approximation of some classes of weighted integrals.

Keywords: Hermite-Hadamard inequality; Fejér inequality; convex functions; numerical integration

MSC 2010: 65D30; 26D15

1 Introduction

Let a , b ∈ R , a < b , and f : [ a , b ] → R . If f is convex, then

(1.1) f a + b 2 ≤ 1 b − a ∫ a b f ( x ) d x ≤ f ( a ) + f ( b ) 2 .

The above double inequality is known in the literature as Hermite-Hadamard inequality. It dates back to an 1883 observation of Hermite [1] with an independent use by Hadamard [2] in 1893. Hermite-Hadamard inequality is very useful in the study of the properties of convex functions and their applications in optimization and approximation theory. One of the first generalizations of (1.1) is the following weighted Hermite-Hadamard inequality obtained by Fejér [3]: Let f : [ a , b ] → R be a convex function and w : [ a , b ] → R be an integrable, nonnegative, and symmetric function with respect to the midpoint a + b 2 , then

(1.2) f a + b 2 ∫ a b w ( x ) d x ≤ ∫ a b f ( x ) w ( x ) d x ≤ f ( a ) + f ( b ) 2 ∫ a b w ( x ) d x .

It is a natural question to ask whether it is possible to derive inequalities of type (1.1) and (1.2) for other classes of functions. Dragomir et al. [4] considered the class of Lipschitzian functions

Lip M ( [ a , b ] ) = { f : ∣ f ( x ) − f ( y ) ∣ ≤ M ∣ x − y ∣ for all a ≤ x , y ≤ b } ,

where M > 0 is a constant. Among other results, they proved that if f ∈ Lip M ( [ a , b ] ) , then

1 b − a ∫ a b f ( x ) d x − f ( a ) + f ( b ) 2 ≤ M 3 ( b − a ) .

From the right inequality of (1.1) and the above inequality, the authors deduced that if f ∈ C 1 ( [ a , b ] ) and f is convex, then

(1.3) 0 ≤ f ( a ) + f ( b ) 2 − 1 b − a ∫ a b f ( x ) d x ≤ ‖ f ′ ‖ ∞ 3 ( b − a ) ,

where ‖ f ′ ‖ ∞ = max a ≤ t ≤ b ∣ f ′ ( t ) ∣ . Iyengar [5] considered the class of differentiable functions f : [ a , b ] → R satisfying ∣ f ′ ( x ) ∣ ≤ M for all x ∈ [ a , b ] , where M > 0 is a constant. For this class of functions, he proved that

1 b − a ∫ a b f ( x ) d x − f ( a ) + f ( b ) 2 ≤ M 4 ( b − a ) − ( f ( b ) − f ( a ) ) 2 4 M ( b − a ) .

Mitrinović et al. (see [6, p. 472, Theorem 4]) studied the class of differentiable functions f : [ a , b ] → R satisfying

f ′ ∈ Lip M ( [ a , b ] ) , f ′ ( a ) = f ′ ( b ) = 0 .

For this class of functions, it was proved that

1 b − a ∫ a b f ( x ) d x − f ( a ) + f ( b ) 2 ≤ M 24 ( b − a ) 2 − ( f ( b ) − f ( a ) ) 2 2 M ( b − a ) 2 .

Dragomir and Agarwal [7] studied the class of differentiable functions f : [ a , b ] → R such that ∣ f ′ ∣ is convex. For this class of functions, they proved that

(1.4) 1 b − a ∫ a b f ( x ) d x − f ( a ) + f ( b ) 2 ≤ ( b − a ) ( ∣ f ′ ( a ) ∣ + ∣ f ′ ( b ) ∣ ) 8 .

Moreover, some error estimates for the trapezoidal formula have been derived. Further results related to inequalities (1.1) and (1.2) for other classes of functions can be found in [8–22] (see also references therein).

In this article, we establish new weighted Hermite-Hadamard-type inequalities for some classes of differentiable functions without requiring any symmetry condition on the weight function w . Next, we apply one of our obtained results to the approximation of weighted integrals for a certain class of functions.

In the next section, some useful lemmas are given. Our results and their proofs are presented in Section 3. Finally, an application to numerical integration is given in Section 4.

2 Preliminaries

In this section, we establish some lemmas that will be used in the proofs of our main results. First, let us fix some notations.

Let a , b ∈ R , a < b and I = [ a , b ] . For a given function ξ : I → R , we denote by Z ( ξ ) the set of zeros of ξ , that is,

Z ( ξ ) = { x ∈ I : ξ ( x ) = 0 } .

We introduce the sets C + ( I ) and C 1 . + ( I ) of real-valued functions defined by

C + ( I ) = { ξ ∈ C ( I ) : ξ ≥ 0 , Z ( ξ ) is at most countable }

and

C 1 . + ( I ) = { ξ ∈ C 1 ( I ) : ξ > 0 } .

Let α ∈ C 1 , + ( I ) and w ∈ C + ( I ) . We introduce the function

G : [ a , b ] × [ a , b ] → R

defined by

G ( x , s ) = ∫ a b 1 α ( t ) d t − 1 × ∫ x b 1 α ( t ) d t ∫ a s 1 α ( t ) d t if a ≤ s ≤ x , ∫ a x 1 α ( t ) d t ∫ s b 1 α ( t ) d t if x ≤ s ≤ b .

Let

(2.1) g ( x ) = ∫ a b G ( x , s ) w ( s ) d s , a ≤ x ≤ b .

Lemma 2.1

The following properties hold:

g ≥ 0 .
g ( a ) = g ( b ) = 0 .
For all a < x < b , we have
(2.2) α ( x ) g ′ ( x ) = ∫ a b 1 α ( t ) d t − 1 ∫ x b ∫ s b 1 α ( t ) d t w ( s ) d s − ∫ a x ∫ a s 1 α ( t ) d t w ( s ) d s

and

(2.3) ( α ( x ) g ′ ( x ) ) ′ = − w ( x ) .

Proof

The statements (i) and (ii) follow immediately from (2.1). On the other hand, for all a < s , x < b , we have

∂ G ∂ x ( x , s ) = ∫ a b 1 α ( t ) d t − 1 × − 1 α ( x ) ∫ a s 1 α ( t ) d t if a ≤ s ≤ x , 1 α ( x ) ∫ s b 1 α ( t ) d t if x ≤ s ≤ b ,

which implies that

g ′ ( x ) = ∫ a b ∂ G ∂ x ( x , s ) w ( s ) d s = ∫ a b 1 α ( t ) d t − 1 − 1 α ( x ) ∫ a x ∫ a s 1 α ( t ) d t w ( s ) d s + 1 α ( x ) ∫ x b ∫ s b 1 α ( t ) d t w ( s ) d s = 1 α ( x ) ∫ a b 1 α ( t ) d t − 1 ∫ x b ∫ s b 1 α ( t ) d t w ( s ) d s − ∫ a x ∫ a s 1 α ( t ) d t w ( s ) d s .

This proves (2.2). Finally, differentiating (2.2), we obtain

( α ( x ) g ′ ( x ) ) ′ = − w ( x ) ,

which proves (2.3).□

Lemma 2.2

There exists a unique ξ ( a , b ) ∈ ] a , b [ such that

g ′ ( ξ ( a , b ) ) = 0 .

Proof

By (2.2), we have

(2.4) α ( a ) g ′ ( a ) = ∫ a b 1 α ( t ) d t − 1 ∫ a b ∫ s b 1 α ( t ) d t w ( s ) d s > 0

and

(2.5) α ( b ) g ′ ( b ) = − ∫ a b 1 α ( t ) d t − 1 ∫ a b ∫ a s 1 α ( t ) d t w ( s ) d s < 0 .

Furthermore, since w ∈ C + ( I ) , it follows from (2.3) that α g ′ is strictly decreasing in [ a , b ] . Consequently, there exists a unique ξ ( a , b ) ∈ ] a , b [ such that

α ( ξ ( a , b ) ) g ′ ( ξ ( a , b ) ) = 0 .

Since α > 0 , the above equation is equivalent to g ′ ( ξ ( a , b ) ) = 0 .□

From Lemma 2.2, (2.4), and (2.5), we deduce the following properties.

Lemma 2.3

We have

g ′ ∣ [ a , ξ ( a , b ) ] ≥ 0 , g ′ ∣ ] ξ ( a , b ) , b ] < 0 .

Lemma 2.4

Let w = α ≡ 1 . Then,

ξ ( a , b ) = a + b 2 , g ( ξ ( a , b ) ) = ( b − a ) 2 8 .

Proof

By (2.1), we have

(2.6) g ( x ) = ∫ a b G ( x , t ) d t = b − x b − a ∫ a x ( t − a ) d t + x − a b − a ∫ x b ( b − t ) d t = ( x − a ) ( b − x ) 2 ,

which implies that

g ′ ( x ) = a + b 2 − x , a < x < b .

Hence, g ′ ( x ) = 0 if and only if x = a + b 2 . This shows that ξ ( a , b ) = a + b 2 . Finally, by (2.6), we obtain g ( ξ ( a , b ) ) = ( b − a ) 2 8 .□

3 Fejér-type inequalities

In this section, we establish Fejér-type inequalities for some classes of differentiable functions.

Let a , b ∈ R , a < b and I = [ a , b ] .

3.1 A class of generalized convex functions

In this subsection, we are concerned with the class of functions

ℱ α ( I ) = { f ∈ C 2 ( I ) : ( α f ′ ) ′ ≥ 0 in I } ,

where α ∈ C 1 , + ( I ) . Observe that if α ≡ 1 , then ℱ α ( I ) reduces to the class of twice continuously differentiable and convex functions on I .

We have the following result.

Theorem 3.1

Let α ∈ C 1 , + ( I ) , f ∈ ℱ α ( I ) , and w ∈ C + ( I ) . Then, it holds that

(3.1) ∫ a b f ( x ) w ( x ) d x ≤ ∫ a b 1 α ( t ) d t − 1 ∫ a b f ( a ) ∫ x b 1 α ( t ) d t + f ( b ) ∫ a x 1 α ( t ) d t w ( x ) d x .

Proof

Let g be the function defined by (2.1). Multiplying ( α g ′ ) ′ by f and integrating two times by parts, we obtain

− ∫ a b ( α ( x ) g ′ ( x ) ) ′ f ( x ) d x = − [ α ( x ) g ′ ( x ) f ( x ) ] x = a b + ∫ a b g ′ ( x ) ( α ( x ) f ′ ( x ) ) d x = − [ α ( x ) g ′ ( x ) f ( x ) ] x = a b + [ g ( x ) α ( x ) f ′ ( x ) ] x = a b − ∫ a b g ( x ) ( α ( x ) f ′ ( x ) ) ′ d x ,

which implies by Lemma 2.1(ii) and (2.3) that

∫ a b f ( x ) w ( x ) d x = α ( a ) g ′ ( a ) f ( a ) − α ( b ) g ′ ( b ) f ( b ) − ∫ a b g ( x ) ( α ( x ) f ′ ( x ) ) ′ d x .

Since f ∈ ℱ α ( I ) and g ≥ 0 by Lemma 2.1(i), it holds that

(3.2) ∫ a b f ( x ) w ( x ) d x ≤ α ( a ) g ′ ( a ) f ( a ) − α ( b ) g ′ ( b ) f ( b ) .

Finally, (3.1) follows from (2.4), (2.5), and (3.2).□

Let us consider the special case of Theorem 3.1 when α ≡ 1 , that is, f is twice continuously differentiable and convex function on I . In this case, taking α ≡ 1 in (3.1), we obtain the following result.

Corollary 3.2

Let α ≡ 1 , f ∈ ℱ α ( I ) , and w ∈ C + ( I ) . Then, it holds that

(3.3) ∫ a b f ( x ) w ( x ) d x ≤ 1 b − a ∫ a b ( f ( a ) ( b − x ) + f ( b ) ( x − a ) ) w ( x ) d x .

Remark 3.3

Let us assume that w is symmetric with respect to the midpoint a + b 2 , that is,

w ( x ) = w ( a + b − x ) , a ≤ x ≤ b .

In this case, by the change of variable y = a + b − x , we have

∫ a b ( b − x ) w ( x ) d x = ∫ a b ( y − a ) w ( a + b − y ) d y = ∫ a b ( x − a ) w ( x ) d x .

Hence, we obtain

∫ a b ( f ( a ) ( b − x ) + f ( b ) ( x − a ) ) w ( x ) d x = ( f ( a ) + f ( b ) ) ∫ a b ( x − a ) w ( x ) d x = f ( a ) + f ( b ) 2 ∫ a b ( x − a ) w ( x ) d x + ∫ a b ( b − x ) w ( x ) d x = ( b − a ) ( f ( a ) + f ( b ) ) 2 ∫ a b w ( x ) d x .

Then, (3.3) reduces to the right inequality of (1.2).

We now consider the class of functions

ℱ 1 , β ( I ) = { f ∈ C 2 ( I ) : f ″ ≥ β in I } ,

where β > 0 is a constant. Observe that ℱ 1 , β ( I ) ⊂ ℱ α ( I ) , where α ≡ 1 . In this case, we obtain the following improvement of (3.3).

Corollary 3.4

Let β > 0 be a constant, f ∈ ℱ 1 , β ( I ) , and w ∈ C + ( I ) . Then, it holds that

(3.4) ∫ a b f ( x ) w ( x ) d x ≤ 1 b − a ∫ a b ( f ( a ) ( b − x ) + f ( b ) ( x − a ) ) w ( x ) d x − β 2 ∫ a b ( x − a ) ( b − x ) w ( x ) d x .

Proof

Let us consider the functions

h ( x ) = β 2 x 2 , x ∈ I

and

u ( x ) = f ( x ) − h ( x ) , x ∈ I .

For all x ∈ I , we have

u ″ ( x ) = f ″ ( x ) − β ≥ 0 ,

which implies that u ∈ ℱ α ( I ) , where α ≡ 1 . Hence, from Corollary 3.2, we deduce that

∫ a b u ( x ) w ( x ) d x ≤ 1 b − a ∫ a b ( u ( a ) ( b − x ) + u ( b ) ( x − a ) ) w ( x ) d x ,

which yields (3.4).□

3.2 The class of continuously differentiable functions

In this subsection, we establish a Fejér-type inequality for functions f ∈ C 1 ( I ) .

We have the following result.

Theorem 3.5

Let f ∈ C 1 ( I ) and w ∈ C + ( I ) . Then, it holds that

(3.5) ∫ a b f ( x ) w ( x ) d x − 1 b − a ∫ a b ( f ( a ) ( b − x ) + f ( b ) ( x − a ) ) w ( x ) d x ≤ 2 g ( ξ ( a , b ) ) ‖ f ′ ‖ ∞ ,

where ‖ f ′ ‖ ∞ = max a ≤ t ≤ b ∣ f ′ ( t ) ∣ , the function g is defined by (2.1) with α ≡ 1 , and ξ ( a , b ) ∈ ] a , b [ is defined by Lemma 2.2.

Proof

Integrating by parts, we obtain

− ∫ a b f ( x ) g ″ ( x ) d x = − [ f ( x ) g ′ ( x ) ] x = a b + ∫ a b f ′ ( x ) g ′ ( x ) d x ,

which implies by (2.2) (with α = 1 ) that

∫ a b f ( x ) w ( x ) d x = 1 b − a ∫ a b ( f ( a ) ( b − x ) + f ( b ) ( x − a ) ) w ( x ) d x + ∫ a b f ′ ( x ) g ′ ( x ) d x .

Hence, it holds that

(3.6) ∫ a b f ( x ) w ( x ) d x − 1 b − a ∫ a b ( f ( a ) ( b − x ) + f ( b ) ( x − a ) ) w ( x ) d x ≤ ‖ f ′ ‖ ∞ ∫ a b ∣ g ′ ( x ) ∣ d x .

On the other hand, by Lemma 2.3, we have

∫ a b ∣ g ′ ( x ) ∣ d x = ∫ a ξ ( a , b ) g ′ ( x ) d x − ∫ ξ ( a , b ) b g ′ ( x ) d x = 2 g ( ξ ( a , b ) ) − g ( a ) − g ( b ) .

Then, due to Lemma 2.1(ii), we obtain

(3.7) ∫ a b ∣ g ′ ( x ) ∣ d x = 2 g ( ξ ( a , b ) ) .

Finally, (3.5) follows from (3.6) and (3.7).□

Let us consider the special case of Theorem 3.5 when w is symmetric with respect to the midpoint a + b 2 . In this case, by Remark 3.3, we have

∫ a b ( f ( a ) ( b − x ) + f ( b ) ( x − a ) ) w ( x ) d x = ( b − a ) ( f ( a ) + f ( b ) ) 2 ∫ a b w ( x ) d x .

Hence, from Theorem 3.5, we deduce the following result.

Corollary 3.6

Let f ∈ C 1 ( I ) and w ∈ C + ( I ) . Assume that w is symmetric with respect to the midpoint a + b 2 . Then, it holds that

∫ a b f ( x ) w ( x ) d x − f ( a ) + f ( b ) 2 ∫ a b w ( x ) d x ≤ 2 g ( ξ ( a , b ) ) ‖ f ′ ‖ ∞ ,

where the function g is defined by (2.1) with α ≡ 1 , and ξ ( a , b ) ∈ ] a , b [ is defined by Lemma 2.2.

We now consider the special case of Corollary 3.6 when w ≡ 1 . From Lemma 2.4 and Corollary 3.6, we deduce the following result.

Corollary 3.7

Let f ∈ C 1 ( I ) . Then, it holds that

1 b − a ∫ a b f ( x ) d x − f ( a ) + f ( b ) 2 ≤ b − a 4 ‖ f ′ ‖ ∞ .

Assume that f is convex in I . Then, by the standard Hermite-Hadamard inequality (the right inequality of (1.1)), we have

f ( a ) + f ( b ) 2 − 1 b − a ∫ a b f ( x ) d x ≥ 0 .

Hence, from Corollary 3.7, we deduce the following improvement of (1.3).

Corollary 3.8

Let f ∈ C 1 ( I ) be a convex function on I. Then, it holds that

(3.8) 0 ≤ f ( a ) + f ( b ) 2 − 1 b − a ∫ a b f ( x ) d x ≤ b − a 4 ‖ f ′ ‖ ∞ .

Remark 3.9

Observe that if ∣ f ′ ∣ is convex, then (1.4) yields the right inequality of (3.8).

We now consider the class of functions

(3.9) G α ( I ) = { f ∈ C 1 ( I ) : α ∣ f ′ ∣ is convex in I } ,

where α ∈ C 1 , + ( I ) . We have the following result.

Theorem 3.10

Let α ∈ C 1 , + ( I ) , f ∈ G α ( I ) , and w ∈ C + ( I ) . Then, it holds that

(3.10) ∫ a b f ( x ) w ( x ) d x − ∫ a b 1 α ( t ) d t − 1 ∫ a b f ( a ) ∫ x b 1 α ( t ) d t + f ( b ) ∫ a x 1 α ( t ) d t w ( x ) d x ≤ 1 b − a 2 ( b − ξ ( a , b ) ) g ( ξ ( a , b ) ) + ∫ a ξ ( a , b ) g ( x ) d x − ∫ ξ ( a , b ) b g ( x ) d x α ( a ) ∣ f ′ ( a ) ∣ + 1 b − a 2 ( ξ ( a , b ) − a ) g ( ξ ( a , b ) ) − ∫ a ξ ( a , b ) g ( x ) d x + ∫ ξ ( a , b ) b g ( x ) d x α ( b ) ∣ f ′ ( b ) ∣ ,

where the function g is defined by (2.1) and ξ ( a , b ) ∈ ] a , b [ is defined by Lemma 2.2.

Proof

Multiplying f by ( α g ′ ) ′ and integrating by parts, we obtain

− ∫ a b ( α ( x ) g ′ ( x ) ) ′ f ( x ) d x = − [ α ( x ) g ′ ( x ) f ( x ) ] x = a b + ∫ a b g ′ ( x ) ( α ( x ) f ′ ( x ) ) d x ,

which implies by (2.3), (2.4), and (2.5) that

(3.11) ∫ a b f ( x ) w ( x ) d x − ∫ a b 1 α ( t ) d t − 1 ∫ a b f ( a ) ∫ x b 1 α ( t ) d t + f ( b ) ∫ a x 1 α ( t ) d t w ( x ) d x ≤ ∫ a b ∣ g ′ ( x ) ∣ α ( x ) ∣ f ′ ( x ) ∣ d x .

On the other hand, since α ∣ f ′ ∣ is convex, we obtain

(3.12) ∫ a b ∣ g ′ ( x ) ∣ α ( x ) ∣ f ′ ( x ) ∣ d x = ∫ a b ∣ g ′ ( x ) ∣ ( α f ′ ) b − x b − a a + x − a b − a b d x ≤ α ( a ) ∣ f ′ ( a ) ∣ ∫ a b b − x b − a ∣ g ′ ( x ) ∣ d x + α ( b ) ∣ f ′ ( b ) ∣ ∫ a b x − a b − a ∣ g ′ ( x ) ∣ d x .

Furthermore, using Lemma 2.1(ii), Lemma 2.3 and integrating by parts, we obtain

(3.13) ∫ a b ( b − x ) ∣ g ′ ( x ) ∣ d x = ∫ a ξ ( a , b ) ( b − x ) g ′ ( x ) d x − ∫ ξ ( a , b ) b ( b − x ) g ′ ( x ) d x = [ ( b − x ) g ( x ) ] x = a ξ ( a , b ) + ∫ a ξ ( a , b ) g ( x ) d x − [ ( b − x ) g ( x ) ] x = ξ ( a , b ) b − ∫ ξ ( a , b ) b g ( x ) d x = 2 ( b − ξ ( a , b ) ) g ( ξ ( a , b ) ) + ∫ a ξ ( a , b ) g ( x ) d x − ∫ ξ ( a , b ) b g ( x ) d x .

Similarly, we obtain

(3.14) ∫ a b ( x − a ) ∣ g ′ ( x ) ∣ d x = 2 ( ξ ( a , b ) − a ) g ( ξ ( a , b ) ) − ∫ a ξ ( a , b ) g ( x ) d x + ∫ ξ ( a , b ) b g ( x ) d x .

Finally, (3.10) follows from (3.11), (3.12), (3.13), and (3.14).□

Let us consider the special case of Theorem 3.10 when α ≡ 1 , i.e., f ∈ C 1 ( I ) and ∣ f ′ ∣ is convex on I . In this case, taking α ≡ 1 in (3.10), we obtain the following result.

Corollary 3.11

Let α ≡ 1 , f ∈ G α ( I ) , and w ∈ C + ( I ) . Then, it holds that

∫ a b f ( x ) w ( x ) d x − 1 b − a ∫ a b ( f ( a ) ( b − x ) + f ( b ) ( x − a ) ) w ( x ) d x ≤ 1 b − a 2 ( b − ξ ( a , b ) ) g ( ξ ( a , b ) ) + ∫ a ξ ( a , b ) g ( x ) d x − ∫ ξ ( a , b ) b g ( x ) d x ∣ f ′ ( a ) ∣ + 1 b − a 2 ( ξ ( a , b ) − a ) g ( ξ ( a , b ) ) − ∫ a ξ ( a , b ) g ( x ) d x + ∫ ξ ( a , b ) b g ( x ) d x ∣ f ′ ( b ) ∣ ,

where the function g is defined by (2.1) and ξ ( a , b ) ∈ ] a , b [ is defined by Lemma 2.2.

In addition to the assumptions of Corollary 3.11, let us suppose that w is symmetric with respect to the midpoint a + b 2 . In this case, from Remark 3.3 and Corollary 3.11, we deduce the following result.

Corollary 3.12

Let α ≡ 1 , f ∈ G α ( I ) , and w ∈ C + ( I ) . Assume that w is symmetric with respect to the midpoint a + b 2 . Then, it holds that

(3.15) ∫ a b f ( x ) w ( x ) d x − f ( a ) + f ( b ) 2 ∫ a b w ( x ) d x ≤ 1 b − a 2 ( b − ξ ( a , b ) ) g ( ξ ( a , b ) ) + ∫ a ξ ( a , b ) g ( x ) d x − ∫ ξ ( a , b ) b g ( x ) d x ∣ f ′ ( a ) ∣ + 1 b − a 2 ( ξ ( a , b ) − a ) g ( ξ ( a , b ) ) − ∫ a ξ ( a , b ) g ( x ) d x + ∫ ξ ( a , b ) b g ( x ) d x ∣ f ′ ( b ) ∣ ,

where the function g is defined by (2.1) and ξ ( a , b ) ∈ ] a , b [ is defined by Lemma 2.2.

Remark 3.13

Taking w ≡ 1 in Corollary 3.12, using Lemma 2.4 and (2.6), (3.15) reduces to

(3.16) ∫ a b f ( x ) w ( x ) d x − ( b − a ) ( f ( a ) + f ( b ) ) 2 ≤ 1 b − a ( b − a ) 3 8 + 1 2 ∫ a a + b 2 ( x − a ) ( b − x ) d x − 1 2 ∫ a + b 2 b ( x − a ) ( b − x ) d x ∣ f ′ ( a ) ∣ + 1 b − a ( b − a ) 3 8 − 1 2 ∫ a a + b 2 ( x − a ) ( b − x ) d x + 1 2 ∫ a + b 2 b ( x − a ) ( b − x ) d x ∣ f ′ ( b ) ∣ .

On the other hand, we have

∫ a a + b 2 ( x − a ) ( b − x ) d x = ∫ a + b 2 b ( x − a ) ( b − x ) d x .

Hence, (3.16) reduces to (1.4).

4 Numerical integration: A convergence result

In this section, we are concerned with approximating weighted integrals of functions f ∈ G α ( I ) , where I = [ a , b ] , α ∈ C 1 , + ( I ) and G α is the class of functions defined by (3.9). Namely, let f ∈ G α ( I ) and w ∈ C + ( I ) . Our aim is to approximate the weighted integral

(4.1) I ( f ) = ∫ a b f ( x ) w ( x ) d x .

For a sufficiently large natural number n , let us consider the subdivision of the interval I given by

x 0 = a < x 1 < x 2 < ⋯ < x n − 1 < x n = b ,

where

x i = a + i b − a n , i ∈ { 0 , 1 , ⋯ , n } .

We introduce the sequence { I n ( f ) } defined by

(4.2) I n ( f ) = ∑ i = 0 n − 1 I ( i , f ) ,

where

I ( i , f ) = ∫ x i x i + 1 1 α ( t ) d t − 1 ∫ x i x i + 1 f ( x i ) ∫ x x i + 1 1 α ( t ) d t + f ( x i + 1 ) ∫ x i x 1 α ( t ) d t w ( x ) d x .

For all u ∈ C ( [ a , b ] ) , let

‖ u ‖ ∞ = max a ≤ t ≤ b ∣ u ( t ) ∣ .

We have the following error estimate.

Theorem 4.1

If α ∈ C 1 , + ( I ) , f ∈ G α ( I ) , and w ∈ C + ( I ) , then

(4.3) ∣ I ( f ) − I n ( f ) ∣ ≤ ‖ α ‖ ∞ 1 α ∞ 2 ‖ w ‖ ∞ max { ( α ∣ f ′ ∣ ) ( a ) , ( α ∣ f ′ ∣ ) ( b ) } ( b − a ) 2 n ,

where I f is the weighted integral of f given by (4.1).

Proof

By Theorem 3.10, for all i = 0 , 1 , ⋯ , n − 1 , we have

(4.4) ∫ x i x i + 1 f ( x ) w ( x ) d x − I ( i , f ) ≤ α ( x i ) ∣ f ′ ( x i ) ∣ A i + α ( x i + 1 ) ∣ f ′ ( x i + 1 ) ∣ B i x i + 1 − x i ,

where

A i = 2 ( x i + 1 − ξ ( x i , x i + 1 ) ) g i ( ξ ( x i , x i + 1 ) ) + ∫ x i ξ ( x i , x i + 1 ) g i ( x ) d x − ∫ ξ ( x i , x i + 1 ) x i + 1 g i ( x ) d x , B i = 2 ( ξ ( x i , x i + 1 ) − x i ) g i ( ξ ( x i , x i + 1 ) ) − ∫ x i ξ ( x i , x i + 1 ) g i ( x ) d x + ∫ ξ ( x i , x i + 1 ) x i + 1 g i ( x ) d x ,

and

g i ( x ) = ∫ x i x i + 1 G i ( x , s ) w ( s ) d s , x i ≤ x ≤ x i + 1 .

Here, G i is defined by

G i ( x , s ) = ∫ x i x i + 1 1 α ( t ) d t − 1 × ∫ x x i + 1 1 α ( t ) d t ∫ x i s 1 α ( t ) d t if x i ≤ s ≤ x , ∫ x i x 1 α ( t ) d t ∫ s x i + 1 1 α ( t ) d t if x ≤ s ≤ x i + 1

and ξ ( x i , x i + 1 ) ∈ ] x i , x i + 1 [ is the unique solution to g ′ ( ξ ( x i , x i + 1 ) ) = 0 (see Lemma 2.2). Note that by Lemma 2.3, g i is increasing in [ x i , ξ ( x i , x i + 1 ) ] and g i is decreasing in [ ξ ( x i , x i + 1 ) , x i + 1 ] . Hence, we have

(4.5) A i ≤ 2 ( x i + 1 − ξ ( x i , x i + 1 ) ) g i ( ξ ( x i , x i + 1 ) ) + ( ξ ( x i , x i + 1 ) − x i ) g i ( ξ ( x i , x i + 1 ) ) = ( 2 x i + 1 − ξ ( x i , x i + 1 ) − x i ) g i ( ξ ( x i , x i + 1 ) ) ≤ 2 ( x i + 1 − x i ) g i ( ξ ( x i , x i + 1 ) ) .

Similarly, we have

(4.6) B i ≤ 2 ( x i + 1 − x i ) g i ( ξ ( x i , x i + 1 ) ) .

On the other hand, for all x i ≤ x , s ≤ x i + 1 , we have

G i ( x , s ) ≤ G i ( s , s ) ≤ ‖ α ‖ ∞ x i + 1 − x i 1 α ∞ 2 ( x i + 1 − s ) ( s − x i ) ≤ ‖ α ‖ ∞ 1 α ∞ 2 x i + 1 − x i 4 ,

which implies that

(4.7) g i ( x ) ≤ ‖ α ‖ ∞ 1 α ∞ 2 x i + 1 − x i 4 ∫ x i x i + 1 w ( s ) d s ≤ 1 4 ‖ α ‖ ∞ 1 α ∞ 2 ‖ w ‖ ∞ ( x i + 1 − x i ) 2 .

Thus, it follows from (4.4), (4.5), (4.6), and (4.7) that

∫ x i x i + 1 f ( x ) w ( x ) d x − I ( i , f ) ≤ 1 x i + 1 − x i ( α ( x i ) ∣ f ′ ( x i ) ∣ A i + α ( x i + 1 ) ∣ f ′ ( x i + 1 ) ∣ B i ) ≤ 2 ( α ( x i ) ∣ f ′ ( x i ) ∣ + α ( x i + 1 ) ∣ f ′ ( x i + 1 ∣ ) ) g i ( ξ ( x i , x i + 1 ) ) ≤ α ( x i ) ∣ f ′ ( x i ) ∣ + α ( x i + 1 ) ∣ f ′ ( x i + 1 ∣ ) 2 ‖ α ‖ ∞ 1 α ∞ 2 ‖ w ‖ ∞ ( x i + 1 − x i ) 2 .

On the other hand, since α ∣ f ′ ∣ is convex, for all a ≤ x ≤ b , one has

α ( x ) ∣ f ′ ( x ) ∣ = ( α ∣ f ′ ∣ ) ( t a + ( 1 − t ) b ) ≤ t ( α ∣ f ′ ∣ ) ( a ) + ( 1 − t ) ( α ∣ f ′ ∣ ) ( b ) ≤ max { ( α ∣ f ′ ∣ ) ( a ) , ( α ∣ f ′ ∣ ) ( b ) } .

Hence, we deduce that

∫ x i x i + 1 f ( x ) w ( x ) d x − I ( i , f ) ≤ α ( x i ) ∣ f ′ ( x i ) ∣ + α ( x i + 1 ) ∣ f ′ ( x i + 1 ∣ ) 2 ‖ α ‖ ∞ 1 α ∞ 2 ‖ w ‖ ∞ ( x i + 1 − x i ) 2 ≤ ‖ α ‖ ∞ 1 α ∞ 2 ‖ w ‖ ∞ max { ( α ∣ f ′ ∣ ) ( a ) , ( α ∣ f ′ ∣ ) ( b ) } ( x i + 1 − x i ) 2 .

Then, thanks to the triangle inequality, we obtain

∫ a b f ( x ) w ( x ) d x − I n ( f ) ≤ ‖ α ‖ ∞ 1 α ∞ 2 ‖ w ‖ ∞ max { ( α ∣ f ′ ∣ ) ( a ) , ( α ∣ f ′ ∣ ) ( b ) } ∑ i = 0 n − 1 ( x i + 1 − x i ) 2 = ‖ α ‖ ∞ 1 α ∞ 2 ‖ w ‖ ∞ max { ( α ∣ f ′ ∣ ) ( a ) , ( α ∣ f ′ ∣ ) ( b ) } ( b − a ) 2 n ,

which proves (4.3).□

From Theorem 4.1, we deduce the following convergence result.

Corollary 4.2

If α ∈ C 1 , + ( I ) , f ∈ G α ( I ) , and w ∈ C + ( I ) , then

lim n → ∞ I n ( f ) = I ( f ) .

5 Conclusion

Making use of some tools from ordinary differential equations, some weighted Hermite-Hadamard-type inequalities are established for certain classes of differentiable functions without any symmetry condition imposed on the weight function. Only the one-dimensional case is considered in this article. It will be interesting to extend the obtained results to the higher-dimensional case, namely, to functions with several variables.

Acknowledgement

Bessem Samet was supported by Researchers Supporting Project number (RSP2024R4), King Saud University, Riyadh, Saudi Arabia.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflicts of interest. Bessem Samet was a Guest Editor of the Open Mathematics journal and was not involved in the review and decision-making process of this article.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

References

[1] C. Hermite, Sur deux limites daune intégrale défine, Mathesis 3 (1883), 1–82. Search in Google Scholar

[2] J. Hadamard, Etude sur les propriétés des fonctions entières et en particulier daune fonction considérée par Riemann, J. Math. Pures Appl. (9) 58 (1893), 171–215. Search in Google Scholar

[3] L. Fejér, Über die Fourierreihen, II, Math. Naturwiss, Anz. Ungar. Akad. Wiss. 24 (1906), 369–390. Search in Google Scholar

[4] S. S. Dragomir, Y. J. Cho, and S. S. Kim, Inequalities of Hadamard’s type for Lipschitzian mappings and their applications, J. Math. Anal. Appl. 245 (2000), no. 2, 489–501, DOI: https://doi.org/10.1006/jmaa.2000.6769. 10.1006/jmaa.2000.6769Search in Google Scholar

[5] K. S. K. Iyengar, Note on an inequality, Math. Student 6 (1938), no. 1, 75–76. Search in Google Scholar

[6] D. S. Mitrinović, J. E. Pecarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic, Dordrecht, 1991. 10.1007/978-94-011-3562-7Search in Google Scholar

[7] S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998), no. 5, 91–95, DOI: https://doi.org/10.1016/S0893-9659(98)00086-X. 10.1016/S0893-9659(98)00086-XSearch in Google Scholar

[8] I. Aldawish, M. Jleli, and B. Samet, On Hermite-Hadamard-type inequalities for functions satisfying second-order differential inequalities, Axioms 12 (2023), no. 5, 443. 10.3390/axioms12050443Search in Google Scholar

[9] H. Budak, T. Tunç, and M. Z. Sarikaya, Fractional Hermite-Hadamard-type inequalities for interval-valued functions, Proc. Amer. Math. Soc. 148 (2020), no. 2, 705–718. 10.1090/proc/14741Search in Google Scholar

[10] H. Chen and U. N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl. 446 (2017), no. 2, 1274–1291, DOI: https://doi.org/10.1016/j.jmaa.2016.09.018. 10.1016/j.jmaa.2016.09.018Search in Google Scholar

[11] M. Rostamian Delavar, On Fejér’s inequality: generalizations and applications, J. Inequal. Appl. 2023 (2023), no. 1, 42. 10.1186/s13660-023-02949-7Search in Google Scholar

[12] S. S. Dragomir and S. Fitzpatrick, The Hadamard inequalities for s-convex functions in the second sense, Demonstr. Math. 32 (1999), no. 4, 687–696. 10.1515/dema-1999-0403Search in Google Scholar

[13] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, Melbourne, 2000. Search in Google Scholar

[14] D. Y. Hwang, Some inequalities for differentiable convex mapping with applications to weighted midpoint formula and higher moments of random variables, Appl. Math. Comput. 217 (2011), no. 23, 9598–9605, DOI: https://doi.org/10.1016/j.amc.2011.04.036. 10.1016/j.amc.2011.04.036Search in Google Scholar

[15] R. Jaksić, L. Kvesić, and J. E. Pecarić, On weighted generalization of the Hermite-Hadamard inequality, Math. Inequal. Appl. 18 (2015), no. 2, 649–665. 10.7153/mia-18-49Search in Google Scholar

[16] M. A. Noor and K. I. Noor, Some new classes of strongly generalized preinvex functions, TWMS J. Pure Appl. Math. 12 (2021), no. 2, 181–192. Search in Google Scholar

[17] C. P. Niculescu and L. E. Persson, Old and new on the Hermite-Hadamard inequality, Real Anal. Exchange 29 (2003), no. 2, 663–685. 10.14321/realanalexch.29.2.0663Search in Google Scholar

[18] S. Obeidat, M. A. Latif, and S. S. Dragomir, Fejeŕ and Hermite-Hadamard type inequalities for differentiable h-convex and quasi convex functions with applications, Miskolc Math. Notes 23 (2022), no. 1, 401–415. 10.18514/MMN.2022.3065Search in Google Scholar

[19] B. Samet, A convexity concept with respect to a pair of functions, Numer. Funct. Anal. Optim. 43 (2022), no. 5, 522–540. 10.1080/01630563.2022.2050753Search in Google Scholar

[20] M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Başak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model. 57 (2013), no. 9–10, 2403–2407. 10.1016/j.mcm.2011.12.048Search in Google Scholar

[21] M. Z. Sarikaya, On new Hermite Hadamard Fejér type integral inequalities, Stud. Univ. Babeş Bolyai Math. 57 (2012), no. 3, 377–386. Search in Google Scholar

[22] K. L. Tseng, S. R. Hwang, and S. S. Dragomir, Fejér-type inequalities (II), Mathematica Slovaca 67 (2017), no. 1, 109–120. 10.1515/ms-2016-0252Search in Google Scholar

Received: 2023-10-26

Revised: 2023-12-21

Accepted: 2024-01-12

Published Online: 2024-03-12

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2023-0178

Keywords for this article

Hermite-Hadamard inequality; Fejér inequality; convex functions; numerical integration

Creative Commons

BY 4.0