Open Access

New general integral inequalities for quasi-geometrically convex functions via fractional integrals

Journal of Inequalities and Applications20132013:491

https://doi.org/10.1186/1029-242X-2013-491

Received: 4 August 2013

Accepted: 9 September 2013

Published: 7 November 2013

Abstract

In this paper, the author introduces the concept of the quasi-geometrically convex functions, gives Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms and defines a new identity for fractional integrals. By using this identity, the author obtains new estimates on generalization of Hadamard et al. type inequalities for quasi-geometrically convex functions via Hadamard fractional integrals.

MSC:26A33, 26A51, 26D15.

Keywords

quasi-geometrically convex function Hermite-Hadamard-type inequalities Hadamard fractional integrals

1 Introduction

Let a real function f be defined on some nonempty interval I of a real line . The function f is said to be convex on I if inequality
f ( t x + ( 1 t ) y ) t f ( x ) + ( 1 t ) f ( y )
(1)

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

We recall that the notion of quasi-convex function generalizes the notion of convex function. More exactly, a function f : [ a , b ] R R is said to be quasi-convex on [ a , b ] if
f ( t x + ( 1 t ) y ) max { f ( x ) , f ( y ) }

for all x , y [ a , b ] and t [ 0 , 1 ] . Clearly, any convex function is a quasi-convex function. Furthermore, there exist quasi-convex functions which are not convex (see [1]).

The following inequalities are well known in the literature as the Hermite-Hadamard inequality, the Ostrowski inequality and the Simpson inequality, respectively.

Theorem 1.1 Let f : I R R be a convex function defined on the interval I of real numbers, and let a , b I with a < b . The following double inequality holds:
f ( a + b 2 ) 1 b a a b f ( x ) d x f ( a ) + f ( b ) 2 .
(2)
Theorem 1.2 Let f : I R R be a mapping differentiable in I , the interior of I, and let a , b I with a < b . If | f ( x ) | M , x [ a , b ] , then the following inequality holds:
| f ( x ) 1 b a a b f ( t ) d t | M b a [ ( x a ) 2 + ( b x ) 2 2 ]

for all x [ a , b ] .

Theorem 1.3 Let f : [ a , b ] R be a four times continuously differentiable mapping on ( a , b ) and f ( 4 ) = sup x ( a , b ) | f ( 4 ) ( x ) | < . Then the following inequality holds:
| 1 3 [ f ( a ) + f ( b ) 2 + 2 f ( a + b 2 ) ] 1 b a a b f ( x ) d x | 1 2 , 880 f ( 4 ) ( b a ) 4 .

The following definitions are well known in the literature.

Definition 1.1 ([2, 3])

A function f : I ( 0 , ) R is said to be GA-convex (geometric-arithmetically convex) if
f ( x t y 1 t ) t f ( x ) + ( 1 t ) f ( y )

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

Definition 1.2 ([2, 3])

A function f : I ( 0 , ) ( 0 , ) is said to be GG-convex (called in [4] a geometrically convex function) if
f ( x t y 1 t ) f ( x ) t f ( y ) ( 1 t )

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

We will now give definitions of the right-hand side and left-hand side Hadamard fractional integrals which are used throughout this paper.

Definition 1.3 Let f L [ a , b ] . The right-hand side and left-hand side Hadamard fractional integrals J a + α f and J b α f of order α > 0 with b > a 0 are defined by
J a + α f ( x ) = 1 Γ ( α ) a x ( ln x t ) α 1 f ( t ) d t t , a < x < b
and
J b α f ( x ) = 1 Γ ( α ) x b ( ln t x ) α 1 f ( t ) d t t , a < x < b ,

respectively, where Γ ( α ) is the Gamma function defined by Γ ( α ) = 0 e t t α 1 d t (see [5]).

In recent years, many authors have studied error estimations for Hermite-Hadamard, Ostrowski and Simpson inequalities; for refinements, counterparts, generalization see [4, 620].

In this paper, the concept of the quasi-geometrically convex function is introduced, Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms are established, and a new identity for Hadamard fractional integrals is defined. By using this identity, author obtains a generalization of Hadamard, Ostrowski and Simpson type inequalities for quasi-geometrically convex functions via Hadamard fractional integrals.

2 Main results

Let f : I ( 0 , ) R be a differentiable function on I , the interior of I, throughout this section we will take
I f ( x , λ , α , a , b ) = ( 1 λ ) [ ln α x a + ln α b x ] f ( x ) + λ [ f ( a ) ln α x a + f ( b ) ln α b x ] Γ ( α + 1 ) [ J x α f ( a ) + J x + α f ( b ) ] ,

where a , b I with a < b , x [ a , b ] , λ [ 0 , 1 ] , α > 0 and Γ is the Euler Gamma function.

Definition 2.1 A function f : I ( 0 , ) R is said to be quasi-geometrically convex on I if
f ( x t y 1 t ) sup { f ( x ) , f ( y ) } ,

for any x , y I and t [ 0 , 1 ] .

Remark 2.1 Clearly, any GA-convex and geometrically convex functions are quasi-geometrically convex functions. Furthermore, there exist quasi-geometrically convex functions which are neither GA-convex nor geometrically convex. In that context, we point out an elementary example. The function f : ( 0 , 4 ] R ,
f ( x ) = { 1 , x ( 0 , 1 ] , ( x 2 ) 2 , x [ 1 , 4 ]

is neither GA-convex nor geometrically convex on ( 0 , 4 ] , but it is a quasi-geometrically convex function on ( 0 , 4 ] .

Proposition 2.1 If f : I ( 0 , ) R is convex and nondecreasing, then it is quasi-geometrically convex on I.

Proof This follows from
f ( x t y 1 t ) f ( t x + ( 1 t ) y ) t f ( x ) + ( 1 t ) f ( y ) sup { f ( x ) , f ( y ) } ,

for all x , y I and t [ 0 , 1 ] . □

Proposition 2.2 If f : I ( 0 , ) R is quasi-convex and nondecreasing, then it is quasi-geometrically convex on I. If f : I ( 0 , ) R is quasi-geometrically convex and nonincreasing, then it is quasi-convex on I.

Proof These conclusions follows from
f ( x t y 1 t ) f ( t x + ( 1 t ) y ) sup { f ( x ) , f ( y ) }
and
f ( t x + ( 1 t ) y ) f ( x t y 1 t ) sup { f ( x ) , f ( y ) }

for all x , y I and t [ 0 , 1 ] , respectively. □

Hermite-Hadamard’s inequalities can be represented for GA-convex functions in fractional integral forms as follows.

Theorem 2.1 Let f : I ( 0 , ) R be a function such that f L [ a , b ] , where a , b I with a < b . If f is a GA-convex function on [ a , b ] , then the following inequalities for fractional integrals hold:
f ( a b ) Γ ( α + 1 ) 2 ( ln b a ) α { J a + α f ( b ) + J b α f ( a ) } f ( a ) + f ( b ) 2
(3)

with α > 0 .

Proof Since f is a GA-convex function on [ a , b ] , we have for all x , y [ a , b ] (with t = 1 / 2 in inequality (1)),
f ( x y ) f ( x ) + f ( y ) 2 .
Choosing x = a t b 1 t , y = b t a 1 t , we get
f ( a b ) f ( a t b 1 t ) + f ( b t a 1 t ) 2 .
(4)
Multiplying both sides of (4) by t α 1 , then integrating the resulting inequality with respect to t over [ 0 , 1 ] , we obtain
f ( a b ) α 2 { 0 1 f ( a t b 1 t ) d t + 0 1 f ( b t a 1 t ) d t } = α 2 { a b ( ln b ln u ln b ln a ) α 1 f ( u ) d u u ln b a + a b ( ln u ln a ln b ln a ) α 1 f ( u ) d u u ln b a } = α Γ ( α ) 2 ( ln b a ) α { J a + α f ( b ) + J b α f ( a ) } = Γ ( α + 1 ) 2 ( ln b a ) α { J a + α f ( b ) + J b α f ( a ) } ,

and the first inequality is proved.

For the proof of the second inequality in (3), we first note that if f is a convex function, then for t [ 0 , 1 ] , it yields
f ( a t b 1 t ) t f ( a ) + ( 1 t ) f ( b )
and
f ( b t a 1 t ) t f ( b ) + ( 1 t ) f ( a ) .
By adding these inequalities, we have
f ( a t b 1 t ) + f ( b t a 1 t ) f ( a ) + f ( b ) .
(5)
Then multiplying both sides of (5) by t α 1 , and integrating the resulting inequality with respect to t over [ 0 , 1 ] , we obtain
0 1 f ( a t b 1 t ) t α 1 d t + 0 1 f ( b t a 1 t ) t α 1 d t [ f ( a ) + f ( b ) ] 0 1 t α 1 d t ,
i.e.,
Γ ( α + 1 ) ( ln b a ) α { J a + α f ( b ) + J b α f ( a ) } f ( a ) + f ( b ) .

The proof is completed. □

In order to prove our main results, we need the following identity.

Lemma 2.1 Let f : I ( 0 , ) R be a differentiable function on I such that f L [ a , b ] , where a , b I with a < b . Then for all x [ a , b ] , λ [ 0 , 1 ] and α > 0 , we have:
I f ( x , λ , α , a , b ) = a ( ln x a ) α + 1 0 1 ( t α λ ) ( x a ) t f ( x t a 1 t ) d t b ( ln b x ) α + 1 0 1 ( t α λ ) ( x b ) t f ( x t b 1 t ) d t .
(6)
Proof By integration by parts and twice changing the variable, for x a , we can state that
a ln x a 0 1 ( t α λ ) ( x a ) t f ( x t a 1 t ) d t = 0 1 ( t α λ ) d f ( x t a 1 t ) = ( t α λ ) f ( x t a 1 t ) | 0 1 α ( ln x a ) α a x ( ln u a ) α 1 f ( u ) u d u = ( 1 λ ) f ( x ) + λ f ( a ) Γ ( α + 1 ) ( ln x a ) α J x α f ( a ) ,
(7)
and for x b , similarly, we get
b ln b x 0 1 ( t α λ ) ( x b ) t f ( x t b 1 t ) d t = 0 1 ( t α λ ) d f ( x t b 1 t ) = ( t α λ ) f ( x t b 1 t ) | 0 1 α ( ln b x ) α x b ( ln b u ) α 1 f ( u ) u d u = ( 1 λ ) f ( x ) + λ f ( b ) Γ ( α + 1 ) ( ln b x ) α J x + α f ( b ) .
(8)
Multiplying both sides of (7) and (8) by ( ln x a ) α and ( ln b x ) α , respectively, and adding the resulting identities, we obtain the desired result. For x = a and x = b , the identities
I f ( a , λ , α ; a , b ) = b ( ln b a ) α + 1 0 1 ( t α λ ) ( a b ) t f ( a t b 1 t ) d t ,
and
I f ( b , λ , α ; a , b ) = a ( ln b a ) α + 1 0 1 ( t α λ ) ( b a ) t f ( b t a 1 t ) ,

can be proved respectively easily by performing an integration by parts in the integrals from the right-hand side and changing the variable. □

Theorem 2.2 Let f : I ( 0 , ) R be a differentiable function on I such that f L [ a , b ] , where a , b I with a < b . If | f | q is quasi-geometrically convex on [ a , b ] for some fixed q 1 , x [ a , b ] , λ [ 0 , 1 ] and α > 0 , then the following inequality for fractional integrals holds:
| I f ( x , λ , α , a , b ) | A 1 1 1 q ( α , λ ) { a ( ln x a ) α + 1 ( sup { | f ( x ) | q , | f ( a ) | q } ) 1 q B 1 1 q ( x , α , λ , q ) + b ( ln b x ) α + 1 ( sup { | f ( x ) | q , | f ( b ) | q } ) 1 q B 2 1 q ( x , α , λ , q ) } ,
(9)
where
A 1 ( α , λ ) = 2 α λ 1 + 1 α + 1 α + 1 λ , B 1 ( x , α , λ , q ) = 0 1 | t α λ | ( x a ) q t d t , B 2 ( x , α , λ , q ) = 0 1 | t α λ | ( x b ) q t d t .
Proof Since | f | q is quasi-geometrically convex on [ a , b ] , for all t [ 0 , 1 ] ,
| f ( x t a 1 t ) | q sup { | f ( x ) | q , | f ( a ) | q }
and
| f ( x t b 1 t ) | q sup { | f ( x ) | q , | f ( b ) | q } .
Hence, using Lemma 2.1 and power mean inequality, we get
| I f ( x , λ , α , a , b ) | a ( ln x a ) α + 1 ( 0 1 | t α λ | d t ) 1 1 q ( 0 1 | t α λ | ( x a ) q t sup { | f ( x ) | q , | f ( a ) | q } d t ) 1 q + b ( ln b x ) α + 1 ( 0 1 | t α λ | d t ) 1 1 q ( 0 1 | t α λ | ( x b ) q t sup { | f ( x ) | q , | f ( b ) | q } d t ) 1 q , | I f ( x , λ , α , a , b ) | ( 0 1 | t α λ | d t ) 1 1 q | I f ( x , λ , α , a , b ) | × { a ( ln x a ) α + 1 ( sup { | f ( x ) | q , | f ( a ) | q } ) 1 q ( 0 1 | t α λ | ( x a ) q t d t ) 1 q | I f ( x , λ , α , a , b ) | + b ( ln b x ) α + 1 ( sup { | f ( x ) | q , | f ( b ) | q } ) 1 q ( 0 1 | t α λ | ( x b ) q t d t ) 1 q } | I f ( x , λ , α , a , b ) | A 1 1 1 q ( α , λ ) { a ( ln x a ) α + 1 ( sup { | f ( x ) | q , | f ( a ) | q } ) 1 q B 1 1 q ( x , α , λ , q ) | I f ( x , λ , α , a , b ) | + b ( ln b x ) α + 1 ( sup { | f ( x ) | q , | f ( b ) | q } ) 1 q B 2 1 q ( x , α , λ , q ) } ,

which completes the proof. □

Corollary 2.1 Under the assumptions of Theorem  2.2 with q = 1 , inequality (9) reduces to the following inequality:
| I f ( x , λ , α , a , b ) | { a ( ln x a ) α + 1 B 1 ( x , α , λ , 1 ) sup { | f ( x ) | , | f ( a ) | } + b ( ln b x ) α + 1 B 2 ( x , α , λ , 1 ) sup { | f ( x ) | , | f ( b ) | } } .
Corollary 2.2 Under the assumptions of Theorem  2.2 with α = 1 , inequality (9) reduces to the following inequality:
( ln b a ) 1 | I f ( x , λ , α , a , b ) | | ( 1 λ ) f ( x ) + λ [ f ( a ) ln x a + f ( b ) ln b x ln b a ] 1 ln b a a b f ( u ) u d u | ( ln b a ) 1 ( 2 λ 2 2 λ + 1 2 ) 1 1 q { a ( ln x a ) 2 B 1 1 q ( x , 1 , λ , q ) ( sup { | f ( x ) | q , | f ( a ) | q } ) 1 q + b ( ln b x ) 2 B 2 1 q ( x , 1 , λ , q ) ( sup { | f ( x ) | q , | f ( b ) | q } ) 1 q } ,
where
B 1 ( x , 1 , λ , q ) = h λ ( ( x a ) q ) , B 2 ( x , 1 , λ , q ) = h λ ( ( x b ) q ) , h ( u , λ ) = 2 u λ u 1 ( ln u ) 2 + ( 1 λ ) u λ ln u , u ( 0 , ) { 1 } ,
(10)
specially for x = a b , we get
| ( 1 λ ) f ( a b ) + λ ( f ( a ) + f ( b ) 2 ) 1 ln b a a b f ( u ) u d u | ln b a 4 ( 2 λ 2 2 λ + 1 2 ) 1 1 q { a h 1 q ( ( b a ) q 2 , λ ) ( sup { | f ( a b ) | q , | f ( a ) | q } ) 1 q + b h 1 q ( ( a b ) q 2 , λ ) ( sup { | f ( a b ) | q , | f ( b ) | q } ) 1 q } .
(11)
Corollary 2.3 In Theorem  2.2,
  1. 1.
    If we take x = a b , λ = 1 3 , then we get the following Simpson-type inequality for fractional integrals:
    | 1 6 [ f ( a ) + 4 f ( a b ) + f ( b ) ] 2 α 1 Γ ( α + 1 ) ( ln b a ) α [ J a b α f ( a ) + J a b + α f ( b ) ] | ln b a 4 A 1 1 1 q ( α , 1 3 ) { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q B 1 1 q ( a b , α , 1 3 , q ) + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q B 2 1 q ( a b , α , 1 3 , q ) } ,
     
specially for α = 1 , we get
| 1 6 [ f ( a ) + 4 f ( a b ) + f ( b ) ] 1 ln b a a b f ( u ) u d u | ln b a 4 ( 5 18 ) 1 1 q { a [ sup { | f ( a b ) | , | f ( a ) | } ] 1 q h 1 q ( ( b a ) q 2 , 1 3 ) + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q h 1 q ( ( a b ) q 2 , 1 3 ) } ,

where h is defined as in (10).

Remark 2.2
  1. 1.
    If we take x = a b , λ = 0 , then we get the following midpoint- type inequality for fractional integrals:
    | f ( a b ) 2 α 1 Γ ( α + 1 ) ( ln b a ) α [ J a b α f ( a ) + J a b + α f ( b ) ] | ln b a 4 ( 1 α + 1 ) 1 1 q { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q B 1 1 q ( a b , 1 , 0 , q ) + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q B 2 1 q ( a b , 1 , 0 , q ) } ,
     
specially for α = 1 , we get
| f ( a b ) 1 ln b a a b f ( u ) u d u | 2 1 q ln b a 8 { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q h 1 q ( ( b a ) q 2 , 0 ) + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q h 1 q ( ( a b ) q 2 , 0 ) } ,
where h is defined as in (10).
  1. 2.
    If we take x = a b , λ = 1 , then we get the following trapezoid-type inequality for fractional integrals:
    | f ( a ) + f ( b ) 2 2 α 1 Γ ( α + 1 ) ( ln b a ) α [ J a b α f ( a ) + J a b + α f ( b ) ] | ln b a 4 ( 1 α + 1 ) 1 1 q { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q B 1 1 q ( a b , α , 1 , q ) + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q B 2 1 q ( a b , α , 1 , q ) } ,
     
specially for α = 1 , we get
| f ( a ) + f ( b ) 2 1 ln b a a b f ( u ) u d u | 2 1 q ln b a 8 { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q h 1 q ( ( b a ) q 2 , 1 ) + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q h 1 q ( ( a b ) q 2 , 1 ) } ,

where h is defined as in (10).

Corollary 2.4 Let the assumptions of Theorem  2.2 hold. If | f ( x ) | M for all x [ a , b ] and λ = 0 , then we get the following Ostrowski-type inequality for fractional integrals from inequality (9):
| [ ( ln x a ) α + ( ln b x ) α ] f ( x ) Γ ( α + 1 ) [ J a b α f ( a ) + J a b + α f ( b ) ] | M ( α + 1 ) 1 1 q [ a ( ln x a ) α + 1 B 1 1 q ( x , α , 0 , q ) + b ( ln b x ) α + 1 B 2 1 q ( x , α , 0 , q ) ] .
Theorem 2.3 Let f : I ( 0 , ) R be a differentiable function on I such that f L [ a , b ] , where a , b I with a < b . If | f | q is quasi-geometrically convex on [ a , b ] for some fixed q > 1 , x [ a , b ] , λ [ 0 , 1 ] and α > 0 , then the following inequality for fractional integrals holds:
| I f ( x , λ , α , a , b ) | A 2 1 p ( α , λ , p ) { a ( ln x a ) α + 1 p ( sup { | f ( x ) | q , | f ( a ) | q } ) 1 q ( x q a q q ) 1 q + b ( ln b x ) α + 1 p ( sup { | f ( x ) | q , | f ( b ) | q } ) 1 q ( b q x q q ) 1 q } ,
(12)
where
A 2 ( α , λ , p ) = { 1 α p + 1 , λ = 0 , λ α p + 1 α α { β ( 1 α , p + 1 ) + ( 1 λ ) p + 1 p + 1 × 2 F 1 ( 1 α + p + 1 , p + 1 , p + 2 ; 1 λ ) } , 0 < λ < 1 , 1 α β ( p + 1 , 1 α ) , λ = 1 ,
F 1 2 is hypergeometric function defined by
F 1 2 ( a , b ; c ; z ) = 1 β ( b , c b ) 0 1 t b 1 ( 1 t ) c b 1 ( 1 z t ) a d t , c > b > 0 , | z | < 1 ( see  [21] ) ,

and 1 p + 1 q = 1 .

Proof Using Lemma 2.1, the Hölder inequality and quasi-geometrical convexity of | f | q , we get
| I f ( x , λ , α , a , b ) | a ( ln x a ) α + 1 ( 0 1 | t α λ | p d t ) 1 p ( 0 1 ( x a ) q t sup { | f ( x ) | q , | f ( a ) | q } d t ) 1 q + b ( ln b x ) α + 1 ( 0 1 | t α λ | p d t ) 1 p ( 0 1 ( x b ) q t sup { | f ( x ) | q , | f ( b ) | q } d t ) 1 q , | I f ( x , λ , α , a , b ) | ( 0 1 | t α λ | p d t ) 1 p | I f ( x , λ , α , a , b ) | × { a ( ln x a ) α + 1 ( sup { | f ( x ) | q , | f ( a ) | q } ) 1 q ( 0 1 ( x a ) q t d t ) 1 q | I f ( x , λ , α , a , b ) | + b ( ln b x ) α + 1 ( sup { | f ( x ) | q , | f ( b ) | q } ) 1 q ( 0 1 ( x b ) q t d t ) 1 q } | I f ( x , λ , α , a , b ) | A 2 1 p ( α , λ , p ) { a ( ln x a ) α + 1 1 q ( sup { | f ( x ) | q , | f ( a ) | q } ) 1 q ( x q a q q ) 1 q | I f ( x , λ , α , a , b ) | + b ( ln b x ) α + 1 1 q ( sup { | f ( x ) | q , | f ( b ) | q } ) 1 q ( b q x q q ) 1 q } ,
here, it is seen by a simple computation that
A 2 ( α , λ , p ) = 0 1 | t α λ | p d t = { 1 α p + 1 , λ = 0 , λ α p + 1 α α { β ( 1 α , p + 1 ) + ( 1 λ ) p + 1 p + 1 × 2 F 1 ( 1 α + p + 1 , p + 1 , 2 + p ; 1 λ ) } , 0 < λ < 1 , 1 α β ( p + 1 , 1 α ) , λ = 1 .

Hence, the proof is completed. □

Corollary 2.5 Under the assumptions of Theorem  2.3 with α = 1 , inequality (12) reduces to the following inequality:
| ( 1 λ ) f ( x ) + λ [ f ( a ) ln x a + f ( b ) ln b x ln b a ] 1 ln b a a b f ( u ) u d u | ( ln b a ) 1 ( λ p + 1 + ( 1 λ ) p + 1 p + 1 ) 1 p × { a ( ln x a ) 1 + 1 p ( sup { | f ( x ) | q , | f ( a ) | q } ) 1 q ( x q a q q ) 1 q + b ( ln b x ) 1 + 1 p ( sup { | f ( x ) | q , | f ( b ) | q } ) 1 q ( b q x q q ) 1 q } ,
specially for x = a b , we get
| ( 1 λ ) f ( a b ) + λ ( f ( a ) + f ( b ) 2 ) 1 ln b a a b f ( u ) u d u | 1 2 ( ln b a ( λ p + 1 + ( 1 λ ) p + 1 ) 2 ( p + 1 ) ) 1 p { a ( sup { | f ( a b ) | q , | f ( a ) | q } ) 1 q ( a b q a q q ) 1 q + b ( sup { | f ( a b ) | q , | f ( b ) | q } ) 1 q ( b q a b q q ) 1 q } .
(13)
Corollary 2.6 In Theorem  2.3,
  1. 1.
    If we take x = a b , λ = 1 3 , then we get the following Simpson-type inequality for fractional integrals:
    | 1 6 [ f ( a ) + 4 f ( a b ) + f ( b ) ] 2 α 1 Γ ( α + 1 ) ( ln b a ) α [ J a b α f ( a ) + J a b + α f ( b ) ] | 1 2 ( ln b a ( 1 + 2 p + 1 ) 3 p + 1 ( p + 1 ) 2 ) 1 p { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q ( a b q a q q ) 1 q + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q ( b q a b q q ) 1 q } ,
     
specially for α = 1 , we get
| 1 6 [ f ( a ) + 4 f ( a b ) + f ( b ) ] 1 ln b a a b f ( u ) u d u | 1 2 ( ln b a ( 1 + 2 p + 1 ) 3 p + 1 ( p + 1 ) 2 ) 1 p × { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q ( a b q a q q ) 1 q + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q ( b q a b q q ) 1 q } .
Remark 2.3
  1. 1.
    If we take x = a b , λ = 0 , then we get the following midpoint- type inequality for fractional integrals:
    | f ( a b ) 2 α 1 Γ ( α + 1 ) ( ln b a ) α [ J a b α f ( a ) + J a b + α f ( b ) ] | 1 2 ( ln b a 2 ( α p + 1 ) ) 1 p { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q ( a b q a q q ) 1 q + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q ( b q a b q q ) 1 q } ,
     
specially for α = 1 , we get
| f ( a b ) 1 ln b a a b f ( u ) u d u | 1 2 ( ln b a 2 ( p + 1 ) ) 1 p { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q ( a b q a q q ) 1 q + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q ( b q a b q q ) 1 q } .
  1. 2.
    If we take x = a b , λ = 1 , then we get the following trapezoid-type inequality for fractional integrals 1 α β ( p + 1 , 1 α ) :
    | f ( a ) + f ( b ) 2 2 α 1 Γ ( α + 1 ) ( ln b a ) α [ J a b α f ( a ) + J a b + α f ( b ) ] | 1 2 ( ln b a β ( p + 1 , 1 α ) 2 α ) 1 p × { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q ( a b q a q q ) 1 q + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q ( b q a b q q ) 1 q } ,
     
specially for α = 1 , we get
| f ( a ) + f ( b ) 2 1 ln b a a b f ( u ) u d u | 1 2 ( ln b a 2 ( p + 1 ) ) 1 p { a [ sup { | f ( a b ) | q , | f ( a ) | q } ] 1 q ( a b q a q q ) 1 q + b [ sup { | f ( a b ) | q , | f ( b ) | q } ] 1 q ( b q a b q q ) 1 q } .
Corollary 2.7 Let the assumptions of Theorem  2.3 hold. If | f ( x ) | M for all x [ a , b ] and λ = 0 , then we get the following Ostrowski-type inequality for fractional integrals from inequality (12):
| [ ( ln x a ) α + ( ln b x ) α ] f ( x ) Γ ( α + 1 ) [ J a b α f ( a ) + J a b + α f ( b ) ] | M ( α p + 1 ) 1 p [ a ( ln x a ) α + 1 p ( x q a q q ) 1 q + b ( ln b x ) α + 1 p ( b q x q q ) 1 q ] .

3 Application to special means

Let us recall the following special means of two nonnegative numbers a, b with b > a :
  1. 1.
    The arithmetic mean
    A = A ( a , b ) : = a + b 2 .
     
  2. 2.
    The geometric mean
    G = G ( a , b ) : = a b .
     
  3. 3.
    The logarithmic mean
    L = L ( a , b ) : = b a ln b ln a .
     
  4. 4.
    The p-logarithmic mean
    L p = L p ( a , b ) : = ( b p + 1 a p + 1 ( p + 1 ) ( b a ) ) 1 p , p R { 1 , 0 } .
     
Proposition 3.1 For b > a > 0 , n > 0 and q 1 , we have
| ( 1 λ ) G n + 1 ( a , b ) + λ A ( a n + 1 , b n + 1 ) ( n + 1 ) L ( a , b ) L n n ( a , b ) | ( n + 1 ) ln b a 4 ( 2 λ 2 2 λ + 1 2 ) 1 1 q { a G n ( a , b ) h 1 q ( ( b a ) q 2 , λ ) + b n + 1 h 1 q ( ( a b ) q 2 , λ ) } ,

where h is defined as in (10).

Proof Let f ( x ) = x n + 1 n + 1 , x > 0 , n > 0 and q 1 . Then the function | f ( x ) | q = x n q is quasi-geometrically convex on ( 0 , ) . Thus, by inequality (11), Proposition 3.1 is proved. □

Proposition 3.2 For b > a > 0 , n > 0 and q > 1 , we have
| ( 1 λ ) G n + 1 ( a , b ) + λ A ( a n + 1 , b n + 1 ) ( n + 1 ) L ( a , b ) L n n ( a , b ) | n + 1 2 ( ln b a ( λ p + 1 + ( 1 λ ) p + 1 ) 2 ( p + 1 ) ) 1 p { a G n ( a , b ) ( G q ( a , b ) a q q ) 1 q + b n + 1 ( b q G q ( a , b ) q ) 1 q } .

Proof Let f ( x ) = x n + 1 n + 1 , x > 0 , n > 0 and q > 1 . Then the function | f ( x ) | q = x n q is quasi-geometrically convex on ( 0 , ) . Thus, by inequality (13), Proposition 3.2 is proved. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences and Arts, Giresun University

References

  1. Ion DA: Some estimates on the Hermite-Hadamard inequality through quasi-convex functions. An. Univ. Craiova, Ser. Mat. Inform. 2007, 34: 82–87.MathSciNetGoogle Scholar
  2. Niculescu CP: Convexity according to the geometric mean. Math. Inequal. Appl. 2000, 3(2):155–167. 10.7153/mia-03-19MathSciNetMATHGoogle Scholar
  3. Niculescu CP: Convexity according to means. Math. Inequal. Appl. 2003, 6(4):571–579. 10.7153/mia-06-53MathSciNetMATHGoogle Scholar
  4. Zhang T-Y, Ji A-P, Qi F: On integral inequalities of Hermite-Hadamard type for s -geometrically convex functions. Abstr. Appl. Anal. 2012., 2012: Article ID 560586 10.1155/2012/560586Google Scholar
  5. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATHGoogle Scholar
  6. Alomari M, Darus M, Dragomir SS, Cerone P: Ostrowski type inequalities for functions whose derivatives are s -convex in the second sense. Appl. Math. Lett. 2010, 23: 1071–1076. 10.1016/j.aml.2010.04.038MathSciNetView ArticleMATHGoogle Scholar
  7. Avci M, Kavurmaci H, Ozdemir ME: New inequalities of Hermite-Hadamard type via s -convex functions in the second sense with applications. Appl. Math. Comput. 2011, 217: 5171–5176. 10.1016/j.amc.2010.11.047MathSciNetView ArticleMATHGoogle Scholar
  8. Dahmani Z: On Minkowski and Hermite-Hadamard integral inequalities via fractional via fractional integration. Ann. Funct. Anal. 2010, 1(1):51–58.MathSciNetView ArticleMATHGoogle Scholar
  9. Iscan I:A new generalization of some integral inequalities for ( α , m ) -convex functions. Math. Sci. 2013., 7: Article ID 22 10.1186/2251-7456-7-22Google Scholar
  10. Iscan I:New estimates on generalization of some integral inequalities for ( α , m ) -convex functions. Contemp. Anal. Appl. Math. 2013, 1(2):253–264.MathSciNetMATHGoogle Scholar
  11. Iscan I: New estimates on generalization of some integral inequalities for s -convex functions and their applications. Int. J. Pure Appl. Math. 2013, 86(4):727–746.View ArticleGoogle Scholar
  12. Iscan I: On generalization of some integral inequalities for quasi-convex functions and their applications. Int. J. Eng. Appl. Sci. 2013, 3(1):37–42.Google Scholar
  13. Park J: Generalization of some Simpson-like type inequalities via differentiable s -convex mappings in the second sense. Int. J. Math. Math. Sci. 2011., 2011: Article ID 493531 10.1155/493531Google Scholar
  14. Sarıkaya MZ, Aktan N: On the generalization of some integral inequalities and their applications. Math. Comput. Model. 2011, 54: 2175–2182. 10.1016/j.mcm.2011.05.026View ArticleMATHMathSciNetGoogle Scholar
  15. Set E: New inequalities of Ostrowski type for mapping whose derivatives are s -convex in the second sense via fractional integrals. Comput. Math. Appl. 2012, 63: 1147–1154. 10.1016/j.camwa.2011.12.023MathSciNetView ArticleMATHGoogle Scholar
  16. Sarıkaya MZ, Ogunmez H: On new inequalities via Riemann-Liouville fractional integration. Abstr. Appl. Anal. 2012., 2012: Article ID 428983 10.1155/2012/428983Google Scholar
  17. Set E, Ozdemir ME, Sarıkaya MZ: On new inequalities of Simpson’s type for quasi-convex functions with applications. Tamkang J. Math. 2012, 43(3):357–364.MathSciNetView ArticleMATHGoogle Scholar
  18. Sarıkaya MZ, Set E, Ozdemir ME: On new inequalities of Simpson’s type for s -convex functions. Comput. Math. Appl. 2010, 60: 2191–2199. 10.1016/j.camwa.2010.07.033MathSciNetView ArticleMATHGoogle Scholar
  19. Sarıkaya MZ, Set E, Yaldız H, Başak N: Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 2013, 57(9–10):2403–2407. 10.1016/j.mcm.2011.12.048View ArticleMATHGoogle Scholar
  20. Zhu C, Feckan M, Wang J: Fractional integral inequalities for differentiable convex mappings and applications to special means and a midpoint formula. J. Appl. Math. Stat. Inform. 2012, 8: 21–28.View ArticleMATHGoogle Scholar
  21. Abramowitz M, Stegun IA (Eds): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York; 1965.MATHGoogle Scholar

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© İşcan; licensee Springer. 2013

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