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Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates

  • Weizhi Hong , Yanran Xu , Jianmiao Ruan EMAIL logo and Xinsheng Ma
Published/Copyright: September 16, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we study some Hermite-Hadamard-type inequalities for strongly h -convex functions on co-ordinates in R n , which extend some known results. Some mappings connected with these inequalities and related applications are also obtained.

Keywords: Hermite-Hadamard inequality; convex function; strongly h-convex function; co-ordinates
MSC 2010: 26A51; 26D07; 26D15

1 Introduction

Let I be an interval in R and h : ( 0 , 1 ) → [ 0 , ∞ ) be a given function. Following Varošanec [1], a function f : I → R is said to be h -convex, provided that

(1.1) f ( t x + ( 1 − t ) y ) ≤ h ( t ) f ( x ) + h ( 1 − t ) f ( y )

holds for all x , y ∈ I and t ∈ ( 0 , 1 ) . This notation unifies and generalizes the known classes of convex functions, s-convex functions, Godunova-Levin functions, and P-functions, which are obtained by taking in (1.1) h ( t ) = t , h ( t ) = t s ( s ∈ ( 0 , 1 ) ) , h ( t ) = 1 ⁄ t , and h ( t ) = 1 , respectively. Many properties of them can be found, for instance, in [2–9].

An important and interesting property with respect to the convex function is the Hermite-Hadamard inequality: if f : I → R is a convex function and a , b ∈ I with a < b , then

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

This inequality was studied extensively and had been extended under various convex-type functions. In 1995, Dragomir et al. [10] established similar results for Godunova-Levin functions and P -functions. In 1999, Dragomir and Fitzpatrick [11] obtained a analogous inequality for the s -convex function. In 2008, Sarikaya et al. [12] proved the following result for the h -convex function.

Theorem A

Let f be a h-convex function on the interval I and h be Lebesgue integrable on ( 0 , 1 ) with h ( 1 ⁄ 2 ) > 0 . If for any a , b ∈ I with a < b , f is Lebesgue integrable on [ a , b ] , then

(1.3) 1 2 h ( 1 ⁄ 2 ) f a + b 2 ≤ 1 b − a ∫ a b f ( x ) d x ≤ ( f ( a ) + f ( b ) ) ∫ 0 1 h ( t ) d t .

It is notable that (1.3) reduces to the results in [10] by h ( t ) = 1 ⁄ t , h ( t ) = 1 and to the one in [11] by h ( t ) = t s , respectively.

Recall that a function f : I → R is called strongly convex with modulus c > 0 , provided that

(1.4) f ( t x + ( 1 − t ) y ) ≤ t f ( x ) + ( 1 − t ) f ( y ) − c t ( 1 − t ) ( x − y ) 2 ,

for all x , y ∈ I and t ∈ ( 0 , 1 ) . Strongly convex functions were introduced by Polyak [13] in 1966, and they played a significant role in optimization theory and mathematical economics (see, e.g., [13–22]). In 2011, Angulo et al. [23] generalized the classes of strongly convex functions and h -convex functions as follows:

Definition 1

Let h : ( 0 , 1 ) → [ 0 , ∞ ) be a given function and c be a positive constant. We say that f : I → R is strongly h -convex with modulus c , or f belongs to the class S X ( h , c , I ) , if

(1.5) f ( t x + ( 1 − t ) y ) ≤ h ( t ) f ( x ) + h ( 1 − t ) f ( y ) − c t ( 1 − t ) ( x − y ) 2 ,

for all x , y ∈ I and t ∈ ( 0 , 1 ) .

Particularly, if f satisfies (1.5) with h ( t ) = t , h ( t ) = t s ( s ∈ ( 0 , 1 ) ) , h ( t ) = 1 ⁄ t , and h ( t ) = 1 , then f is said to be strongly convex functions, strongly s-convex functions, strongly Godunova-Levin functions, and strongly P-function, respectively. Moreover, it is not difficult to check that h ( 1 ⁄ 2 ) > 0 if f ∈ S X ( h , c , I ) and f ≥ 0 . As an application, Angulo et al. [23] established the following Hermite-Hadamard inequality.

Theorem B

Let f ∈ S X ( h , c , I ) and h be Lebesgue integrable on ( 0 , 1 ) with h ( 1 ⁄ 2 ) > 0 . If for any a , b ∈ I with a < b and f is Lebesgue integrable on [ a , b ] , then

(1.6) 1 2 h ( 1 ⁄ 2 ) f a + b 2 + c 12 ( b − a ) 2 ≤ 1 b − a ∫ a b f ( x ) d x ≤ ( f ( a ) + f ( b ) ) ∫ 0 1 h ( t ) d t − c 6 ( b − a ) 2 .

It is easy to see that Theorem B reduces to Theorem A with c → 0 .

The main purpose of this article is to give new Hermite-Hadamard-type inequalities under the cases of strongly h -convex functions on multi-dimension spaces and study some related mappings.

2 Preliminaries

In the sequel, unless otherwise specified, R n denotes the Euclidean space of dimension n and R 1 = R . [ a , b ] ⊂ R n denotes the usual Cartesian product by [ a , b ] = [ a 1 , b 1 ] × [ a 2 , b 2 ] × … × [ a n , b n ] and the Lebesgue measure of it by ∣ [ a , b ] ∣ = ∏ i = 1 n ( b i − a i ) . Denote by L 1 ( E ) the set of Lebesgue integrable functions on the measurable set E ⊂ R n . For any points x = ( x 1 , x 2 , … , x n ) , y = ( y 1 , y 2 , … , y n ) ∈ R n , define the product of vectors by

x ⊗ y = ( x 1 y 1 , x 2 y 2 , … , x n y n ) ,

the linear combination of vectors by

a x + b y = ( a x 1 + b y 1 , a x 2 + b y 2 , … , a x n + b y n ) , a , b ∈ R ,

and the norm of x by

∣ x ∣ = x 1 2 + x 2 2 + … + x n 2 .

A mapping f : [ a , b ] ⊂ R n → R is said to be an h -convex function on [ a , b ] if

f ( t x + ( 1 − t ) y ) ≤ h ( t ) f ( x ) + h ( 1 − t ) f ( y )

holds for all x , y ∈ [ a , b ] and t ∈ ( 0 , 1 ) .

A function f : [ a , b ] ⊂ R n → R is called coordinated h-convex on [ a , b ] if for every i ∈ { 1 , 2 , … , n } the partial mapping f i : [ a i , b i ] → R , f i ( u ) = f ( x 1 , … , x i − 1 , u , x i + 1 , … , x n ) is h -convex for all given x j ∈ [ a j , b j ] , j ≠ i .

Particularly, taking h ( t ) = t , h ( t ) = t s ( s ∈ ( 0 , 1 ) ) , h ( t ) = 1 ⁄ t , and h ( t ) = 1 , then the corresponding function f : [ a , b ] → R is called the coordinated convex, coordinated s-convex, coordinated Godunova-Levin function, and coordinated P-function on [ a , b ] , respectively.

It is notable to point out that Latif and Alomari [24] proved that if f : [ a , b ] ⊂ R n → R is convex, then f is also coordinated convex on [ a , b ] ( n ≥ 2 ), but the converse is not necessarily true.

One interesting work related to Hermite-Hadamard-type inequalities is to extend them to high-dimension spaces. In 2001, Dragomir [25] proved the following Hermite-Hadamard-type inequality for coordinated convex functions.

Theorem C

Let f : [ a , b ] ⊂ R 2 → R be a coordinated convex on [ a , b ] and f ∈ L 1 ( [ a , b ] ) . Then

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

Thereafter, Alomari and Darus [26] extended similar results for coordinated s -convex functions. Latif and Alomari [24] considered the case of coordinated h -convex functions and they obtained the following theorem.

Theorem D

Let h : ( 0 , 1 ) → [ 0 , ∞ ) with h ( 1 ⁄ 2 ) > 0 and h ∈ L 1 ( ( 0 , 1 ) ) . Suppose that f : [ a , b ] ⊂ R 2 → R be a coordinated h-convex function on [ a , b ] and f ∈ L 1 ( [ a , b ] ) . Then

1 4 h 2 ( 1 ⁄ 2 ) f a 1 + b 1 2 , a 2 + b 2 2 ≤ 1 ( b 1 − a 1 ) ( b 2 − a 2 ) ∫ a 2 b 2 ∫ a 1 b 1 f ( x 1 , x 2 ) d x 1 d x 2 ≤ [ f ( a 1 , a 2 ) + f ( a 1 , b 2 ) + f ( b 1 , a 2 ) + f ( b 1 , b 2 ) ] ∫ 0 1 h ( t ) d t 2 .

Definition 2

Let h : ( 0 , 1 ) → [ 0 , ∞ ) . A function f : [ a , b ] ⊂ R n → R is called strongly h -convex with modulus c > 0 if

f ( t x + ( 1 − t ) y ) ≤ h ( t ) f ( x ) + h ( 1 − t ) f ( y ) − c t ( 1 − t ) ∣ x − y ∣ 2

holds for any x , y ∈ [ a , b ] and t ∈ ( 0 , 1 ) .

Definition 3

Let h : ( 0 , 1 ) → [ 0 , ∞ ) . A function f : [ a , b ] ⊂ R n → R is called coordinated strongly h -convex with modulus c > 0 on [ a , b ] , provided that the partial mapping f i is strongly h -convex with modulus c > 0 on the interval [ a i , b i ] for every i ∈ { 1 , 2 , … , n } and all given x j ∈ [ a j , b j ] , j ≠ i .

Another interesting topic is to provide some applications of the Hermite-Hadamard inequalities. Let f : [ a , b ] ⊂ R n → R , define the mapping H : [ 0 , 1 ] = [ 0 , 1 ] n ⊂ R n → R by

(2.1) H ( t ) = 1 ∣ [ a , b ] ∣ ∫ [ a , b ] f t ⊗ x + ( 1 − t ) ⊗ a + b 2 d x .

It is easy to see that

H ( 0 ) = H ( 0 , … , 0 ) = f a + b 2 , H ( 1 ) = H ( 1 , … , 1 ) = 1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x .

In 2001, Dragomir [25] studied some properties of the mapping connected to the Hermite-Hadamard-type inequalities of coordinated convex functions on the plane.

Theorem E

[25] If f : [ a , b ] ⊂ R 2 → R is coordinated convex on [ a , b ] , then:

  1. The mapping H : [ 0 , 1 ] = [ 0 , 1 ] 2 → R is coordinated convex on [ 0 , 1 ] 2 .

  2. sup t ∈ [ 0 , 1 ] H ( t ) = H ( 1 ) , inf t ∈ [ 0 , 1 ] H ( t ) = H ( 0 ) .

In 2008, Alomari and Darus [27] extended Theorem E to the case of coordinated s -convex functions. In 2013, Matłoka [28] obtained similar results for coordinated h -convex functions.

Theorem F

[28] Let h 1 , h 2 : ( 0 , 1 ) → [ 0 , ∞ ) with h 1 , h 2 ∈ L 1 ( ( 0 , 1 ) ) and h 1 ( 1 ⁄ 2 ) h 2 ( 1 ⁄ 2 ) > 0 . Suppose that the function f : [ a , b ] ⊂ R 2 → R , f ∈ L ( [ a , b ] ) and its partial mappings f ( ⋅ , x 2 ) and f ( x 1 , ⋅ ) are h 1 -convex on [ a 1 , b 1 ] and h 2 -convex on [ a 2 , b 2 ] , respectively. H : [ 0 , 1 ] = [ 0 , 1 ] 2 → R is defined by (2.1), then

  1. The partial mappings H ( ⋅ , t 2 ) and H ( t 1 , ⋅ ) are h 1 -convex and h 2 -convex on [ 0 , 1 ] , respectively.

  2. 1 4 h 1 ( 1 ⁄ 2 ) h 2 ( 1 ⁄ 2 ) H ( 0 , 0 ) ≤ H ( t 1 , t 2 )

    holds for all ( t 1 , t 2 ) ∈ [ 0 , 1 ] × [ 0 , 1 ] .

Another interesting mapping connected closely with Hermite-Hadamard’s inequalities F : [ 0 , 1 ] = [ 0 , 1 ] n ⊂ R n → R is given by

(2.2) F ( t ) = 1 ∣ [ a , b ] ∣ 2 ∫ [ a , b ] ∫ [ a , b ] f ( t ⊗ x + ( 1 − t ) ⊗ y ) d x d y .

Clearly,

F ( 0 ) ≔ F ( 0 , … , 0 ) = F ( 1 ) ≔ F ( 1 , … , 1 ) = 1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x , F ( 1⁄2 ) ≔ F ( 1 ⁄ 2 , … , 1 ⁄ 2 ) = 1 ∣ [ a , b ] ∣ 2 ∫ [ a , b ] ∫ [ a , b ] f x + y 2 d x d y .

Dragomir and Pearce [29] obtained the following results.

Theorem G

If f : [ a , b ] ⊂ R 2 → R is coordinated convex on [ a , b ] , then:

  1. The mapping F is coordinated convex on [ 0 , 1 ] 2 .

  2. sup t ∈ [ 0 , 1 ] 2 F ( t ) = F ( 0 ) = F ( 1 ) , inf t ∈ [ 0 , 1 ] 2 F ( t ) = F ( 1⁄2 ) .

Throughout the article, we assume that the function h in the above definitions is always Lebesgue integrable on the interval [ 0 , 1 ] and satisfies h ( 1 ⁄ 2 ) > 0 . In the next section, we will generalize the above theorems for strongly h -convex functions on R n .

3 Main results

Theorem 1

Let f : [ a , b ] ⊂ R n → R and f ∈ L 1 ( [ a , b ] ) . If the partial mapping f i is a strongly h i -convex function with modulus c i > 0 on [ a i , b i ] , i.e., f i ∈ S X ( h i , c i , [ a i , b i ] ) , for i = 1 , 2 , … , n , respectively, then

1 2 n ∏ i = 1 n h i ( 1 ⁄ 2 ) f a + b 2 + A ˜ 1 ≤ 1 2 n ∏ i = 1 n h i ( 1 ⁄ 2 ) f a + b 2 + A 1 ≤ 1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x ≤ ∏ i = 1 n ∫ 0 1 h i ( t ) d t ∑ d i = a i o r b i , i = 1 , … , n f ( d 1 , … , d n ) − ℬ 1 ≤ ∏ i = 1 n ∫ 0 1 h i ( t ) d t ∑ d i = a i o r b i , i = 1 , … , n f ( d 1 , … , d n ) − ℬ ˜ 1 ,

where

A 1 = max λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ k = 1 n 1 2 k ∏ i = 1 k h λ i ( 1 ⁄ 2 ) c λ k 12 ( b λ k − a λ k ) 2 , A ˜ 1 = 1 n ! ∑ λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ k = 1 n 1 2 k ∏ i = 1 k h λ i ( 1 ⁄ 2 ) c λ k 12 ( b λ k − a λ k ) 2 , ℬ 1 = max λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ k = 0 n − 1 2 k ∏ i = 0 k ∫ 0 1 h λ i ( t ) d t c λ k + 1 6 ( b λ k + 1 − a λ k + 1 ) 2 , ℬ ˜ 1 = 1 n ! ∑ λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ k = 0 n − 1 2 k ∏ i = 0 k ∫ 0 1 h λ i ( t ) d t c λ k + 1 6 ( b λ k + 1 − a λ k + 1 ) 2 ;

here we denote h λ 0 ≡ 1 .

Proof

Since the first and fourth inequalities are obvious, we will pay more attention to prove the second and third inequalities. It follows from Fubini’s theorem and Theorem B that, for each λ 1 ∈ { 1 , 2 , … , n } ,

1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x = 1 ∏ j = 1 n ( b j − a j ) ∫ a n b n … ∫ a λ 1 + 1 b λ 1 + 1 ∫ a λ 1 − 1 b λ 1 − 1 … ∫ a 1 b 1 ∫ a λ 1 b λ 1 × f ( x 1 , … , x λ 1 − 1 , x λ 1 , x λ 1 + 1 , … , x n ) d x λ 1 d x 1 … d x λ 1 − 1 d x λ 1 + 1 … d x n ≤ ∫ 0 1 h λ 1 ( t ) d t ∏ j ≠ λ 1 ( b j − a j ) ∫ a n b n … ∫ a λ 1 + 1 b λ 1 + 1 ∫ a λ 1 − 1 b λ 1 − 1 … ∫ a 1 b 1 [ f ( x 1 , … , x λ 1 − 1 , a λ 1 , x λ 1 + 1 , … , x n ) + f ( x 1 , … , x λ 1 − 1 , b λ 1 , x λ 1 + 1 , … , x n ) ] d x 1 … d x λ 1 − 1 d x λ 1 + 1 … d x n − c λ 1 6 ( b λ 1 − a λ 1 ) 2 .

A similar argument as above tells us that, for any λ 2 ∈ { 1 , 2 , … , n } \ { λ 1 } ,

∫ 0 1 h λ 1 ( t ) d t ∏ j ≠ λ 1 ( b j − a j ) ∫ a n b n … ∫ a λ 1 + 1 b λ 1 + 1 ∫ a λ 1 − 1 b λ 1 − 1 … ∫ a 1 b 1 f ( x 1 , … , x λ 1 − 1 , c λ 1 , x λ 1 + 1 , … , x n ) d x 1 … d x λ 1 − 1 d x λ 1 + 1 … d x n ≤ ∫ 0 1 h λ 1 ( t ) d t ∫ 0 1 h λ 2 ( t ) d t ∏ j ≠ λ 1 , λ 2 ( b j − a j ) ∫ a n b n … ∫ a λ 2 + 1 b λ 2 + 1 ∫ a λ 2 − 1 b λ 2 − 1 … ∫ a λ 1 + 1 b λ 1 + 1 ∫ a λ 1 − 1 b λ 1 − 1 … ∫ a 1 b 1 × [ f ( x 1 , … , x λ 1 − 1 , d λ 1 , x λ 1 + 1 , … , x λ 2 − 1 , a λ 2 , x λ 2 + 1 , … , x n ) + f ( x 1 , … , x λ 1 − 1 , d λ 1 , x λ 1 + 1 , … , x λ 2 − 1 , b λ 2 , x λ 2 + 1 , … , x n ) ] d x 1 … d x λ 1 − 1 d x λ 1 + 1 … d x λ 2 − 1 d x λ 2 + 1 … d x n − ∫ 0 1 h λ 1 ( t ) d t c λ 2 6 ( b λ 2 − a λ 2 ) 2 ,

where d λ 1 = a λ 1 or b λ 1 . Then, by induction, we have

1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x ≤ ∏ i = 1 n ∫ 0 1 h i ( t ) d t ∑ d i = a i or b i , i = 1 , … , n f ( d 1 , … , d n ) − c λ 1 6 ( b λ 1 − a λ 1 ) 2 − 2 ∫ 0 1 h λ 1 ( t ) d t c λ 2 6 ( b λ 2 − a λ 2 ) 2 − 2 2 ∏ i = 1 2 ∫ 0 1 h λ i ( t ) d t c λ 3 6 ( b λ 3 − a λ 3 ) 2 − … − 2 n − 1 ∏ i = 1 n − 1 ∫ 0 1 h λ i ( t ) d t c λ n 6 ( b λ n − a λ n ) 2 ,

where λ i ∈ { 1 , 2 , … , n } , i = 1 , 2 , … , n , and λ i ≠ λ j if i ≠ j . This completes the proof of the third inequality.

On the other hand, according to Fubini’s theorem and Theorem B again, for any λ 1 ∈ { 1 , 2 , … , n } , we have

1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x ≥ 1 2 h λ 1 ( 1 ⁄ 2 ) ∏ j ≠ λ 1 ( b j − a j ) ∫ a n b n … ∫ a λ 1 + 1 b λ 1 + 1 ∫ a λ 1 − 1 b λ 1 − 1 … ∫ a 1 b 1 × f x 1 , … , x λ 1 − 1 , a λ 1 + b λ 1 2 , x λ 1 + 1 , … , x n d x 1 … d x λ 1 − 1 d x λ 1 + 1 … d x n + 1 2 h λ 1 ( 1 ⁄ 2 ) c λ 1 12 ( b λ 1 − a λ 1 ) 2 .

Then for any λ 2 ∈ { 1 , 2 , … , n } \ { λ 1 } , similar discussion shows that

1 2 h λ 1 ( 1 ⁄ 2 ) ∏ j ≠ λ 1 ( b j − a j ) ∫ a n b n … ∫ a λ 1 + 1 b λ 1 + 1 ∫ a λ 1 − 1 b λ 1 − 1 … ∫ a 1 b 1 × f x 1 , … , x λ 1 − 1 , a λ 1 + b λ 1 2 , x λ 1 + 1 , … , x n d x 1 … d x λ 1 − 1 d x λ 1 + 1 … d x n ≥ 1 2 2 h λ 1 ( 1 ⁄ 2 ) h λ 2 ( 1 ⁄ 2 ) ∏ j ≠ λ 1 , λ 2 ( b j − a j ) ∫ a n b n … ∫ a λ 2 + 1 b λ 2 + 1 ∫ a λ 2 − 1 b λ 2 − 1 … ∫ a λ 1 + 1 b λ 1 + 1 ∫ a λ 1 − 1 b λ 1 − 1 … ∫ a 1 b 1 × f x 1 , … , x λ 1 − 1 , a λ 1 + b λ 1 2 , x λ 1 + 1 , … , x λ 2 − 1 , a λ 2 + b λ 2 2 , x λ 2 + 1 , … , x n × d x 1 … d x λ 1 − 1 d x λ 1 + 1 … d x λ 2 − 1 d x λ 2 + 1 … d x n + 1 2 2 h λ 1 ( 1 ⁄ 2 ) h λ 2 ( 1 ⁄ 2 ) c λ 2 12 ( b λ 2 − a λ 2 ) 2 .

Therefore, by induction,

1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x ≥ 1 2 n ∏ i = 1 n h i ( 1 ⁄ 2 ) f a + b 2 + 1 2 h λ 1 ( 1 ⁄ 2 ) c λ 1 12 ( b λ 1 − a λ 1 ) 2 + 1 2 2 ∏ i = 1 2 h λ i ( 1 ⁄ 2 ) c λ 2 12 ( b λ 2 − a λ 2 ) 2 + … + 1 2 n ∏ i = 1 n h λ i ( 1 ⁄ 2 ) c λ n 12 ( b λ n − a λ n ) 2 ,

where { λ i } are as before, i.e., λ i ∈ { 1 , 2 , … , n } , i = 1 , 2 , … , n and λ i ≠ λ j if i ≠ j . Thus, we complete the proof of the theorem.□

In particular, taking c 1 = c 2 = … = c n and h 1 = h 2 = … = h n , we have the following conclusion.

Corollary 1

If f : [ a , b ] ⊂ R n → R is a coordinated strongly h-convex function with modulus c > 0 on [ a , b ] and f ∈ L 1 ( [ a , b ] ) , then

1 2 n h n ( 1 ⁄ 2 ) f a + b 2 + A ˜ 2 ≤ 1 2 n h n ( 1 ⁄ 2 ) f a + b 2 + A 2 ≤ 1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x ≤ ∫ 0 1 h ( t ) d t n ∑ d i = a i or b i , i = 1 , … , n f ( d 1 , … , d n ) − ℬ 2 ≤ ∫ 0 1 h ( t ) d t n ∑ d i = a i or b i , i = 1 , … , n f ( d 1 , … , d n ) − ℬ ˜ 2 ,

where

A 2 = max λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j c 12 ∑ k = 1 n 1 ( 2 h ( 1 ⁄ 2 ) ) k ( b λ k − a λ k ) 2 , A ˜ 2 = 1 n ! ∑ λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j c 12 ∑ k = 1 n 1 ( 2 h ( 1 ⁄ 2 ) ) k ( b λ k − a λ k ) 2 , ℬ 2 = max λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j c 6 ∑ k = 0 n − 1 2 ∫ 0 1 h ( t ) d t k ( b λ k + 1 − a λ k + 1 ) 2 , ℬ ˜ 2 = 1 n ! ∑ λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j c 6 ∑ k = 0 n − 1 2 ∫ 0 1 h ( t ) d t k ( b λ k + 1 − a λ k + 1 ) 2 .

Particularly, if [ a , b ] is a cube, i.e., [ a , b ] = [ a , b ] × … × [ a , b ] , we have

1 2 n h n ( 1 ⁄ 2 ) f a + b 2 + c ( b − a ) 2 12 ∑ k = 1 n 1 ( 2 h ( 1 ⁄ 2 ) ) k ≤ 1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x ≤ ∫ 0 1 h ( t ) d t n ∑ d i = a o r b , i = 1 , … , n f ( d 1 , … , d n ) − c ( b − a ) 2 6 ∑ k = 0 n − 1 2 ∫ 0 1 h ( t ) d t k .

It is not difficult to check that Corollary 1 reduces to Theorem D if letting c → 0 with n = 2 and, as a consequence, we have the following result.

Corollary 2

If f : [ a , b ] ⊂ R n → R is a coordinated strongly convex function with modulus c > 0 on [ a , b ] and f ∈ L 1 ( [ a , b ] ) , then

f a + b 2 + c 12 ∣ b − a ∣ 2 ≤ 1 ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x ≤ 1 2 n ∑ d i = a i or b i , i = 1 , … , n f ( d 1 , … , d n ) − c 6 ∣ b − a ∣ 2 .

Obviously, Theorem C is obtained by letting c → 0 in Corollary 2 for n = 2 .

Next we will give some applications of the Hermite-Hadamard inequalities. To prove our main results, we first introduce two key lemmas.

Lemma 1

Let f ∈ S X ( h , c , [ a , b ] ) . Define the mapping G : [ 0 , 1 ] ⊂ R → R by

G ( t ) = 1 b − a ∫ a b f t x + ( 1 − t ) a + b 2 d x .

Then:

  1. G ∈ S X h , c ( b − a ) 2 12 , [ 0 , 1 ] .

  2. For any t ∈ [ 0 , 1 ] , we have

    (3.1) 1 2 h ( 1 ⁄ 2 ) f a + b 2 + c ( b − a ) 2 12 t 2 ≤ G ( t ) ≤ G ( 1 ) [ h ( t ) + 2 h ( 1 ⁄ 2 ) h ( 1 − t ) ] − c ( b − a ) 2 12 [ h ( 1 − t ) + t ( 1 − t ) ] .

    Particularly, if f is a strongly convex function with modulus c on [ a , b ] , we have

    (3.2) f a + b 2 + c ( b − a ) 2 12 t 2 ≤ G ( t ) ≤ G ( 1 ) − c ( b − a ) 2 6 ( 1 − t 2 ) .

Proof

(i) Let t 1 , t 2 , ξ , η ∈ [ 0 , 1 ] and ξ + η = 1 . For any x ∈ [ a , b ] , we have

f ( ξ t 1 + η t 2 ) x + [ 1 − ( ξ t 1 + η t 2 ) ] a + b 2 = f ξ t 1 x + ( 1 − t 1 ) a + b 2 + η t 2 x + ( 1 − t 2 ) a + b 2 ≤ h ( ξ ) f t 1 x + ( 1 − t 1 ) a + b 2 + h ( η ) f t 2 x + ( 1 − t 2 ) a + b 2 − c ξ η ( t 2 − t 1 ) 2 x − a + b 2 2 .

Therefore,

G ( ξ t 1 + η t 2 ) ≤ h ( ξ ) G ( t 1 ) + h ( η ) G ( t 2 ) − c ξ η ( t 2 − t 1 ) 2 b − a ∫ a b x − a + b 2 2 d x = h ( ξ ) G ( t 1 ) + h ( η ) G ( t 2 ) − ξ η c ( b − a ) 2 12 ( t 2 − t 1 ) 2 ,

which completes the proof of (i).

(ii) A change of variable shows that

G ( t ) = 1 t ( b − a ) ∫ ( 1 + t ) a + ( 1 − t ) b 2 ( 1 − t ) a + ( 1 + t ) b 2 f ( ξ ) d ξ .

Since

( 1 + t ) a + ( 1 − t ) b 2 − ( 1 − t ) a + ( 1 + t ) b 2 = t ( b − a ) , ( 1 + t ) a + ( 1 − t ) b 2 + ( 1 − t ) a + ( 1 + t ) b 2 = a + b ,

by Theorem B,

(3.3) G ( t ) ≥ 1 2 h ( 1 ⁄ 2 ) f a + b 2 + c ( b − a ) 2 12 t 2

holds for all t ∈ [ 0 , 1 ] .

On the other hand, by Definition 1 and (3.3),

G ( t ) ≤ h ( t ) G ( 1 ) + h ( 1 − t ) f a + b 2 − c t ( 1 − t ) b − a ∫ a b x − a + b 2 2 d x ≤ h ( t ) G ( 1 ) + h ( 1 − t ) 2 h ( 1 ⁄ 2 ) G ( 1 ) − c ( b − a ) 2 12 − c ( b − a ) 2 12 t ( 1 − t ) = G ( 1 ) [ h ( t ) + 2 h ( 1 ⁄ 2 ) h ( 1 − t ) ] − c ( b − a ) 2 12 [ h ( 1 − t ) + t ( 1 − t ) ] .

Thus, we complete the proof of Lemma 1.□

Lemma 2

Let f ∈ S X ( h , c , [ a , b ] ) . Define the mapping P : [ 0 , 1 ] ⊂ R → R by

P ( t ) = 1 ( b − a ) 2 ∫ a b ∫ a b f ( t x + ( 1 − t ) y ) d x d y .

Then:

  1. P ∈ S X h , c ( b − a ) 2 6 , [ 0 , 1 ] .

  2. For any t ∈ [ 0 , 1 ] , we have

    (3.4) 1 2 h ( 1 ⁄ 2 ) P ( 1 ⁄ 2 ) + c ( b − a ) 2 24 ( 1 − 2 t ) 2 ≤ P ( t ) ≤ P ( 0 ) [ h ( t ) + h ( 1 − t ) ] − c ( b − a ) 2 6 t ( 1 − t ) = P ( 1 ) [ h ( t ) + h ( 1 − t ) ] − c ( b − a ) 2 6 t ( 1 − t ) .

Particularly, if f is a strongly convex function with modulus c on [ a , b ] , we have

(3.5) P ( 1 ⁄ 2 ) + c ( b − a ) 2 24 ( 1 − 2 t ) 2 ≤ P ( t ) ≤ P ( 0 ) − c ( b − a ) 2 6 t ( 1 − t ) = P ( 1 ) − c ( b − a ) 2 6 t ( 1 − t ) .

Proof

(i) Let t 1 , t 2 , ξ , η ∈ [ 0 , 1 ] and ξ + η = 1 . Similar to the proof of Lemma 1, for any x ∈ [ a , b ] , we have

f ( ( ξ t 1 + η t 2 ) x + [ 1 − ( ξ t 1 + η t 2 ) ] y ) ≤ h ( ξ ) f ( t 1 x + ( 1 − t 1 ) y ) + h ( η ) f ( t 2 x + ( 1 − t 2 ) y ) − c ξ η ( t 2 − t 1 ) 2 ( x − y ) 2 .

Therefore,

P ( ξ t 1 + η t 2 ) ≤ h ( ξ ) P ( t 1 ) + h ( η ) P ( t 2 ) − c ξ η ( t 2 − t 1 ) 2 ( b − a ) 2 ∫ a b ∫ a b ( x − y ) 2 d x d y = h ( ξ ) P ( t 1 ) + h ( η ) P ( t 2 ) − ξ η c ( b − a ) 2 6 ( t 2 − t 1 ) 2 ,

which completes the proof of (i).

(ii) Since

f x + y 2 ≤ h ( 1 ⁄ 2 ) [ f ( t x + ( 1 − t ) y ) + f ( ( 1 − t ) x + t y ) ] − c 4 ( 1 − 2 t ) 2 ( x − y ) 2 ,

Fubini’s theorem yields that

(3.6) P ( 1 ⁄ 2 ) = 1 ( b − a ) 2 ∫ a b ∫ a b f x + y 2 d x d y ≤ 2 h ( 1 ⁄ 2 ) ( b − a ) 2 ∫ a b ∫ a b f ( t x + ( 1 − t ) y ) d x d y − c ( 1 − 2 t ) 2 4 ( b − a ) 2 ∫ a b ∫ a b ( x − y ) 2 d x d y = 2 h ( 1 ⁄ 2 ) P ( t ) − c ( b − a ) 2 24 ( 1 − 2 t ) 2 .

On the other hand, we easily infer from Definition 2 that

P ( t ) ≤ h ( t ) ( b − a ) 2 ∫ a b ∫ a b f ( x ) d x d y + h ( 1 − t ) ( b − a ) 2 ∫ a b ∫ a b f ( y ) d x d y − c t ( 1 − t ) ( b − a ) 2 ∫ a b ∫ a b ( x − y ) 2 d x d y = [ h ( t ) + h ( 1 − t ) ] P ( 1 ) − c ( b − a ) 2 6 t ( 1 − t ) = [ h ( t ) + h ( 1 − t ) ] P ( 0 ) − c ( b − a ) 2 6 t ( 1 − t ) ,

which combining with (3.6), obtains the desired results.□

It is notable that Lemmas 1 and 2 extend Theorems 1 and 2 in [30] as h ( t ) = t and c → 0 , respectively. And using these results, Dragomir [30] provided applications on some interesting and basic inequalities.

Theorem 2

Let f : [ a , b ] ⊂ R n → R be as in Theorem 1. Then:

  1. For any j ∈ { 1 , 2 , … , n } and any given t k ∈ [ 0 , 1 ] , k ≠ j , the partial mappings H j ( ⋅ ) = H ( t 1 , … , t j − 1 , ⋅ , t j + 1 , … , t n ) ∈ S X h j , c j ( b − a ) 2 12 , [ 0 , 1 ] respectively.

  2. For any t ∈ [ 0 , 1 ] ⊂ R n ,

    (3.7) 1 2 n ∏ j = 1 n h j ( 1 ⁄ 2 ) H ( 0 ) + A ˜ 3 ( t ) ≤ 1 2 n ∏ j = 1 n h j ( 1 ⁄ 2 ) H ( 0 ) + A 3 ( t ) ≤ H ( t ) ≤ ∏ j = 1 n [ h j ( t j ) + 2 h j ( 1 ⁄ 2 ) h j ( 1 − t j ) ] H ( 1 ) − ℬ 3 ( t ) ≤ ∏ j = 1 n [ h j ( t j ) + 2 h j ( 1 ⁄ 2 ) h j ( 1 − t j ) ] H ( 1 ) − ℬ ˜ 3 ( t ) ,

    where

    A 3 ( t ) = max λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ j = 1 n 1 2 j ∏ i = 1 j h λ i ( 1 ⁄ 2 ) c λ j ( b λ j − a λ j ) 2 12 t λ j 2 , A ˜ 3 ( t ) = 1 n ! ∑ λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ j = 1 n 1 2 j ∏ i = 1 j h λ i ( 1 ⁄ 2 ) c λ j ( b λ j − a λ j ) 2 12 t λ j 2 , ℬ 3 ( t ) = max λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ j = 1 n c λ j ( b λ j − a λ j ) 2 12 [ h λ j ( 1 − t λ j ) + t λ j ( 1 − t λ j ) ] ∏ i = 0 j − 1 [ h λ i ( t λ i ) + 2 h λ i ( 1 ⁄ 2 ) h λ i ( 1 − t λ i ) ] , ℬ ˜ 3 ( t ) = 1 n ! ∑ λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ j = 1 n c λ j ( b λ j − a λ j ) 2 12 [ h λ j ( 1 − t λ j ) + t λ j ( 1 − t λ j ) ] ∏ i = 0 j − 1 [ h λ i ( t λ i ) + 2 h λ i ( 1 ⁄ 2 ) h λ i ( 1 − t λ i ) ] ;

    here we denote h λ 0 ( t λ 0 ) + 2 h λ 0 ( 1 ⁄ 2 ) h λ 0 ( 1 − t λ 0 ) ≡ 1 .

Proof

The first and fourth inequalities of the proceeding theorem are clear, and we have to prove only the second and third inequalities.

(i) Without loss of generality, we just prove that H 1 ( ⋅ ) is a strongly h 1 -convex function with modulus c 1 ( b 1 − a 1 ) 2 12 on [ 0 , 1 ] ⊂ R , the other follows the same procedure. For any ξ , η , u , v ∈ [ 0 , 1 ] and ξ + η = 1 , Fubini’s theorem and Definition 2 tell us that

H 1 ( ξ u + η v ) = 1 ∏ j = 1 n ( b j − a j ) ∫ a n b n … ∫ a 1 b 1 f ( ξ u + η v ) x 1 + [ 1 − ( ξ u + η v ) ] a 1 + b 1 2 , t 2 x 2 + ( 1 − t 2 ) a 2 + b 2 2 , … , t n x n + ( 1 − t n ) a n + b n 2 d x 1 d x 2 … d x n = 1 ∏ j = 1 n ( b j − a j ) ∫ a n b n … ∫ a 1 b 1 f ξ u x 1 + ( 1 − u ) a 1 + b 1 2 + η v x 1 + ( 1 − v ) a 1 + b 1 2 , t 2 x 2 + ( 1 − t 2 ) a 2 + b 2 2 , … , t n x n + ( 1 − t n ) a n + b n 2 d x 1 d x 2 … d x n ≤ h 1 ( ξ ) H 1 ( u ) + h 1 ( η ) H 1 ( v ) − c 1 ξ η ( u − v ) 2 1 b 1 − a 1 ∫ a 1 b 1 x 1 − a 1 + b 1 2 2 d x 1 = h 1 ( ξ ) H 1 ( u ) + h 1 ( η ) H 1 ( v ) − c 1 ( b 1 − a 1 ) 2 ξ η 12 ( u − v ) 2 ,

which means that H 1 ( ⋅ ) ∈ S X h 1 , c 1 ( b 1 − a 1 ) 2 12 , [ 0 , 1 ] . This proves Theorem 2 (i).

(ii) It follows from (3.1) and Fubini’s theorem that

H ( t ) ≤ h 1 ( t 1 ) + 2 h 1 ( 1 ⁄ 2 ) h 1 ( 1 − t 1 ) ∏ j = 1 n ( b j − a j ) × ∫ a n b n … ∫ a 1 b 1 f x 1 , t 2 x 2 + ( 1 − t 2 ) a 2 + b 2 2 , … , t n x n + ( 1 − t n ) a n + b n 2 d x 1 d x 2 … d x n − c 1 ( b 1 − a 1 ) 2 12 [ h 1 ( 1 − t 1 ) + t 1 ( 1 − t 1 ) ] ≤ ∏ j = 1 2 [ h j ( t j ) + 2 h j ( 1 ⁄ 2 ) h j ( 1 − t j ) ] ∏ j = 1 n ( b j − a j ) × ∫ a n b n … ∫ a 1 b 1 f x 1 , x 2 , t 3 x 3 + ( 1 − t 3 ) a 3 + b 3 2 , … , t n x n + ( 1 − t n ) a n + b n 2 d x 1 d x 2 d x 3 … d x n − c 2 ( b 2 − a 2 ) 2 12 [ h 2 ( 1 − t 2 ) + t 2 ( 1 − t 2 ) ] [ h 1 ( t 1 ) + 2 h 1 ( 1 ⁄ 2 ) h 1 ( 1 − t 1 ) ] − c 1 ( b 1 − a 1 ) 2 12 [ h 1 ( 1 − t 1 ) + t 1 ( 1 − t 1 ) ] ≤ … ≤ ∏ j = 1 n [ h j ( t j ) + 2 h j ( 1 ⁄ 2 ) h j ( 1 − t j ) ] H ( 1 ) − ∑ j = 1 n c j ( b j − a j ) 2 12 [ h j ( 1 − t j ) + t j ( 1 − t j ) ] ∏ i = 0 j − 1 [ h i ( t i ) + 2 h i ( 1 ⁄ 2 ) h i ( 1 − t i ) ] ,

here we denote h 0 ( t 0 ) + 2 h 0 ( 1 ⁄ 2 ) h 0 ( 1 − t 0 ) ≡ 1 . Changing the integral order and taking similar argument as above, for any class { λ 1 , … , λ n } = { 1 , … , n } , where λ i ≠ λ j if i ≠ j , we also have

H ( t ) ≤ ∏ j = 1 n [ h j ( t j ) + 2 h j ( 1 ⁄ 2 ) h j ( 1 − t j ) ] H ( 1 ) − ∑ j = 1 n c λ j ( b λ j − a λ j ) 2 12 [ h λ j ( 1 − t λ j ) + t λ j ( 1 − t λ j ) ] ∏ i = 0 j − 1 [ h λ i ( t λ i ) + 2 h λ i ( 1 ⁄ 2 ) h λ i ( 1 − t λ i ) ] ,

where we define h λ 0 ( t λ 0 ) + 2 h λ 0 ( 1 ⁄ 2 ) h λ 0 ( 1 − t λ 0 ) ≡ 1 . Then the third inequality in (3.7) is immediately obtained by the proceeding inequality.

On the other hand, using Lemma 1 and Fubini’s theorem again, we derive that

H ( t ) ≥ 1 2 h 1 ( 1 ⁄ 2 ) 1 ∏ j = 2 n ( b j − a j ) ∫ a n b n … ∫ a 2 b 2 × f a 1 + b 1 2 , t 2 x 2 + ( 1 − t 2 ) a 2 + b 2 2 , … , t n x n + ( 1 − t n ) a n + b n 2 d x 2 … d x n + 1 2 h 1 ( 1 ⁄ 2 ) c 1 ( b 1 − a 1 ) 2 12 t 1 2 ≥ 1 2 2 h 1 ( 1 ⁄ 2 ) h 2 ( 1 ⁄ 2 ) 1 ∏ j = 3 n ( b j − a j ) ∫ a n b n … ∫ a 3 b 3 f a 1 + b 1 2 , a 2 + b 2 2 , t 3 x 3 + ( 1 − t 3 ) a 3 + b 3 2 , … , t n x n + ( 1 − t n ) a n + b n 2 d x 3 … d x n + 1 2 2 h 1 ( 1 ⁄ 2 ) h 2 ( 1 ⁄ 2 ) c 2 ( b 2 − a 2 ) 2 12 t 2 2 + 1 2 h 1 ( 1 ⁄ 2 ) c 1 ( b 1 − a 1 ) 2 12 t 1 2 ≥ … ≥ 1 2 n ∏ j = 1 n h j ( 1 ⁄ 2 ) f a + b 2 + ∑ j = 1 n 1 2 j ∏ i = 1 j h i ( 1 ⁄ 2 ) c j ( b j − a j ) 2 12 t j 2 .

A similar discussion as above, for any class { λ 1 , … , λ n } = { 1 , … , n } , here λ i ≠ λ j if i ≠ j , we also have

(3.8) H ( t ) ≥ 1 2 n ∏ j = 1 n h j ( 1 ⁄ 2 ) f a + b 2 + ∑ j = 1 n 1 2 j ∏ i = 1 j h λ i ( 1 ⁄ 2 ) c λ j ( b λ j − a λ j ) 2 12 t λ j 2 .

Therefore, the proof of Theorem 2 is easily completed by (3.8).□

Obviously, Theorem 2 is an extension of Theorem F and, consequently, we have the following conclusion.

Corollary 3

Let f : [ a , b ] ⊂ R n → R be a coordinated strongly convex function with modulus c. Then:

  1. The partial mappings H j ( ⋅ ) are strongly convex functions with modulus c ( b j − a j ) 2 12 , j = 1 , 2 … , n , respectively.

  2. For any t ∈ [ 0 , 1 ] ⊂ R n ,

    H ( 0 ) + c 12 ∣ t ⊗ ( b − a ) ∣ 2 ≤ H ( t ) ≤ H ( 1 ) − c 12 [ ∣ b − a ∣ 2 − ∣ t ⊗ ( b − a ) ∣ 2 ] .

Theorem H

Let f : [ a , b ] ⊂ R n → R be as in Theorem 2. Then:

  1. For any j ∈ { 1 , 2 , … , n } and any given t k ∈ [ 0 , 1 ] , k ≠ j , the partial mappings F j ( ⋅ ) = F ( t 1 , … , t j − 1 , ⋅ , t j + 1 , … , t n ) ∈ S X h j , c j ( b j − a j ) 2 6 , [ 0 , 1 ] , respectively.

  2. For any t ∈ [ 0 , 1 ] ,

    F ( 1⁄2 ) 2 n ∏ j = 1 n h j ( 1 ⁄ 2 ) + A ˜ 4 ( t ) ≤ F ( 1⁄2 ) 2 n ∏ j = 1 n h j ( 1 ⁄ 2 ) + A 4 ( t ) ≤ F ( t ) ≤ ∏ j = 1 n [ h j ( t j ) + h j ( 1 − t j ) ] F ( 1 ) − ℬ 4 ( t ) = ∏ j = 1 n [ h j ( t j ) + h j ( 1 − t j ) ] F ( 0 ) − ℬ 4 ( t ) ≤ ∏ j = 1 n [ h j ( t j ) + h j ( 1 − t j ) ] F ( 1 ) − ℬ ˜ 4 ( t ) = ∏ j = 1 n [ h j ( t j ) + h j ( 1 − t j ) ] F ( 0 ) − ℬ ˜ 4 ( t ) ,

    where

    A 4 ( t ) = max λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ j = 1 n 1 2 j ∏ i = 1 j h λ i ( 1 ⁄ 2 ) c λ i ( b λ j − a λ j ) 2 24 ( 1 − 2 t λ j ) 2 , A ˜ 4 ( t ) = 1 n ! ∑ λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ j = 1 n 1 2 j ∏ i = 1 j h λ i ( 1 ⁄ 2 ) c λ i ( b λ j − a λ j ) 2 24 ( 1 − 2 t λ j ) 2 , ℬ 4 ( t ) = max λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ j = 1 n c λ j ( b λ j − a λ j ) 2 6 t λ j ( 1 − t λ j ) ∏ i = 0 j − 1 [ h λ i ( t λ i ) + h λ i ( 1 − t λ i ) ] , ℬ ˜ 4 ( t ) = 1 n ! ∑ λ i ∈ { 1 , 2 , … , n } λ i ≠ λ j , i ≠ j ∑ j = 1 n c λ j ( b λ j − a λ j ) 2 6 t λ j ( 1 − t λ j ) ∏ i = 0 j − 1 [ h λ i ( t λ i ) + h λ i ( 1 − t λ i ) ] ;

    here we denote h λ 0 ( t λ 0 ) + h λ 0 ( 1 − t λ 0 ) ≡ 1 .

Proof

Similar to the proof of Theorem 2 (i), Theorem 3 (i) is easily obtained by Lemma 2 (i). Now we turn to prove the part of Theorem 3 (ii). The first and fourth inequalities are obvious, and we have to prove only the second and third inequalities.

(i) It follows from (3.4) and Fubini’s theorem that

F ( t ) ≤ h 1 ( t 1 ) + h 1 ( 1 − t 1 ) ( b 1 − a 1 ) ∏ j = 2 n ( b i − a i ) 2 ∫ [ a , b ] ∫ a n b n … ∫ a 2 b 2 f ( x 1 , t 2 x 2 + ( 1 − t 2 ) y 2 , … , t n x n + ( 1 − t n ) y n ) d x d y 2 … d y n − c 1 ( b 1 − a 1 ) 2 6 t 1 ( 1 − t 1 ) ≤ ∏ j = 1 2 [ h j ( t j ) + h j ( 1 − t j ) ] ( b 1 − a 1 ) ( b 2 − a 2 ) ∏ j = 3 n ( b i − a i ) 2 × ∫ [ a , b ] ∫ a n b n … ∫ a 3 b 3 f ( x 1 , x 2 , t 3 x 3 + ( 1 − t 3 ) y 3 , … , t n x n + ( 1 − t n ) y n ) d x d y 3 … d y n − c 2 ( b 2 − a 2 ) 2 6 t 2 ( 1 − t 2 ) [ h 1 ( t 1 ) + h 1 ( 1 − t 1 ) ] − c 1 ( b 1 − a 1 ) 2 6 t 1 ( 1 − t 1 ) ≤ … ≤ ∏ j = 1 n [ h j ( t j ) + h j ( 1 − t j ) ] ∏ j = 1 n ( b i − a i ) ∫ [ a , b ] f ( x ) d x − ∑ j = 1 n c j ( b j − a j ) 2 6 t j ( 1 − t j ) ∏ i = 0 j − 1 [ h i ( t i ) + h i ( 1 − t i ) ] ,

where h 0 ( t 0 ) + h 0 ( 1 − t 0 ) ≡ 1 . That is,

F ( t ) ≤ ∏ j = 1 n [ h j ( t j ) + h j ( 1 − t j ) ] ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x − ∑ j = 1 n c j ( b j − a j ) 2 6 t j ( 1 − t j ) ∏ i = 0 j − 1 [ h i ( t i ) + h i ( 1 − t i ) ] .

Changing the integral order and taking similar argument as above, for any class { λ 1 , … , λ n } = { 1 , … , n } , here λ i ≠ λ j if i ≠ j , we also have

(3.9) F ( t ) ≤ ∏ j = 1 n [ h j ( t j ) + h j ( 1 − t j ) ] ∣ [ a , b ] ∣ ∫ [ a , b ] f ( x ) d x − ∑ j = 1 n c λ j ( b λ j − a λ j ) 2 6 t λ j ( 1 − t λ j ) ∏ i = 0 j − 1 [ h λ i ( t λ i ) + h λ i ( 1 − t λ i ) ] = ∏ j = 1 n [ h j ( t j ) + h j ( 1 − t j ) ] F ( 1 ) − ∑ j = 1 n c λ j ( b λ j − a λ j ) 2 6 t λ j ( 1 − t λ j ) ∏ i = 0 j − 1 [ h λ i ( t λ i ) + h λ i ( 1 − t λ i ) ] ,

here we denote h λ 0 ( t λ 0 ) + h λ 0 ( 1 − t λ 0 ) ≡ 1 . Noting that F ( 1 ) = F ( 0 ) , we conclude the second inequality of (ii) by (3.9).

(ii) On the other hand, by Lemma 2 and Fubini’s theorem, we derive that

F ( t ) ≥ 1 2 h 1 ( 1 ⁄ 2 ) ∣ [ a , b ] ∣ 2 ∫ [ a , b ] ∫ [ a , b ] f x 1 + y 1 2 , t 2 x 2 + ( 1 − t 2 ) y 2 , … , t n x n + ( 1 − t n ) y n d x d y + 1 2 h 1 ( 1 ⁄ 2 ) c 1 ( b 1 − a 1 ) 2 24 ( 1 − 2 t 1 ) 2 ≥ 1 2 2 h 1 ( 1 ⁄ 2 ) h 2 ( 1 ⁄ 2 ) ∣ [ a , b ] ∣ 2 ∫ [ a , b ] ∫ [ a , b ] f x 1 + y 1 2 , x 2 + y 2 2 , t 3 x 3 + ( 1 − t 3 ) y 3 , … , t n x n + ( 1 − t n ) y n d x d y + 1 2 2 h 1 ( 1 ⁄ 2 ) h 2 ( 1 ⁄ 2 ) c 2 ( b 2 − a 2 ) 2 24 ( 1 − 2 t 2 ) 2 + 1 2 h 1 ( 1 ⁄ 2 ) c 1 ( b 1 − a 1 ) 2 24 ( 1 − 2 t 1 ) 2 ≥ … ≥ F ( 1⁄2 ) 2 n ∏ j = 1 n h j ( 1 ⁄ 2 ) + ∑ j = 1 n 1 2 j ∏ i = 1 j h i ( 1 ⁄ 2 ) c j ( b j − a j ) 2 24 ( 1 − 2 t j ) 2 .

A similar argument shows that, for any class { λ 1 , … , λ n } = { 1 , … , n } , λ i ≠ λ j if i ≠ j ,

(3.10) F ( t ) ≥ F ( 1⁄2 ) 2 n ∏ j = 1 n h j ( 1 ⁄ 2 ) + ∑ j = 1 n 1 2 j ∏ i = 1 j h λ i ( 1 ⁄ 2 ) c λ i ( b λ j − a λ j ) 2 24 ( 1 − 2 t λ j ) 2 .

Therefore, the proof of Theorem 3 is completed by (3.10).□

Corollary 4

Let f : [ a , b ] ⊂ R n → R be a coordinated strongly convex function with modulus c. Then:

  1. The partial mappings F j ( ⋅ ) are strongly convex functions with modulus c ( b j − a j ) 2 6 , j = 1 , 2 , … , n , respectively.

  2. For any t ∈ [ 0 , 1 ] ,

    F ( 1⁄2 ) + c 24 ∑ j = 1 n [ ( 1 − 2 t j ) 2 ( b j − a j ) 2 ] ≤ F ( t ) ≤ F ( 1 ) − c 6 ∑ j = 1 n [ t j ( 1 − t j ) ( b j − a j ) 2 ] = F ( 0 ) − c 6 ∑ j = 1 n [ t j ( 1 − t j ) ( b j − a j ) 2 ] .

Acknowledgement

The authors would like to express their deep thanks to the referees for many helpful comments and suggestions.

  1. Funding information: The research was supported by the National Natural Science Foundation of China (No. 11771358) and the Teaching Reform Project of Zhejiang Province (No. jg20180252).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflicts of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2024-02-26
Revised: 2024-07-15
Accepted: 2024-07-30
Published Online: 2024-09-16

© 2024 the author(s), published by De Gruyter

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

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  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
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  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
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  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
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  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
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  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
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  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
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  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
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  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
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  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
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  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
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  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
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  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
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  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
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  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0054
Keywords for this article
Hermite-Hadamard inequality; convex function; strongly h-convex function; co-ordinates
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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