On some inequalities for relative semi-convex functions

  • Muhammad Aslam Noor
  • Muhammad Uzair Awan
  • Khalida Inayat Noor
Open AccessResearch

DOI: 10.1186/1029-242X-2013-332

Cite this article as:
Noor, M.A., Awan, M.U. & Noor, K.I. J Inequal Appl (2013) 2013: 332. doi:10.1186/1029-242X-2013-332

Abstract

We consider and study a new class of convex functions that are called relative semi-convex functions. Some Hermite-Hadamard inequalities for the relative semi-convex function and its variant forms are derived. Several special cases are also discussed. Results proved in this paper may stimulate further research in this area.

MSC:26D15, 26A51, 49J40.

Keywords

relative semi-convex function convex set Hermite-Hadamard inequality fractional integral 

1 Introduction

Convexity plays a central and fundamental role in the fields of mathematical finance, economics, engineering, management sciences, and optimization theory. In recent years, the concept of convexity has been extended and generalized in several directions using the novel and innovative ideas; see, for example, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and the references therein. A significant generalization of a convex set and a convex function was the introduction of a relative convex (g-convex) set and a relative convex (g-convex) function by Youness [13]. Noor [14] showed that the optimality condition for a relative convex function on the relative convex set can be characterized by a class of variational inequalities known as general variational inequalities. Motivated by the work of Youness [13] and Noor [14], Chen [2] introduced and studied a new class of functions called relative semi-convex functions. Noor et al. [15] derived Hermite-Hadamard inequalities for differentiable relative semi-convex functions. For useful details on Hermite-Hadamard inequalities, see [1, 5, 6, 7, 8, 10, 15, 16, 17, 18, 19, 20, 21, 22].

Niculescu [7] introduced the concept of relative convexity and proved various properties and generalizations of classical results for relative convexity. Mercer [6] has also proved some useful results for relative convexity.

In this paper, we derive some Hermite-Hadamard inequalities for the relative semi-convex function and the logarithmic relative semi-convex function. The ideas of this paper may stimulate further research in this area.

2 Preliminaries

In this section, we recall some basic results and concepts, which are useful in proving our results. Let R n Open image in new window be a finite dimensional space, the inner product of which is denoted by , Open image in new window.

Definition 2.1 [13]

A set M R n Open image in new window is said to be a relative convex (g-convex) set if and only if there exists an arbitrary function g : R n R n Open image in new window such that
( 1 t ) g ( x ) + t g ( y ) M , x , y R n : g ( x ) , g ( y ) M , t [ 0 , 1 ] . Open image in new window
(2.1)

It is known [23] that if M is a relative convex set, then it may not be a classical convex set. For example, for M = [ 1 , 1 2 ] [ 0 , 1 ] Open image in new window and g ( x ) = x 2 Open image in new window, x R Open image in new window. Clearly, this is a relative convex set but not a classical convex set.

Definition 2.2 [13]

A function f is said to be a relative convex (g-convex) function on the relative convex set M if and only if there exists a function g : R n R n Open image in new window such that
f ( ( 1 t ) g ( x ) + t g ( y ) ) ( 1 t ) f ( g ( x ) ) + t f ( g ( y ) ) , x , y R n : g ( x ) , g ( y ) M , t [ 0 , 1 ] . Open image in new window
(2.2)

Every convex function f on a convex set is a relative convex function. However, the converse is not true. There are functions which are relative convex functions but may not be convex functions in the classical sense, see [13].

Definition 2.3 [2]

A function f is said to be a relative semi-convex function if and only if there exists an arbitrary function g : R n R n Open image in new window such that
f ( ( 1 t ) g ( x ) + t g ( y ) ) ( 1 t ) f ( x ) + t f ( y ) , x , y M , t [ 0 , 1 ] . Open image in new window
(2.3)

Remark 2.1 A relative semi-convex function on a relative convex set is not necessarily a relative convex function, see [2].

Definition 2.4 [3]

A function f : M R + Open image in new window is said to be relative logarithmic semi-convex on a relative convex set M if
f ( ( 1 t ) g ( x ) + t g ( y ) ) [ f ( x ) ] 1 t [ f ( y ) ] t , x , y M , t [ 0 , 1 ] . Open image in new window
(2.4)
From Definition 2.4 it follows that
f ( ( 1 t ) g ( x ) + t g ( y ) ) [ f ( x ) ] 1 t [ f ( y ) ] t ( 1 t ) f ( x ) + t f ( y ) , Open image in new window

which shows that every relative logarithmic semi-convex function is a relative semi-convex function, but the converse is not true.

Definition 2.5 [24]

Let f L 1 [ a , b ] Open image in new window. The generalized Riemann-Liouville fractional integrals J a + α f Open image in new window and J b α f Open image in new window of order α > 0 Open image in new window with p 0 Open image in new window are defined by
J p , a + α f ( x ) = ( p + 1 ) 1 α Γ ( α ) a x ( x p + 1 t p + 1 ) α 1 t p f ( t ) d t , x > a , Open image in new window
and
J p , b α f ( x ) = ( p + 1 ) 1 α Γ ( α ) x b ( t p + 1 x p + 1 ) α 1 t p f ( t ) d t , x < b , Open image in new window

respectively, where Γ ( α ) = 0 e t x α 1 d x Open image in new window is the gamma function.

If p = 0 Open image in new window, then Definition 2.5 reduces to the definition for classical Riemann-Liouville integrals. See also [25, 26].

Definition 2.6 [20]

Two functions f and g are said to be similarly ordered on I R Open image in new window if
f ( x ) f ( y ) , g ( x ) g ( y ) 0 , x , y I . Open image in new window

Let M = I = [ g ( a ) , g ( b ) ] Open image in new window be a relative semi-convex set. We now define a relative semi-convex function on I, which appears to be a new one.

Definition 2.7 Let I = [ g ( a ) , g ( b ) ] Open image in new window, then f is called a relative semi-convex function if and only if
| 1 1 1 g ( a ) g ( x ) g ( b ) f ( a ) f ( g ( x ) ) f ( b ) | 0 ; g ( a ) g ( x ) g ( b ) . Open image in new window
One can easily show that the following are equivalent:
  1. 1.

    f is a relative semi-convex function on a relative convex set.

     
  2. 2.

    f ( g ( x ) ) f ( a ) + f ( b ) f ( a ) g ( b ) g ( a ) ( g ( x ) g ( a ) ) Open image in new window.

     
  3. 3.

    f ( g ( x ) ) f ( a ) g ( x ) g ( a ) f ( b ) f ( a ) g ( b ) g ( a ) f ( b ) f ( g ( x ) ) g ( b ) g ( x ) Open image in new window.

     
  4. 4.

    f ( a ) ( g ( x ) g ( a ) ) ( g ( b ) g ( a ) ) + f ( g ( x ) ) ( g ( b ) g ( x ) ) ( g ( x ) g ( a ) ) f ( b ) ( g ( b ) g ( a ) ) ( g ( b ) g ( x ) ) 0 Open image in new window.

     
  5. 5.

    ( g ( b ) g ( x ) ) f ( a ) ( g ( b ) g ( a ) ) f ( g ( x ) ) + ( g ( x ) g ( a ) ) f ( b ) 0 Open image in new window,

     

where g ( x ) = ( 1 t ) g ( a ) + t g ( b ) M Open image in new window, t [ 0 , 1 ] Open image in new window.

For the applications of the relative convex functions, see [27].

Remark 2.2 We note that if f is a differentiable relative semi-convex function, then
f ( g ( y ) ) f ( x ) f ( x ) g ( x ) , g ( y ) g ( x ) , g ( y ) ( g ( a ) , g ( b ) ) . Open image in new window

3 Main results

In this section we discuss our main results.

Essentially using the techniques of [7], one can prove the following results for relative semi-convexity.

Lemma 3.1Letfbe a relative semi-convex function. Ifgis not a constant function, then
g ( a ) = g ( x ) implies f ( a ) = f ( g ( x ) ) . Open image in new window
Lemma 3.2Let f : I R Open image in new windowbe a relative semi-convex function, where I = [ g ( a ) , g ( b ) ] Open image in new window. If g ( x ) { g ( a ) , g ( b ) } Open image in new window, then
f ( b ) f ( g ( x ) ) g ( b ) g ( x ) f ( a ) f ( g ( x ) ) g ( a ) g ( x ) . Open image in new window
Lemma 3.3Letfbe a relative semi-convex function. Consider g ( x 1 ) , g ( x 2 ) , , g ( x n ) I Open image in new window, g ( y 1 ) , g ( y 2 ) , , g ( y n ) I Open image in new windowand weights ω 1 , ω 2 , , ω n R Open image in new windowsuch that:
  1. (i)

    g ( x 1 ) g ( x 2 ) g ( x n ) Open image in new windowand g ( y 1 ) g ( y 2 ) g ( y n ) Open image in new window,

     
  2. (ii)

    k = 1 r ω k g ( x k ) k = 1 r ω k g ( y k ) Open image in new window, r = 1 , , n Open image in new window,

     
  3. (iii)

    k = 1 n ω k g ( x k ) = k = 1 n ω k g ( y k ) Open image in new window,

     
then we have
k = 1 n ω k f ( g ( x k ) ) k = 1 n ω k f ( y k ) . Open image in new window
Lemma 3.4Letfbe a relative semi-convex function. Consider g ( x 1 ) , g ( x 2 ) , , g ( x n ) I Open image in new window, g ( y 1 ) , g ( y 2 ) , , g ( y n ) I Open image in new windowand weights ω 1 , ω 2 , , ω n R Open image in new windowsuch that
  1. (i)

    g ( x 1 ) g ( x 2 ) g ( x n ) Open image in new windowand g ( y 1 ) g ( y 2 ) g ( y n ) Open image in new window,

     
  2. (ii)

    k = 1 r ω k g ( x k ) k = 1 r ω k g ( y k ) Open image in new window, r = 1 , , n Open image in new window,

     
  3. (iii)

    f ( x ) f ( y ) , g ( x ) g ( y ) 0 Open image in new window,

     
then we have
k = 1 n ω k f ( g ( x k ) ) k = 1 n ω k f ( y k ) . Open image in new window
Lemma 3.5Letfbe a relative semi-convex function, then, for all g ( a ) < g ( c ) < g ( d ) < g ( b ) Open image in new window, we have
f ( a ) + f ( b ) 2 f ( g ( a ) + g ( b ) 2 ) f ( c ) + f ( d ) 2 f ( g ( c ) + g ( d ) 2 ) . Open image in new window

Theorem 3.6Letfandwbe two relative semi-convex functions. Then the product offandwwill be a relative semi-convex function iffandware similarly ordered functions.

Proof Since f and w are relative semi-convex functions, so we have
f ( ( 1 t ) g ( a ) + t g ( b ) ) w ( ( 1 t ) g ( a ) + t g ( b ) ) [ ( 1 t ) f ( a ) + t f ( b ) ] [ ( 1 t ) w ( a ) + t w ( b ) ] = [ 1 t ] 2 f ( a ) w ( a ) + t ( 1 t ) f ( a ) w ( b ) + t ( 1 t ) f ( b ) w ( a ) + [ t ] 2 f ( b ) w ( b ) = ( 1 t ) f ( a ) w ( a ) + t f ( b ) w ( b ) t ( 1 t ) [ f ( a ) w ( a ) + f ( b ) w ( b ) f ( b ) w ( a ) f ( a ) w ( b ) ] ( 1 t ) f ( a ) w ( a ) + t f ( b ) w ( b ) , Open image in new window

where we have used the fact that f and w are similarly ordered. This completes the proof. □

We now obtain some Hermite-Hadamard inequalities for relative semi-convex functions.

Theorem 3.7Let f : I R R Open image in new windowbe a relative semi-convex function on I = [ g ( a ) , g ( b ) ] Open image in new windowwith g ( a ) < g ( b ) Open image in new window, then we have
f ( g ( a ) + g ( b ) 2 ) 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) d g ( x ) f ( a ) + f ( b ) 2 . Open image in new window
(3.1)
Proof Let f be relative semi-convex. Then
f ( g ( a ) + g ( b ) 2 ) = 0 1 f ( g ( a ) + g ( b ) 2 ) d t = 0 1 f ( ( 1 t ) g ( a ) + t g ( b ) + t g ( a ) + ( 1 t ) g ( b ) 2 ) d t 1 2 0 1 [ f ( ( 1 t ) g ( a ) + t g ( b ) ) + f ( t g ( a ) + ( 1 t ) g ( b ) ) ] d t = 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) d g ( x ) = 0 1 f ( ( 1 t ) g ( a ) + t g ( b ) ) d t 0 1 ( ( 1 t ) f ( a ) + t f ( b ) ) d t = f ( a ) + f ( b ) 2 . Open image in new window

 □

Using the technique of [21], we can prove the following result.

Lemma 3.8Letfbe a semi-relative convex function. Then, for any g ( x ) [ g ( a ) , g ( b ) ] Open image in new window, we have
f ( g ( a ) + g ( b ) g ( x ) ) f ( a ) + f ( b ) f ( g ( x ) ) . Open image in new window
Theorem 3.9Letfbe a relative semi-convex function and let w : [ g ( a ) , g ( b ) ] R Open image in new windowbe nonnegative, integrable and symmetric about g ( a ) + g ( b ) 2 Open image in new window. Then
f ( g ( a ) + g ( b ) 2 ) g ( a ) g ( b ) w ( g ( x ) ) d g ( x ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) f ( a ) + f ( b ) 2 g ( a ) g ( b ) w ( g ( x ) ) d g ( x ) . Open image in new window
(3.2)
Proof Since f is a relative semi-convex function and w : [ g ( a ) , g ( b ) ] R Open image in new window is nonnegative, integrable and symmetric about g ( a ) + g ( b ) 2 Open image in new window, we have
f ( g ( a ) + g ( b ) 2 ) g ( a ) g ( b ) w ( g ( x ) ) d g ( x ) = g ( a ) g ( b ) f ( g ( a ) + g ( b ) 2 ) w ( g ( x ) ) d g ( x ) g ( a ) g ( b ) [ 1 2 ( f ( g ( a ) + g ( b ) g ( x ) ) + f ( g ( x ) ) ) ] w ( g ( x ) ) d g ( x ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) = 1 2 g ( a ) g ( b ) f ( g ( a ) + g ( b ) g ( x ) ) w ( g ( x ) ) d g ( x ) + 1 2 g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) 1 2 g ( a ) g ( b ) { f ( a ) + f ( b ) f ( g ( x ) ) } w ( g ( x ) ) d g ( x ) + 1 2 a g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) = f ( a ) + f ( b ) 2 g ( a ) g ( b ) w ( g ( x ) ) d g ( x ) . Open image in new window

This completes the proof. □

Theorem 3.10Let f , w : I R R Open image in new windowbe relative semi-convex functions onIwith g ( a ) < g ( b ) Open image in new window. Then, for all t [ 0 , 1 ] Open image in new window, we have
2 f ( g ( a ) + g ( b ) 2 ) w ( g ( a ) + g ( b ) 2 ) [ 1 6 M ( a , b ) + 1 2 N ( a , b ) ] 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) 1 3 M ( a , b ) + 1 6 N ( a , b ) , Open image in new window
where
M ( a , b ) = f ( a ) w ( a ) + f ( b ) w ( b ) , Open image in new window
(3.3)
N ( a , b ) = f ( a ) w ( b ) + f ( b ) w ( a ) . Open image in new window
(3.4)
Proof Let f and w be relative semi-convex functions. Then
f ( g ( a ) + g ( b ) 2 ) w ( g ( a ) + g ( b ) 2 ) = f ( t g ( a ) + ( 1 t ) g ( b ) + ( 1 t ) g ( a ) + t g ( b ) 2 ) × w ( t g ( a ) + ( 1 t ) g ( b ) + ( 1 t ) g ( a ) + t g ( b ) 2 ) 1 2 [ f ( t g ( a ) + ( 1 t ) g ( b ) ) + f ( ( 1 t ) g ( a ) + t g ( b ) ) ] × 1 2 [ w ( t g ( a ) + ( 1 t ) g ( b ) ) + w ( ( 1 t ) g ( a ) + t g ( b ) ) ] = 1 4 [ f ( t g ( a ) + ( 1 t ) g ( b ) ) w ( t g ( a ) + ( 1 t ) g ( b ) ) + f ( ( 1 t ) g ( a ) + t g ( b ) ) w ( ( 1 t ) g ( a ) + t g ( b ) ) ] + 1 4 [ f ( t g ( a ) + ( 1 t ) g ( b ) ) w ( ( 1 t ) g ( a ) + t g ( b ) ) + f ( ( 1 t ) g ( a ) + t g ( b ) ) w ( t g ( a ) + ( 1 t ) g ( b ) ) ] 1 4 [ f ( t g ( a ) + ( 1 t ) g ( b ) ) w ( t g ( a ) + ( 1 t ) g ( b ) ) + f ( ( 1 t ) g ( a ) + t g ( b ) ) w ( ( 1 t ) g ( a ) + t g ( b ) ) ] + 1 4 [ 2 t ( 1 t ) ( f ( a ) w ( a ) + f ( b ) w ( b ) ) + ( t 2 + ( 1 t ) 2 ) ( f ( b ) w ( a ) + f ( a ) w ( b ) ) ] . Open image in new window
Integrating with respect to t on [ 0 , 1 ] Open image in new window, we have
f ( g ( a ) + g ( b ) 2 ) w ( g ( a ) + g ( b ) 2 ) 1 4 [ 2 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) ] + 1 2 [ 1 6 M ( a , b ) + 1 3 N ( a , b ) ] . Open image in new window
This implies that
2 f ( g ( a ) + g ( b ) 2 ) w ( g ( a ) + g ( b ) 2 ) [ 1 6 M ( a , b ) + 1 3 N ( a , b ) ] 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) = 0 1 f ( t g ( a ) + ( 1 t ) g ( b ) ) w ( t g ( a ) + ( 1 t ) g ( b ) ) d t 0 1 [ t f ( a ) + ( 1 t ) f ( b ) ] [ t w ( a ) + ( 1 t ) w ( b ) ] d t = 1 3 M ( a , b ) + 1 6 N ( a , b ) . Open image in new window

This completes the proof. □

Theorem 3.11Let f , w : I R R Open image in new windowbe relative semi-convex functions onIwith g ( a ) < g ( b ) Open image in new window. Ifwis symmetric about g ( a ) + g ( b ) 2 Open image in new window, then for all t [ 0 , 1 ] Open image in new windowwe have
1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( a ) + g ( b ) g ( x ) ) d g ( x ) 1 6 M ( a , b ) + 1 3 N ( a , b ) , Open image in new window

where M ( a , b ) Open image in new windowand N ( a , b ) Open image in new windoware given by (3.3) and (3.4), Θ ( a , b ) = [ f ( a ) ] 2 + [ f ( b ) ] 2 + [ w ( a ) ] 2 + [ w ( b ) ] 2 Open image in new window.

Proof Since f and w are relative semi-convex functions, then we have
1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( a ) + g ( b ) g ( x ) ) d g ( x ) = 0 1 f ( t g ( a ) + ( 1 t ) g ( b ) ) w ( ( 1 t ) g ( a ) + t g ( b ) ) d t 1 2 0 1 { [ f ( t g ( a ) + ( 1 t ) g ( b ) ) ] 2 + [ w ( ( 1 t ) g ( a ) + t g ( b ) ) ] 2 } d t 1 2 0 1 { [ t f ( a ) + ( 1 t ) f ( b ) ] 2 + [ ( 1 t ) w ( a ) + t w ( b ) ] 2 } d t = 1 6 { [ f ( a ) ] 2 + [ f ( b ) ] 2 + f ( a ) f ( b ) + [ w ( a ) ] 2 + [ w ( b ) ] 2 + w ( a ) w ( b ) } 1 4 { [ f ( a ) ] 2 + [ f ( b ) ] 2 + [ w ( a ) ] 2 + [ w ( b ) ] 2 } = 1 4 Θ ( a , b ) 0 1 ( t f ( a ) + ( 1 t ) f ( b ) ) ( ( 1 t ) w ( a ) + t w ( b ) ) d t = 1 6 f ( a ) w ( a ) + 1 3 f ( a ) w ( b ) + 1 3 f ( b ) w ( a ) + 1 6 f ( b ) w ( b ) = 1 6 M ( a , b ) + 1 3 N ( a , b ) . Open image in new window

The desired result. □

Theorem 3.12Let f , w : I R R Open image in new windowbe similarly ordered and relative semi-convex functions onIwith g ( a ) < g ( b ) Open image in new window. Then, for all t [ 0 , 1 ] Open image in new window, we have
2 f ( g ( a ) + g ( b ) 2 ) w ( g ( a ) + g ( b ) 2 ) 1 4 M ( a , b ) 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) f ( a ) w ( a ) + f ( b ) w ( b ) 2 , Open image in new window

where M ( a , b ) Open image in new windowis given by (3.3).

Proof Since f and w are similarly ordered functions, the proof follows from Theorem 3.10. □

Theorem 3.13Letfbe a relative semi-convex function, then for all λ ( 0 , 1 ) Open image in new windowwe have
f ( g ( a ) + g ( b ) 2 ) Δ 1 ( λ ) 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) d g ( x ) Δ 2 ( λ ) f ( a ) + f ( b ) 2 , Open image in new window
(3.5)
where
Δ 1 ( λ ) = λ f ( ( 2 λ ) g ( a ) + λ g ( b ) 2 ) + ( 1 λ ) f ( ( 1 λ ) g ( a ) + ( 1 + λ ) g ( b ) 2 ) Open image in new window
and
Δ 2 ( λ ) = f ( ( 1 λ ) g ( a ) + λ g ( b ) ) + λ f ( a ) + ( 1 λ ) f ( b ) 2 . Open image in new window
Proof We divide the interval [ g ( a ) , g ( b ) ] Open image in new window into [ g ( a ) , ( 1 λ ) g ( a ) + λ g ( b ) ] Open image in new window and [ ( 1 λ ) g ( a ) + λ g ( b ) , g ( b ) ] Open image in new window. Using the left-hand side of (3.1), we have
f ( ( 2 λ ) g ( a ) + λ g ( b ) 2 ) 1 λ ( g ( b ) g ( a ) ) g ( a ) ( 1 λ ) g ( a ) + λ g ( b ) f ( g ( x ) ) d g ( x ) , Open image in new window
(3.6)
f ( ( 1 λ ) g ( a ) + ( 1 + λ g ( b ) ) 2 ) 1 ( 1 λ ) ( g ( b ) g ( a ) ) ( 1 λ ) g ( a ) + λ g ( b ) g ( b ) f ( g ( x ) ) d g ( x ) . Open image in new window
(3.7)
Multiplying (3.6) by λ and (3.7) by ( 1 λ ) Open image in new window, and then adding the resultant, we have
Δ 1 ( λ ) = λ f ( ( 2 λ ) g ( a ) + λ g ( b ) 2 ) + ( 1 λ ) f ( ( 1 λ ) g ( a ) + ( 1 + λ g ( b ) ) 2 ) 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) d g ( x ) . Open image in new window
(3.8)
Now, using the right-hand side of (3.1), we have
1 λ ( g ( b ) g ( a ) ) g ( a ) ( 1 λ ) g ( a ) + λ g ( b ) f ( g ( x ) ) d g ( x ) f ( g ( a ) ) + f ( ( 1 λ ) g ( a ) + λ g ( b ) ) 2 1 λ ( g ( b ) g ( a ) ) g ( a ) ( 1 λ ) g ( a ) + λ g ( b ) f ( g ( x ) ) d g ( x ) f ( a ) + f ( ( 1 λ ) g ( a ) + λ g ( b ) ) 2 , Open image in new window
(3.9)
1 ( 1 λ ) ( g ( b ) g ( a ) ) ( 1 λ ) g ( a ) + λ g ( b ) g ( b ) f ( g ( x ) ) d g ( x ) f ( ( 1 λ ) g ( a ) + λ g ( b ) ) + f ( g ( b ) ) 2 1 ( 1 λ ) ( g ( b ) g ( a ) ) ( 1 λ ) g ( a ) + λ g ( b ) g ( b ) f ( g ( x ) ) d g ( x ) f ( ( 1 λ ) g ( a ) + λ g ( b ) ) + f ( b ) 2 . Open image in new window
(3.10)
Multiplying (3.9) by λ and (3.10) by ( 1 λ ) Open image in new window and adding the resultant, we have
1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) d g ( x ) f ( ( 1 λ ) g ( a ) + λ g ( b ) ) + λ f ( a ) + ( 1 λ ) f ( b ) 2 = Δ 2 ( λ ) . Open image in new window
(3.11)
Now, using the fact that f is a relative semi-convex function, and also every convex function is a relative semi-convex function, we have
f ( g ( a ) + g ( b ) 2 ) = f ( λ ( 2 λ ) g ( a ) + λ g ( b ) 2 + ( 1 λ ) ( 1 λ ) g ( a ) + ( 1 + λ ) g ( b ) 2 ) λ f ( ( 2 λ ) g ( a ) + λ g ( b ) 2 ) + ( 1 λ ) f ( ( 1 λ ) g ( a ) + ( 1 + λ ) g ( b ) 2 ) = Δ 1 ( λ ) 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) d g ( x ) 1 2 [ λ f ( ( 1 λ ) g ( a ) + λ g ( b ) ) + λ f ( a ) + ( 1 λ ) f ( ( 1 λ ) g ( a ) + λ g ( b ) ) + ( 1 λ ) f ( b ) ] = 1 2 [ f ( ( 1 λ ) g ( a ) + λ g ( b ) ) + λ f ( a ) + ( 1 λ ) f ( b ) ] = Δ 2 ( λ ) 1 2 [ ( 1 λ ) f ( a ) + λ f ( b ) + λ f ( a ) + ( 1 λ ) f ( b ) ] = f ( a ) + f ( b ) 2 , Open image in new window
(3.12)

the required result. □

Remark 3.1 For suitable and different choices of λ ( 0 , 1 ) Open image in new window and g = I Open image in new window in Theorem 3.13, one can obtain several new and previously known results for various classes of convex functions.

We now prove the Hermite-Hadamard type inequalities for relative semi-convex functions via fractional integrals.

Theorem 3.14Letfbe a relative semi-convex function. Then
J p , g ( a ) + α f ( g ( b ) ) + J p , g ( b ) α f ( g ( a ) ) [ f ( a ) + f ( b ) ] [ J p , g ( a ) + α ( 1 ) + J p , g ( b ) α ( 1 ) ] , α > 0 , p 0 . Open image in new window
Proof Since f is a relative semi-convex function on M, so
( p + 1 ) 1 α Γ ( α ) 0 1 ( [ g ( b ) ] p + 1 [ ( 1 t ) g ( a ) + t g ( b ) ] p + 1 ) α 1 × [ ( 1 t ) g ( a ) + t g ( b ) ] p f ( ( 1 t ) g ( a ) + t g ( b ) ) d t ( p + 1 ) 1 α Γ ( α ) f ( a ) 0 1 ( [ g ( b ) ] p + 1 [ ( 1 t ) g ( a ) + t g ( b ) ] p + 1 ) α 1 × [ ( 1 t ) g ( a ) + t g ( b ) ] p ( 1 t ) d t + ( p + 1 ) 1 α Γ ( α ) f ( b ) 0 1 ( [ g ( b ) ] p + 1 [ ( 1 t ) g ( a ) + t g ( b ) ] p + 1 ) α 1 × [ ( 1 t ) g ( a ) + t g ( b ) ] p ( t ) d t . Open image in new window
Let g ( x ) = ( 1 t ) g ( a ) + t g ( b ) Open image in new window, then d t = d g ( x ) g ( b ) g ( a ) Open image in new window. Take t = g ( x ) g ( a ) g ( b ) g ( a ) Open image in new window, 1 t = g ( b ) g ( x ) g ( b ) g ( a ) Open image in new window. Then we have
( p + 1 ) 1 α Γ ( α ) ( g ( b ) g ( a ) ) g ( a ) g ( b ) ( [ g ( b ) ] p + 1 [ g ( x ) ] p + 1 ) α 1 [ g ( x ) ] p f ( g ( x ) ) d g ( x ) ( p + 1 ) 1 α Γ ( α ) f ( a ) g ( b ) g ( a ) g ( a ) g ( b ) ( [ g ( b ) ] p + 1 [ g ( x ) ] p + 1 ) α 1 [ g ( x ) ] p g ( b ) g ( x ) g ( b ) g ( a ) d g ( x ) + ( p + 1 ) 1 α Γ ( α ) f ( b ) g ( b ) g ( a ) × g ( a ) g ( b ) ( [ g ( b ) ] p + 1 [ g ( x ) ] p + 1 ) α 1 [ g ( x ) ] p g ( x ) g ( a ) g ( b ) g ( a ) d g ( x ) [ f ( a ) + f ( b ) ] ( p + 1 ) 1 α Γ ( α ) g ( a ) g ( b ) ( [ g ( b ) ] p + 1 [ g ( x ) ] p + 1 ) α 1 [ g ( x ) ] p d g ( x ) . Open image in new window
This implies that
J p , g ( a ) + α f ( g ( b ) ) [ f ( a ) + f ( b ) ] J p , g ( a ) + α ( 1 ) . Open image in new window
(3.13)
Also
( p + 1 ) 1 α Γ ( α ) 0 1 ( [ ( 1 t ) g ( a ) + t g ( b ) ] p + 1 [ g ( a ) ] p + 1 ) α 1 × [ ( 1 t ) g ( a ) + t g ( b ) ] p f ( ( 1 t ) g ( a ) + t g ( b ) ) d t ( p + 1 ) 1 α Γ ( α ) f ( a ) 0 1 ( [ ( 1 t ) g ( a ) + t g ( b ) ] p + 1 [ g ( a ) ] p + 1 ) α 1 × [ ( 1 t ) g ( a ) + t g ( b ) ] p ( 1 t ) d t + ( p + 1 ) 1 α Γ ( α ) f ( b ) 0 1 ( [ ( 1 t ) g ( a ) + t g ( b ) ] p + 1 [ g ( a ) ] p + 1 ) α 1 × [ ( 1 t ) g ( a ) + t g ( b ) ] p ( t ) d t . Open image in new window
This implies that
( p + 1 ) 1 α Γ ( α ) ( g ( b ) g ( a ) ) g ( a ) g ( b ) ( [ g ( x ) ] p + 1 [ g ( a ) ] p + 1 ) α 1 [ g ( x ) ] p f ( g ( x ) ) d g ( x ) ( p + 1 ) 1 α Γ ( α ) f ( a ) g ( b ) g ( a ) g ( a ) g ( b ) ( [ g ( x ) ] p + 1 [ g ( a ) ] p + 1 ) α 1 [ g ( x ) ] p g ( b ) g ( x ) g ( b ) g ( a ) d g ( x ) + ( p + 1 ) 1 α Γ ( α ) f ( b ) g ( b ) g ( a ) g ( a ) g ( b ) ( [ g ( x ) ] p + 1 [ g ( a ) ] p + 1 ) α 1 [ g ( x ) ] p g ( x ) g ( a ) g ( b ) g ( a ) d g ( x ) [ f ( a ) + f ( b ) ] ( p + 1 ) 1 α Γ ( α ) g ( a ) g ( b ) ( [ g ( x ) ] p + 1 [ g ( a ) ] p + 1 ) α 1 [ g ( x ) ] p d g ( x ) . Open image in new window
This implies that
J p , g ( b ) α f ( g ( a ) ) [ f ( a ) + f ( b ) ] J p , g ( b ) α ( 1 ) . Open image in new window
(3.14)

Combining (3.13) and (3.14), we have the required result. □

Remark 3.2 We can prove the Hermite-Hadamard inequality for the classical Riemann-Liouville integrals as follows:
f ( g ( a ) + g ( b ) 2 ) Γ ( α + 1 ) 2 ( g ( b ) g ( a ) ) α [ J g ( a ) + α f ( g ( b ) ) + J g ( b ) α f ( g ( a ) ) ] f ( a ) + f ( b ) 2 . Open image in new window

We now derive the Hermite-Hadamard inequalities for the class of relative logarithmic semi-convex functions.

Theorem 3.15Let f : I R R Open image in new windowbe a relative logarithmic semi-convex function, then for all t [ 0 , 1 ] Open image in new windowwe have
f ( g ( a ) + g ( b ) 2 ) exp [ 1 g ( b ) g ( a ) g ( a ) g ( b ) log f ( g ( x ) ) d g ( x ) ] f ( a ) f ( b ) . Open image in new window
Theorem 3.16Let f : I R R Open image in new windowbe a relative logarithmic semi-convex function, then for all t [ 0 , 1 ] Open image in new window,
f ( g ( a ) + g ( b ) 2 ) exp [ 1 g ( b ) g ( a ) g ( a ) g ( b ) log f ( g ( x ) ) d g ( x ) ] 1 g ( b ) g ( a ) g ( a ) g ( b ) G ( f ( g ( x ) ) , f ( g ( a ) + g ( b ) g ( x ) ) ) d g ( x ) 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) d g ( x ) L [ f ( b ) , f ( a ) ] f ( a ) + f ( b ) 2 , Open image in new window

where L [ f ( b ) , f ( a ) ] = f ( b ) f ( a ) log f ( b ) log f ( a ) Open image in new window, and G [ f ( a ) , f ( b ) ] = f ( a ) f ( b ) Open image in new window.

Proof The proof of the first inequality follows directly from Theorem 3.15. For the second inequality, we consider
1 g ( b ) g ( a ) g ( a ) g ( b ) G ( f ( g ( x ) ) , f ( g ( a ) + g ( b ) g ( x ) ) ) d g ( x ) = 1 g ( b ) g ( a ) g ( a ) g ( b ) exp [ log G ( f ( g ( x ) ) , f ( g ( a ) + g ( b ) g ( x ) ) ) ] d g ( x ) exp [ 1 g ( b ) g ( a ) g ( a ) g ( b ) log G ( f ( g ( x ) ) , f ( g ( a ) + g ( b ) g ( x ) ) ) d g ( x ) ] = exp [ 1 g ( b ) g ( a ) g ( a ) g ( b ) log f ( g ( x ) ) + log f ( g ( a ) + g ( b ) g ( x ) ) 2 d g ( x ) ] = exp [ 1 g ( b ) g ( a ) g ( a ) g ( b ) log f ( g ( x ) ) d g ( x ) ] . Open image in new window
Using the AM-GM inequality, we have
G ( f ( g ( x ) ) , f ( g ( a ) + g ( b ) g ( x ) ) ) f ( g ( x ) ) + f ( g ( a ) + g ( b ) g ( x ) ) 2 . Open image in new window
Integrating the above inequality with respect to x on [ g ( a ) , g ( b ) ] Open image in new window, we have
1 g ( b ) g ( a ) g ( a ) g ( b ) G ( f ( g ( x ) ) , f ( g ( a ) + g ( b ) g ( x ) ) ) d g ( x ) 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) d g ( x ) . Open image in new window

Now, using the fact that f is a relative semi-convex function and applying the change of variable technique on the right-hand side of the above inequality completes the proof. □

Theorem 3.17Let f , w : I R R Open image in new windowbe relative logarithmic semi-convex functions, then we have
1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) L [ f ( a ) w ( b ) , f ( a ) w ( a ) ] f ( a ) w ( a ) + f ( b ) w ( b ) 2 1 4 Θ ( a , b ) , Open image in new window

where Θ ( a , b ) = [ f ( a ) ] 2 + [ f ( b ) ] 2 + [ w ( a ) ] 2 + [ w ( b ) ] 2 Open image in new window.

Proof Let f and w be relative logarithmic semi-convex functions. Then
1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) = 0 1 f ( ( 1 t ) g ( a ) + t g ( b ) ) w ( ( 1 t ) g ( a ) + t g ( b ) ) 0 1 [ f ( a ) w ( a ) ] 1 t [ f ( b ) w ( b ) ] t d t = f ( b ) w ( b ) f ( a ) w ( a ) log f ( b ) w ( b ) log f ( a ) w ( a ) = L [ f ( b ) w ( b ) , f ( a ) w ( a ) ] f ( a ) w ( a ) + f ( b ) w ( b ) 2 1 2 0 1 [ { f ( ( 1 t ) g ( a ) + t g ( b ) ) } 2 + { w ( ( 1 t ) g ( a ) + t g ( b ) ) } 2 ] d t 1 2 0 1 [ { [ f ( a ) ] 1 t [ f ( b ) ] t } 2 + { [ w ( a ) ] 1 t [ w ( b ) ] t } 2 ] d t = 1 4 [ [ f ( a ) + f ( b ) ] [ f ( b ) f ( a ) ] log f ( b ) log f ( a ) + [ w ( a ) + w ( b ) ] [ w ( b ) w ( a ) ] log w ( b ) log w ( a ) ] 1 8 [ [ f ( a ) + f ( b ) ] 2 + [ w ( a ) + w ( b ) ] 2 ] 1 4 Θ ( a , b ) . Open image in new window

 □

Theorem 3.18Let f , w : I R R Open image in new windowbe relative logarithmic semi-convex functions, then
log w ( g ( a ) + g ( b ) 2 ) 1 g ( b ) g ( a ) g ( a ) g ( b ) log w ( g ( x ) ) d g ( x ) 1 g ( b ) g ( a ) g ( a ) g ( b ) log f ( g ( x ) ) d g ( x ) log f ( g ( a ) + g ( b ) 2 ) . Open image in new window
Proof Let f and w be relative logarithmic semi-convex functions. Then
log f ( g ( a ) + g ( b ) 2 ) w ( g ( a ) + g ( b ) 2 ) = log [ f ( ( 1 t ) g ( a ) + t g ( b ) + t g ( a ) + ( 1 t ) g ( b ) 2 ) × w ( ( 1 t ) g ( a ) + t g ( b ) + t g ( a ) + ( 1 t ) g ( b ) 2 ) ] log [ [ f ( ( 1 t ) g ( a ) + t g ( b ) ) f ( t g ( a ) + ( 1 t ) g ( b ) ) ] 1 2 × [ w ( ( 1 t ) g ( a ) + t g ( b ) ) w ( t g ( a ) + ( 1 t ) g ( b ) ) ] 1 2 ] = 1 2 [ log f ( ( 1 t ) g ( a ) + t g ( b ) ) + log f ( t g ( a ) + ( 1 t ) g ( b ) ) ] + 1 2 [ log w ( ( 1 t ) g ( a ) + t g ( b ) ) + log w ( t g ( a ) + ( 1 t ) g ( b ) ) ] . Open image in new window

Integrating both sides of the above inequality with respect to t on [ 0 , 1 ] Open image in new window, we have the required result. □

Theorem 3.19Let f , w : I R R Open image in new windowbe relative logarithmic semi-convex functions, then
1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( a ) + g ( b ) g ( x ) ) d g ( x ) f ( a ) w ( b ) f ( b ) w ( a ) log f ( a ) w ( b ) log f ( b ) w ( a ) 1 4 Θ ( a , b ) , Open image in new window

where Θ ( a , b ) = [ f ( a ) ] 2 + [ f ( b ) ] 2 + [ w ( a ) ] 2 + [ w ( b ) ] 2 Open image in new window.

Proof Since f, w are relative logarithmic semi-convex functions, then we have
1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( a ) + g ( b ) g ( x ) ) d g ( x ) = 0 1 f ( t g ( a ) + ( 1 t ) g ( b ) ) w ( ( 1 t ) g ( a ) + t g ( b ) ) d t 0 1 [ f ( a ) ] t [ f ( b ) ] 1 t [ w ( a ) ] 1 t [ w ( b ) ] t d t = f ( a ) w ( b ) f ( b ) w ( a ) log f ( a ) w ( b ) log f ( b ) w ( a ) = L [ f ( a ) w ( b ) , f ( b ) w ( a ) ] f ( a ) w ( b ) + f ( b ) w ( a ) 2 1 2 0 1 { [ f ( t g ( a ) + ( 1 t ) g ( b ) ) ] 2 + [ w ( ( 1 t ) g ( a ) + t g ( b ) ) ] 2 } d t 1 2 0 1 { [ f ( a ) ] t [ f ( b ) ] 1 t } 2 d t + 1 2 0 1 { [ w ( a ) ] 1 t [ w ( b ) ] t } 2 d t = 1 4 [ f ( a ) ] 2 [ f ( b ) ] 2 log f ( a ) log f ( b ) + 1 4 [ w ( a ) ] 2 [ w ( b ) ] 2 log w ( a ) log w ( b ) = 1 2 [ f ( a ) + f ( b ) 2 f ( a ) f ( b ) log f ( a ) log f ( b ) ] + 1 2 [ w ( a ) + w ( b ) 2 w ( a ) w ( b ) log w ( a ) log w ( b ) ] 1 2 [ f ( a ) + f ( b ) 2 f ( a ) + f ( b ) 2 ] + 1 2 [ w ( a ) + w ( b ) 2 w ( a ) + w ( b ) 2 ] 1 4 Θ ( a , b ) , Open image in new window

which is the required result. □

Theorem 3.20Let f , w : I ( 0 , ) Open image in new windowbe increasing and relative logarithmic semi-convex functions onIwith g ( a ) , g ( b ) I Open image in new window. Then we have
f ( g ( a ) + g ( b ) 2 ) L [ w ( a ) , w ( b ) ] + w ( g ( a ) + g ( b ) 2 ) L [ f ( a ) , f ( b ) ] 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) + L [ f ( a ) w ( a ) , f ( b ) w ( b ) ] . Open image in new window
Proof Let f and w be relative logarithmic semi-convex functions. Then
f ( t g ( a ) + ( 1 t ) g ( b ) ) [ f ( a ) ] t [ f ( b ) ] 1 t , w ( t g ( a ) + ( 1 t ) g ( b ) ) [ w ( a ) ] t [ w ( b ) ] 1 t . Open image in new window
Now, using x 1 x 2 , x 3 x 4 0 Open image in new window ( x 1 , x 2 , x 3 , x 4 R Open image in new window) and x 1 < x 2 < x 3 < x 4 Open image in new window, we have
f ( t g ( a ) + ( 1 t ) g ( b ) ) [ w ( a ) ] t [ w ( b ) ] 1 t + w ( t g ( a ) + ( 1 t ) g ( b ) ) [ f ( a ) ] t [ f ( b ) ] 1 t f ( t g ( a ) + ( 1 t ) g ( b ) ) w ( t g ( a ) + ( 1 t ) g ( b ) ) + [ f ( a ) ] t [ f ( b ) ] 1 t [ w ( a ) ] t [ w ( b ) ] 1 t . Open image in new window
Integrating the above inequalities with respect to t on [ 0 , 1 ] Open image in new window, we have
0 1 f ( t g ( a ) + ( 1 t ) g ( b ) ) [ w ( a ) ] t [ w ( b ) ] 1 t d t + 0 1 w ( t g ( a ) + ( 1 t ) g ( b ) ) [ f ( a ) ] t [ f ( b ) ] 1 t d t 0 1 f ( t g ( a ) + ( 1 t ) g ( b ) ) w ( t g ( a ) + ( 1 t ) g ( b ) ) d t + 0 1 [ f ( a ) ] t [ f ( b ) ] 1 t [ w ( a ) ] t [ w ( b ) ] 1 t d t . Open image in new window
Now, since f and w are increasing, using Chebyshev inequalities [28], we have
0 1 f ( t g ( a ) + ( 1 t ) g ( b ) ) d t 0 1 [ w ( a ) ] t [ w ( b ) ] 1 t d t + 0 1 w ( t g ( a ) + ( 1 t ) g ( b ) ) d t 0 1 [ f ( a ) ] t [ f ( b ) ] 1 t d t 0 1 f ( t g ( a ) + ( 1 t ) g ( b ) ) w ( t g ( a ) + ( 1 t ) g ( b ) ) d t + 0 1 [ f ( a ) ] t [ f ( b ) ] 1 t [ w ( a ) ] t [ w ( b ) ] 1 t d t . Open image in new window
Now calculating the simple integration, we have
1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) d g ( x ) L [ w ( a ) , w ( b ) ] + 1 g ( b ) g ( a ) g ( a ) g ( b ) w ( g ( x ) ) d g ( x ) L [ f ( a ) , f ( b ) ] 1 g ( b ) g ( a ) g ( a ) g ( b ) f ( g ( x ) ) w ( g ( x ) ) d g ( x ) + L [ f ( a ) w ( a ) , f ( b ) , w ( b ) ] . Open image in new window

Now, using the left-hand side of Hermite-Hadamard’s inequality for relative logarithmic semi-convex functions, we have the required result. □

Acknowledgements

The authors would like to thank Dr. SM Junaid Zaidi, Rector of COMSATS Institute of Information Technology, Pakistan, for providing excellent research and academic environment. We are grateful to the referees and the editor for their constructive comments and suggestions.

Copyright information

© Noor et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Muhammad Aslam Noor
    • 1
  • Muhammad Uzair Awan
    • 1
  • Khalida Inayat Noor
    • 1
  1. 1.Mathematics DepartmentCOMSATS Institute of Information TechnologyIslamabadPakistan