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

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.

1 Introduction

It is common knowledge in mathematical analysis that a function f:IRR is said to be convex on an interval I if the inequality

f ( λ x + ( 1 λ ) y ) λf(x)+(1λ)f(y)
(1.1)

is valid for all x,yI and λ[0,1]. If f:IRR is a convex function on I and a,bI with a<b, then the double inequality

f ( a + b 2 ) 1 b a a b f(x)dx 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 RR 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 RR 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:IR[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,yI and λ[0,1].

Theorem 1.3 ([[4], Theorem 2])

Let f:IRR 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,yI 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 q1 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:IRR 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:IRR be three times differentiable on I and f L([a,b]) for a,b I with a<b. If | f | q for q1 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:IRR 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 qr and s0, 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 dt.
(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:IRR 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 q0, 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 } .

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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.

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Chun, L., Qi, F. Integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex. J Inequal Appl 2013, 451 (2013). https://doi.org/10.1186/1029-242X-2013-451

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Keywords

  • Hermite-Hadamard’s integral inequality
  • convex function
  • third derivative