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Integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex

Journal of Inequalities and Applications20132013:451

https://doi.org/10.1186/1029-242X-2013-451

Received: 30 November 2012

Accepted: 26 September 2013

Published: 7 November 2013

Abstract

In the paper, the authors establish some new inequalities of Hermite-Hadamard type for functions whose third derivatives are convex.

MSC: 26D15, 26A51, 41A55.

Keywords

Hermite-Hadamard’s integral inequality convex function third derivative

1 Introduction

It is common knowledge in mathematical analysis that a function f : I R R is said to be convex on an interval I if the inequality
f ( λ x + ( 1 λ ) y ) λ f ( x ) + ( 1 λ ) f ( y )
(1.1)
is valid for all x , y I and λ [ 0 , 1 ] . If f : I R R is a convex function on I and a , b I with a < b , then the double inequality
f ( a + b 2 ) 1 b a a b f ( x ) d x f ( a ) + f ( b ) 2
(1.2)

holds. This double inequality is known in the literature as Hermite-Hadamard’s integral inequality for convex functions. The definition of convex functions and Hermite-Hadamard’s integral inequality (1.2) have been generalized, refined, and extended by many mathematicians in a lot of references. Some of them may be recited as follows.

Theorem 1.1 ([[1], Theorems 2.2 and 2.3])

Let f : I R R be differentiable on I and a , b I with a < b .
  1. (1)
    If | f ( x ) | is a convex function on [ a , b ] , then
    | f ( a ) + f ( b ) 2 1 b a a b f ( x ) d x | b a 8 ( | f ( a ) | + | f ( b ) | ) .
    (1.3)
     
  2. (2)
    If | f ( x ) | p / ( p 1 ) for p > 1 is a convex function on [ a , b ] , then
    | f ( a ) + f ( b ) 2 1 b a a b f ( x ) d x | b a 2 ( p + 1 ) 1 / p [ | f ( a ) | p / ( p 1 ) + | f ( b ) | p / ( p 1 ) 2 ] ( p 1 ) / p .
    (1.4)
     

Theorem 1.2 ([[2], Theorems 2.2 and 2.3])

Let f : I R R be a differentiable mapping on I and a , b I with a < b . If | f | p / ( p 1 ) for p > 1 is convex on [ a , b ] , then
| f ( a + b 2 ) 1 b a a b f ( x ) d x | b a 16 ( 4 p + 1 ) 1 / p [ ( | f ( a ) | p / ( p 1 ) + 3 | f ( b ) | p / ( p 1 ) ) ( p 1 ) / p + ( 3 | f ( a ) | p / ( p 1 ) + | f ( b ) | p / ( p 1 ) ) ( p 1 ) / p ]
(1.5)
and
| f ( a + b 2 ) 1 b a a b f ( x ) d x | b a 4 ( 4 p + 1 ) 1 / p [ | f ( a ) | + | f ( b ) | ] .
(1.6)

Definition 1.1 ([3])

A function f : I R [ 0 , ) is said to be quasi-convex if
f ( λ x + ( 1 λ ) y ) sup { f ( x ) , f ( y ) }
(1.7)

holds for all x , y I and λ [ 0 , 1 ] .

Theorem 1.3 ([[4], Theorem 2])

Let f : I R R be differentiable on I such that f L ( [ a , b ] ) and a , b I with a < b . If | f | is quasi-convex on [ a , b ] , then
| a b f ( x ) d x b a 6 [ f ( a ) + 4 f ( a + b 2 ) + f ( b ) ] | ( b a ) 4 1152 [ max { | f ( a ) | , | f ( a + b 2 ) | } + max { | f ( a + b 2 ) | , | f ( b ) | } ] .
(1.8)

Definition 1.2 ([5])

Let s ( 0 , 1 ] . A function f : R 0 R 0 is said to be s-convex in the second sense if
f ( λ x + ( 1 λ ) y ) λ s f ( x ) + ( 1 λ ) s f ( y )
(1.9)

for all x , y I and λ [ 0 , 1 ] .

Theorem 1.4 ([[6], Theorem 3.1])

Let f : I R 0 R be differentiable on I , a , b I with a < b , and f L ( [ a , b ] ) . If q 1 and | f | is s-convex in the second sense on [ a , b ] for s ( 0 , 1 ] , then
| f ( a ) + f ( b ) 2 1 b a a b f ( x ) d x b a 12 [ f ( b ) f ( a ) ] | ( b a ) 3 192 [ 2 2 s ( s + 6 + 2 s + 2 s ) ( s + 2 ) ( s + 3 ) ( s + 4 ) ] 1 / q [ | f ( a ) | q + | f ( b ) | q ] 1 / q .
(1.10)

For more information on Hermite-Hadamard type inequalities, please refer to [719], for example, and to monographs [20, 21] and related references therein.

In this paper, we will create some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex.

2 Lemma

For establishing some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex, we need an integral identity below.

Lemma 2.1 Let f : I R R be a three times differentiable mapping on I and a , b I with a < b . If f L ( [ a , b ] ) , then
f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] = ( b a ) 3 96 [ 0 1 t ( 1 t ) ( 2 t ) f ( 1 t 2 a + 1 + t 2 b ) d t 0 1 t ( 1 t ) ( 2 t ) f ( 1 + t 2 a + 1 t 2 b ) d t ] .
(2.1)
Proof Integrating by part and changing variable of definite integral yield
0 1 t ( 1 t ) ( 2 t ) f ( 1 t 2 a + 1 + t 2 b ) d t = 2 b a 0 1 ( 3 t 2 6 t + 2 ) f ( 1 t 2 a + 1 + t 2 b ) d t = 4 ( b a ) 2 [ f ( b ) + 2 f ( a + b 2 ) ] + 48 ( b a ) 3 0 1 ( t 1 ) d f ( 1 t 2 a + 1 + t 2 b ) = 4 ( b a ) 2 [ f ( b ) + 2 f ( a + b 2 ) ] + 48 ( b a ) 3 f ( a + b 2 ) 48 ( b a ) 3 0 1 f ( 1 t 2 a + 1 + t 2 b ) d t
and
0 1 t ( 1 t ) ( 2 t ) f ( 1 + t 2 a + 1 t 2 b ) d t = 2 b a 0 1 ( 3 t 2 6 t + 2 ) f ( 1 + t 2 a + 1 t 2 b ) d t = 4 ( b a ) 2 [ f ( a ) + 2 f ( a + b 2 ) ] 48 ( b a ) 3 0 1 ( t 1 ) d f ( 1 + t 2 a + 1 t 2 b ) = 4 ( b a ) 2 [ f ( a ) + 2 f ( a + b 2 ) ] 48 ( b a ) 3 f ( a + b 2 ) + 48 ( b a ) 3 0 1 f ( 1 + t 2 a + 1 t 2 b ) d t .

Lemma 2.1 is thus proved. □

3 Hermite-Hadamard type inequalities for convex functions

Basing on Lemma 2.1, we now start out to establish some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex.

Theorem 3.1 Let f : I R R be three times differentiable on I and f L ( [ a , b ] ) for a , b I with a < b . If | f | q for q 1 is convex on [ a , b ] , then
| f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 384 { [ 4 | f ( a ) | q + 11 | f ( b ) | q 15 ] 1 / q + [ 11 | f ( a ) | q + 4 | f ( b ) | q 15 ] 1 / q } .
(3.1)
Proof Since | f | q is convex on [ a , b ] , by Lemma 2.1 and Hölder’s inequality, we have
| f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 { 0 1 t ( 1 t ) ( 2 t ) | f ( 1 t 2 a + 1 + t 2 b ) | d t + 0 1 t ( 1 t ) ( 2 t ) | f ( 1 + t 2 a + 1 t 2 b ) | d t } ( b a ) 3 96 [ 0 1 t ( 1 t ) ( 2 t ) d t ] 1 1 / q × { [ 0 1 t ( 1 t ) ( 2 t ) | f ( 1 t 2 a + 1 + t 2 b ) | q d t ] 1 / q + [ 0 1 t ( 1 t ) ( 2 t ) | f ( 1 + t 2 a + 1 t 2 b ) | q d t ] 1 / q } ( b a ) 3 96 ( 1 4 ) 1 1 / q { [ 1 2 0 1 t ( 1 t ) 2 ( 2 t ) | f ( a ) | q d t + 1 2 0 1 t ( 1 t 2 ) ( 2 t ) | f ( b ) | q d t ] 1 / q + [ 1 2 0 1 t ( 1 t 2 ) ( 2 t ) | f ( a ) | q d t + 1 2 0 1 t ( 1 t ) 2 ( 2 t ) | f ( b ) | q d t ] 1 / q } = ( b a ) 3 384 { [ 4 | f ( a ) | q + 11 | f ( b ) | q 15 ] 1 / q + [ 11 | f ( a ) | q + 4 | f ( b ) | q 15 ] 1 / q } .

The proof of Theorem 3.1 is complete. □

Corollary 3.1 Under conditions of Theorem  3.1, if q = 1 , we have
| f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 384 [ | f ( a ) | + | f ( b ) | ] .
(3.2)
Theorem 3.2 Let f : I R R be three times differentiable on I and f L ( [ a , b ] ) for a , b I with a < b . If | f | q for q > 1 is convex on [ a , b ] and if q r and s 0 , then
| f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 [ 2 B ( 2 q r 1 q 1 , 2 q s 1 q 1 ) B ( 3 q r 2 q 1 , 2 q s 1 q 1 ) ] 1 1 / q × ( 1 2 ) 1 / q { ( [ 2 B ( r + 1 , s + 2 ) B ( r + 2 , s + 2 ) ] | f ( a ) | q + [ 2 B ( r + 1 , s + 1 ) + B ( r + 2 , s + 1 ) B ( r + 3 , s + 1 ) ] | f ( b ) | q ) 1 / q + ( [ 2 B ( r + 1 , s + 1 ) + B ( r + 2 , s + 1 ) B ( r + 3 , s + 1 ) ] | f ( a ) | q + [ 2 B ( r + 1 , s + 2 ) B ( r + 2 , s + 2 ) ] | f ( b ) | q ) 1 / q } ,
where B ( x , y ) is the classical Beta function, which may be defined for Re ( x ) > 0 and Re ( y ) > 0 by
B ( x , y ) = 0 1 t x 1 ( 1 t ) y 1 d t .
(3.3)
Proof By Lemma 2.1, Hölder’s inequality, and the convexity of | f | q on [ a , b ] , we have
| f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 { 0 1 t ( 1 t ) ( 2 t ) | f ( 1 t 2 a + 1 + t 2 b ) | d t + 0 1 t ( 1 t ) ( 2 t ) | f ( 1 + t 2 a + 1 t 2 b ) | d t } ( b a ) 3 96 ( 0 1 t ( q r ) / ( q 1 ) ( 1 t ) ( q s ) / ( q 1 ) ( 2 t ) d t ) 1 1 / q × { ( 0 1 t r ( 1 t ) s ( 2 t ) | f ( 1 t 2 a + 1 + t 2 b ) | q d t ) 1 / q + ( 0 1 t r ( 1 t ) s ( 2 t ) | f ( 1 + t 2 a + 1 t 2 b ) | q d t ) 1 / q } ( b a ) 3 96 [ 2 B ( 2 q r 1 q 1 , 2 q s 1 q 1 ) B ( 3 q r 2 q 1 , 2 q s 1 q 1 ) ] 1 1 / q × { [ 1 2 0 1 t r ( 1 t ) s + 1 ( 2 t ) | f ( a ) | q d t + 1 2 0 1 t r ( 1 + t ) ( 1 t ) s ( 2 t ) | f ( b ) | q d t ] 1 / q + [ 1 2 0 1 t r ( 1 + t ) ( 1 t ) s ( 2 t ) | f ( a ) | q d t + 1 2 0 1 t r ( 1 t ) s + 1 ( 2 t ) | f ( b ) | q d t ] 1 / q } = ( b a ) 3 96 [ 2 B ( 2 q r 1 q 1 , 2 q s 1 q 1 ) B ( 3 q r 2 q 1 , 2 q s 1 q 1 ) ] 1 1 / q × ( 1 2 ) 1 / q { [ [ 2 B ( r + 1 , s + 2 ) B ( r + 2 , s + 2 ) ] | f ( a ) | q + [ 2 B ( r + 1 , s + 1 ) + B ( r + 2 , s + 1 ) B ( r + 3 , s + 1 ) ] | f ( b ) | q ] 1 / q + [ [ 2 B ( r + 1 , s + 1 ) + B ( r + 2 , s + 1 ) B ( r + 3 , s + 1 ) ] | f ( a ) | q + [ 2 B ( r + 1 , s + 2 ) B ( r + 2 , s + 2 ) ] | f ( b ) | q ] 1 / q } .

The proof of Theorem 3.2 is completed. □

Corollary 3.2 Under conditions of Theorem  3.2,
  1. (1)
    if r = 0 , we have
    | f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 [ 2 B ( 2 q 1 q 1 , 2 q s 1 q 1 ) B ( 3 q 2 q 1 , 2 q s 1 q 1 ) ] 1 1 / q ( 1 2 ( s + 1 ) ( s + 2 ) ( s + 3 ) ) 1 / q × { [ ( s + 1 ) ( 2 s + 5 ) | f ( a ) | q + ( 2 s 2 + 11 s + 13 ) | f ( b ) | q ] 1 / q + [ ( 2 s 2 + 11 s + 13 ) | f ( a ) | q + ( s + 1 ) ( 2 s + 5 ) | f ( b ) | q ] 1 / q } ;
     
  2. (2)
    if s = 0 , we have
    | f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 [ 2 B ( 2 q r 1 q 1 , 2 q 1 q 1 ) B ( 3 q r 2 q 1 , 2 q 1 q 1 ) ] 1 1 / q [ 1 2 ( r + 1 ) ( r + 2 ) ( r + 3 ) ] 1 / q × { [ ( r + 5 ) | f ( a ) | q + ( 2 r 2 + 11 r + 13 ) | f ( b ) | q ] 1 / q + [ ( 2 r 2 + 11 r + 13 ) | f ( a ) | q + ( r + 5 ) | f ( b ) | q ] 1 / q } ;
     
  3. (3)
    if r = s = 0 , we have
    | f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 [ 2 B ( 2 q 1 q 1 , 2 q 1 q 1 ) B ( 3 q 2 q 1 , 2 q 1 q 1 ) ] 1 1 / q ( 3 2 ) 1 / q × { [ 5 | f ( a ) | q + 13 | f ( b ) | q 18 ] 1 / q + [ 13 | f ( a ) | q + 5 | f ( b ) | q 18 ] 1 / q } ;
     
  4. (4)
    if r = q , we have
    | f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 [ ( 5 q 2 s 3 ) ( q 1 ) ( q s ) 2 + ( 5 q 3 s 2 ) ( q 1 ) ] 1 1 / q ( 1 2 ) 1 / q × { [ [ 2 B ( q + 1 , s + 2 ) B ( q + 2 , s + 2 ) ] | f ( a ) | q + [ 2 B ( q + 1 , s + 1 ) + B ( q + 2 , s + 1 ) B ( q + 3 , s + 1 ) ] | f ( b ) | q ] 1 / q + [ [ 2 B ( q + 1 , s + 1 ) + B ( q + 2 , s + 1 ) B ( q + 3 , s + 1 ) ] | f ( a ) | q + [ 2 B ( q + 1 , s + 2 ) B ( q + 2 , s + 2 ) ] | f ( b ) | q ] 1 / q } ;
     
  5. (5)
    if s = q , we have
    | f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 ( ( q 1 ) ( 4 q r 3 ) ( 2 q r 1 ) ( 3 q r 2 ) ) 1 1 / q ( 1 2 ) 1 / q × { [ [ 2 B ( r + 1 , q + 2 ) B ( r + 2 , q + 2 ) ] | f ( a ) | q + [ 2 B ( r + 1 , q + 1 ) + B ( r + 2 , q + 1 ) B ( r + 3 , q + 1 ) ] | f ( b ) | q ] 1 / q + [ [ 2 B ( r + 1 , q + 1 ) + B ( r + 2 , q + 1 ) B ( r + 3 , q + 1 ) ] | f ( a ) | q + [ 2 B ( r + 1 , q + 2 ) B ( r + 2 , q + 2 ) ] | f ( b ) | q ] 1 / q } ;
     
  6. (6)
    if r = s = q , we have
    | f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | 3 1 / q ( b a ) 3 64 { [ [ 2 B ( q + 1 , q + 2 ) B ( q + 2 , q + 2 ) ] | f ( a ) | q + [ 2 B ( q + 1 , q + 1 ) + B ( q + 2 , q + 1 ) B ( q + 3 , q + 1 ) ] | f ( b ) | q ] 1 / q + [ [ 2 B ( q + 1 , q + 1 ) + B ( q + 2 , q + 1 ) B ( q + 3 , q + 1 ) ] | f ( a ) | q + [ 2 B ( q + 1 , q + 2 ) B ( q + 2 , q + 2 ) ] | f ( b ) | q ] 1 / q } .
     
Theorem 3.3 Let f : I R R be three times differentiable on I and f L ( [ a , b ] ) for a , b I with a < b . If | f | q is convex on [ a , b ] for q > 1 and q 0 , then
| f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 [ ( 2 ξ + 2 + 1 ) ξ 2 ξ + 2 + 5 ξ 3 + 6 ξ 2 + 11 ξ + 6 ] 1 1 / q [ 1 ( + 1 ) ( + 2 ) ( + 3 ) ( + 4 ) ] 1 / q × { [ ( 2 + 1 2 ( 2 + 1 + 1 ) + 2 + 3 7 ) | f ( a ) | q + ( ( 2 + 1 + 1 ) 2 + 2 ( 7 × 2 + 5 ) 3 × 2 + 3 + 27 ) | f ( b ) | q ] 1 / q + [ ( ( 2 + 1 + 1 ) 2 + 2 ( 7 × 2 + 5 ) 3 × 2 + 3 + 27 ) | f ( a ) | q + ( 2 + 1 2 ( 2 + 1 + 1 ) + 2 + 3 7 ) | f ( b ) | q ] 1 / q } ,

where ξ = q q 1 .

Proof By Lemma 2.1, Hölder’s inequality, and the convexity of | f | q on [ a , b ] , we have
| f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 { 0 1 t ( 1 t ) ( 2 t ) | f ( 1 t 2 a + 1 + t 2 b ) | d t + 0 1 t ( 1 t ) ( 2 t ) | f ( 1 + t 2 a + 1 t 2 b ) | d t } ( b a ) 3 96 ( 0 1 t ( 1 t ) ( 2 t ) ( q ) / ( q 1 ) d t ) 1 1 / q × { ( 0 1 t ( 1 t ) ( 2 t ) | f ( 1 t 2 a + 1 + t 2 b ) | q d t ) 1 / q + ( 0 1 t ( 1 t ) ( 2 t ) | f ( 1 + t 2 a + 1 t 2 b ) | q d t ) 1 / q } ( b a ) 3 96 ( 0 1 t ( 1 t ) ( 2 t ) ( q ) / ( q 1 ) d t ) 1 1 / q × { [ 1 2 0 1 t ( 1 t ) ( 2 t ) [ ( 1 t ) | f ( a ) | q + ( 1 + t ) | f ( b ) | q ] d t ] 1 / q + [ 1 2 0 1 t ( 1 t ) ( 2 t ) [ ( 1 + t ) | f ( a ) | q + ( 1 t ) | f ( b ) | q ] d t ] 1 / q } = ( b a ) 3 96 [ ( 2 ξ + 2 + 1 ) ξ 2 ξ + 2 + 5 ξ 3 + 6 ξ 2 + 11 ξ + 6 ] 1 1 / q [ 1 ( + 1 ) ( + 2 ) ( + 3 ) ( + 4 ) ] 1 / q × { [ ( 2 + 1 2 ( 2 + 1 + 1 ) + 2 + 3 7 ) | f ( a ) | q + ( ( 2 + 1 + 1 ) 2 + 2 ( 7 × 2 + 5 ) 3 × 2 + 3 + 27 ) | f ( b ) | q ] 1 / q + [ ( ( 2 + 1 + 1 ) 2 + 2 ( 7 × 2 + 5 ) 3 × 2 + 3 + 27 ) | f ( a ) | q + ( 2 + 1 2 ( 2 + 1 + 1 ) + 2 + 3 7 ) | f ( b ) | q ] 1 / q } .

The proof of Theorem 3.3 is complete. □

Corollary 3.3 Under conditions of Theorem  3.3.
  1. (1)
    if = 0 , we have
    | f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 ( 1 6 ) 1 / q [ ( q 1 ) 2 ( 6 q + 2 ( 3 q 2 ) / ( q 1 ) 5 ) ( 2 q 1 ) ( 3 q 2 ) ( 4 q 3 ) ] 1 1 / q × { [ | f ( a ) | q + 3 | f ( b ) | q 4 ] 1 / q + [ 3 | f ( a ) | q + | f ( b ) | q 4 ] 1 / q } ,
    (3.4)
     
  2. (2)
    if = q , we have
    | f ( a + b 2 ) 1 b a a b f ( x ) d x + b a 24 [ f ( b ) f ( a ) ] | ( b a ) 3 96 ( 1 6 ) 1 1 / q [ 1 ( q + 1 ) ( q + 2 ) ( q + 3 ) ( q + 4 ) ] 1 / q × { [ ( 2 q + 1 q 2 ( 2 q + 1 + 1 ) q + 2 q + 3 7 ) | f ( a ) | q + ( ( 2 q + 1 + 1 ) q 2 + 2 ( 7 × 2 q + 5 ) q 3 × 2 q + 3 + 27 ) | f ( b ) | q ] 1 / q + [ ( ( 2 q + 1 + 1 ) q 2 + 2 ( 7 × 2 q + 5 ) q 3 × 2 q + 3 + 27 ) | f ( a ) | q + ( 2 q + 1 q 2 ( 2 q + 1 + 1 ) q + 2 q + 3 7 ) | f ( b ) | q ] 1 / q } .
     

Declarations

Acknowledgements

The authors appreciate the editor and anonymous referees for their careful reading, helpful comments on, and valuable suggestions to the original version of this manuscript. This work was partially supported by the NNSF of China under Grant No. 11361038, by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159, and by the Science Research Funding of the Inner Mongolia University for Nationalities under Grant No. NMD1225, China.

Authors’ Affiliations

(1)
College of Mathematics, Inner Mongolia University for Nationalities
(2)
Department of Mathematics, College of Science, Tianjin Polytechnic University

References

  1. Dragomir SS: Two mappings on connection to Hadamard’s inequality. J. Math. Anal. Appl. 1992, 167(1):49–56. Available online at 10.1016/0022-247X(92)90233-4MathSciNetView ArticleGoogle Scholar
  2. Kirmaci US: Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula. Appl. Math. Comput. 2004, 147(1):137–146. Available online at 10.1016/S0096-3003(02)00657-4MathSciNetView ArticleGoogle Scholar
  3. Dragomir SS, Pečarić J, Persson L-E: Some inequalities of Hadamard type. Soochow J. Math. 1995, 21(3):335–341.MathSciNetGoogle Scholar
  4. Alomari M, Hussain S: Two inequalities of Simpson type for quasi-convex functions and applications. Appl. Math. E-Notes 2011, 11: 110–117.MathSciNetGoogle Scholar
  5. Hudzik H, Maligranda L: Some remarks on s -convex functions. Aequ. Math. 1994, 48(1):100–111. Available online at 10.1007/BF01837981MathSciNetView ArticleGoogle Scholar
  6. Chun L, Qi F: Integral inequalities of Hermite-Hadamard type for functions whose 3rd derivatives are s -convex. Appl. Math. 2012, 3(11):1680–1685. Available online at 10.4236/am.2012.311232View ArticleGoogle Scholar
  7. Bai R-F, Qi F, Xi B-Y: Hermite-Hadamard type inequalities for the m - and ( α , m ) -logarithmically convex functions. Filomat 2013, 27(1):1–7. Available online at 10.2298/FIL1301001BMathSciNetView ArticleGoogle Scholar
  8. Bai S-P, Qi F:Some inequalities for ( s 1 , m 1 ) - ( s 2 , m 2 ) -convex functions on the co-ordinates. Glob. J. Math. Anal. 2013, 1(1):22–28.MathSciNetGoogle Scholar
  9. Bakula MK, Özdemir ME, Pečarić J: Hadamard type inequalities for m -convex and ( α , m ) -convex functions. J. Inequal. Pure Appl. Math. 2008., 9(4): Article ID 96. Available online at http://www.emis.de/journals/JIPAM/article1032.htmlGoogle Scholar
  10. Kavurmaci, H, Avci, M, Özdemir, ME: New inequalities of Hermite-Hadamard type for convex functions with applications. Available online at arXiv:1006.1593. e-printatarXiv.org
  11. Qi F, Wei Z-L, Yang Q: Generalizations and refinements of Hermite-Hadamard’s inequality. Rocky Mt. J. Math. 2005, 35(1):235–251. Available online at 10.1216/rmjm/1181069779MathSciNetView ArticleGoogle Scholar
  12. Shuang Y, Yin H-P, Qi F: Hermite-Hadamard type integral inequalities for geometric-arithmetically s -convex functions. Analysis 2013, 33(2):197–208. Available online at 10.1524/anly.2013.1192MathSciNetView ArticleGoogle Scholar
  13. Xi B-Y, Qi F: Some Hermite-Hadamard type inequalities for differentiable convex functions and applications. Hacet. J. Math. Stat. 2013, 42(3):243–257.MathSciNetGoogle Scholar
  14. Xi B-Y, Qi F: Hermite-Hadamard type inequalities for functions whose derivatives are of convexities. Nonlinear Funct. Anal. Appl. 2013, 18(2):163–176.Google Scholar
  15. Xi B-Y, Qi F: Some inequalities of Hermite-Hadamard type for h -convex functions. Adv. Inequal. Appl. 2013, 2(1):1–15.MathSciNetGoogle Scholar
  16. Xi B-Y, Wang Y, Qi F:Some integral inequalities of Hermite-Hadamard type for extended ( s , m ) -convex functions. Transylv. J. Math. Mech. 2013, 5(1):69–84.MathSciNetGoogle Scholar
  17. Zhang T-Y, Ji A-P, Qi F: Integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions. Proc. Jangjeon Math. Soc. 2013, 16(3):399–407.MathSciNetGoogle Scholar
  18. Zhang T-Y, Ji A-P, Qi F: On integral inequalities of Hermite-Hadamard type for s -geometrically convex functions. Abstr. Appl. Anal. 2013., 2013: Article ID 560586. Available online at 10.1155/2012/560586Google Scholar
  19. Zhang T-Y, Ji A-P, Qi F: Some inequalities of Hermite-Hadamard type for GA-convex functions with applications to means. Matematiche 2013, 68(1):229–239. Available online at 10.4418/2013.68.1.17MathSciNetGoogle Scholar
  20. Dragomir SS, Pearce CEM RGMIA Monographs. In Selected Topics on Hermite-Hadamard Type Inequalities and Applications. Victoria University, Melbourne; 2000. Available online at http://rgmia.org/monographs/hermite_hadamard.htmlGoogle Scholar
  21. Niculescu CP, Persson L-E: Convex functions and their applications. In CMS Books in Mathematics. Springer, Berlin; 2005.Google Scholar

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© Chun and Qi; licensee Springer. 2013

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