Journal of Inequalities and Applications

, 2013:332

On some inequalities for relative semi-convex functions

Authors

    • Mathematics DepartmentCOMSATS Institute of Information Technology
  • Muhammad Uzair Awan
    • Mathematics DepartmentCOMSATS Institute of Information Technology
  • Khalida Inayat Noor
    • Mathematics DepartmentCOMSATS Institute of Information Technology
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 functionconvex setHermite-Hadamard inequalityfractional 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, [113] 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, 58, 10, 1522].

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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq1_HTML.gif be a finite dimensional space, the inner product of which is denoted by , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq2_HTML.gif.

Definition 2.1 [13]

A set M R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq3_HTML.gif is said to be a relative convex (g-convex) set if and only if there exists an arbitrary function g : R n R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq4_HTML.gif such that
( 1 t ) g ( x ) + t g ( y ) M , x , y R n : g ( x ) , g ( y ) M , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ1_HTML.gif
(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 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq5_HTML.gif and g ( x ) = x 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq6_HTML.gif, x R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq7_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq4_HTML.gif 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 ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ2_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq4_HTML.gif such that
f ( ( 1 t ) g ( x ) + t g ( y ) ) ( 1 t ) f ( x ) + t f ( y ) , x , y M , t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ3_HTML.gif
(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 + https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq8_HTML.gif 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 ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ4_HTML.gif
(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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equa_HTML.gif

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 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq9_HTML.gif. The generalized Riemann-Liouville fractional integrals J a + α f https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq10_HTML.gif and J b α f https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq11_HTML.gif of order α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq12_HTML.gif with p 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq13_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equb_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equc_HTML.gif

respectively, where Γ ( α ) = 0 e t x α 1 d x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq14_HTML.gif is the gamma function.

If p = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq15_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq16_HTML.gif if
f ( x ) f ( y ) , g ( x ) g ( y ) 0 , x , y I . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equd_HTML.gif

Let M = I = [ g ( a ) , g ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq17_HTML.gif 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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq18_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Eque_HTML.gif
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 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq19_HTML.gif.

     
  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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq20_HTML.gif.

     
  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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq21_HTML.gif.

     
  5. 5.

    ( g ( b ) g ( x ) ) f ( a ) ( g ( b ) g ( a ) ) f ( g ( x ) ) + ( g ( x ) g ( a ) ) f ( b ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq22_HTML.gif,

     

where g ( x ) = ( 1 t ) g ( a ) + t g ( b ) M https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq23_HTML.gif, t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq24_HTML.gif.

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 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equf_HTML.gif

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 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equg_HTML.gif
Lemma 3.2Let f : I R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq25_HTML.gifbe a relative semi-convex function, where I = [ g ( a ) , g ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq18_HTML.gif. If g ( x ) { g ( a ) , g ( b ) } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq26_HTML.gif, then
f ( b ) f ( g ( x ) ) g ( b ) g ( x ) f ( a ) f ( g ( x ) ) g ( a ) g ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equh_HTML.gif
Lemma 3.3Letfbe a relative semi-convex function. Consider g ( x 1 ) , g ( x 2 ) , , g ( x n ) I https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq27_HTML.gif, g ( y 1 ) , g ( y 2 ) , , g ( y n ) I https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq28_HTML.gifand weights ω 1 , ω 2 , , ω n R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq29_HTML.gifsuch that:
  1. (i)

    g ( x 1 ) g ( x 2 ) g ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq30_HTML.gifand g ( y 1 ) g ( y 2 ) g ( y n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq31_HTML.gif,

     
  2. (ii)

    k = 1 r ω k g ( x k ) k = 1 r ω k g ( y k ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq32_HTML.gif, r = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq33_HTML.gif,

     
  3. (iii)

    k = 1 n ω k g ( x k ) = k = 1 n ω k g ( y k ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq34_HTML.gif,

     
then we have
k = 1 n ω k f ( g ( x k ) ) k = 1 n ω k f ( y k ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equi_HTML.gif
Lemma 3.4Letfbe a relative semi-convex function. Consider g ( x 1 ) , g ( x 2 ) , , g ( x n ) I https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq27_HTML.gif, g ( y 1 ) , g ( y 2 ) , , g ( y n ) I https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq28_HTML.gifand weights ω 1 , ω 2 , , ω n R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq29_HTML.gifsuch that
  1. (i)

    g ( x 1 ) g ( x 2 ) g ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq30_HTML.gifand g ( y 1 ) g ( y 2 ) g ( y n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq31_HTML.gif,

     
  2. (ii)

    k = 1 r ω k g ( x k ) k = 1 r ω k g ( y k ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq32_HTML.gif, r = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq33_HTML.gif,

     
  3. (iii)

    f ( x ) f ( y ) , g ( x ) g ( y ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq35_HTML.gif,

     
then we have
k = 1 n ω k f ( g ( x k ) ) k = 1 n ω k f ( y k ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equj_HTML.gif
Lemma 3.5Letfbe a relative semi-convex function, then, for all g ( a ) < g ( c ) < g ( d ) < g ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq36_HTML.gif, we have
f ( a ) + f ( b ) 2 f ( g ( a ) + g ( b ) 2 ) f ( c ) + f ( d ) 2 f ( g ( c ) + g ( d ) 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equk_HTML.gif

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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equl_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq37_HTML.gifbe a relative semi-convex function on I = [ g ( a ) , g ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq38_HTML.gifwith g ( a ) < g ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq39_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ5_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equm_HTML.gif

 □

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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq40_HTML.gif, we have
f ( g ( a ) + g ( b ) g ( x ) ) f ( a ) + f ( b ) f ( g ( x ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equn_HTML.gif
Theorem 3.9Letfbe a relative semi-convex function and let w : [ g ( a ) , g ( b ) ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq41_HTML.gifbe nonnegative, integrable and symmetric about g ( a ) + g ( b ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq42_HTML.gif. 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ6_HTML.gif
(3.2)
Proof Since f is a relative semi-convex function and w : [ g ( a ) , g ( b ) ] R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq41_HTML.gif is nonnegative, integrable and symmetric about g ( a ) + g ( b ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq42_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equo_HTML.gif

This completes the proof. □

Theorem 3.10Let f , w : I R R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq43_HTML.gifbe relative semi-convex functions onIwith g ( a ) < g ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq39_HTML.gif. Then, for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq24_HTML.gif, 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equp_HTML.gif
where
M ( a , b ) = f ( a ) w ( a ) + f ( b ) w ( b ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ7_HTML.gif
(3.3)
N ( a , b ) = f ( a ) w ( b ) + f ( b ) w ( a ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ8_HTML.gif
(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 ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equq_HTML.gif
Integrating with respect to t on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq44_HTML.gif, 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 ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equr_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equs_HTML.gif

This completes the proof. □

Theorem 3.11Let f , w : I R R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq43_HTML.gifbe relative semi-convex functions onIwith g ( a ) < g ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq39_HTML.gif. Ifwis symmetric about g ( a ) + g ( b ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq42_HTML.gif, then for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq24_HTML.gifwe 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equt_HTML.gif

where M ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq45_HTML.gifand N ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq46_HTML.gifare given by (3.3) and (3.4), Θ ( a , b ) = [ f ( a ) ] 2 + [ f ( b ) ] 2 + [ w ( a ) ] 2 + [ w ( b ) ] 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq47_HTML.gif.

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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equu_HTML.gif

The desired result. □

Theorem 3.12Let f , w : I R R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq43_HTML.gifbe similarly ordered and relative semi-convex functions onIwith g ( a ) < g ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq39_HTML.gif. Then, for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq24_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equv_HTML.gif

where M ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq45_HTML.gifis 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq48_HTML.gifwe 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ9_HTML.gif
(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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equw_HTML.gif
and
Δ 2 ( λ ) = f ( ( 1 λ ) g ( a ) + λ g ( b ) ) + λ f ( a ) + ( 1 λ ) f ( b ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equx_HTML.gif
Proof We divide the interval [ g ( a ) , g ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq49_HTML.gif into [ g ( a ) , ( 1 λ ) g ( a ) + λ g ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq50_HTML.gif and [ ( 1 λ ) g ( a ) + λ g ( b ) , g ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq51_HTML.gif. 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ10_HTML.gif
(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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ11_HTML.gif
(3.7)
Multiplying (3.6) by λ and (3.7) by ( 1 λ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq52_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ12_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ13_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ14_HTML.gif
(3.10)
Multiplying (3.9) by λ and (3.10) by ( 1 λ ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq52_HTML.gif 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ15_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ16_HTML.gif
(3.12)

the required result. □

Remark 3.1 For suitable and different choices of λ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq48_HTML.gif and g = I https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq53_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equy_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equz_HTML.gif
Let g ( x ) = ( 1 t ) g ( a ) + t g ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq54_HTML.gif, then d t = d g ( x ) g ( b ) g ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq55_HTML.gif. Take t = g ( x ) g ( a ) g ( b ) g ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq56_HTML.gif, 1 t = g ( b ) g ( x ) g ( b ) g ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq57_HTML.gif. 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equaa_HTML.gif
This implies that
J p , g ( a ) + α f ( g ( b ) ) [ f ( a ) + f ( b ) ] J p , g ( a ) + α ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ17_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equab_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equac_HTML.gif
This implies that
J p , g ( b ) α f ( g ( a ) ) [ f ( a ) + f ( b ) ] J p , g ( b ) α ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equ18_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equad_HTML.gif

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

Theorem 3.15Let f : I R R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq37_HTML.gifbe a relative logarithmic semi-convex function, then for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq24_HTML.gifwe 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equae_HTML.gif
Theorem 3.16Let f : I R R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq37_HTML.gifbe a relative logarithmic semi-convex function, then for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq24_HTML.gif,
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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equaf_HTML.gif

where L [ f ( b ) , f ( a ) ] = f ( b ) f ( a ) log f ( b ) log f ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq58_HTML.gif, and G [ f ( a ) , f ( b ) ] = f ( a ) f ( b ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq59_HTML.gif.

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 ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equag_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equah_HTML.gif
Integrating the above inequality with respect to x on [ g ( a ) , g ( b ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq49_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equai_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq43_HTML.gifbe 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equaj_HTML.gif

where Θ ( a , b ) = [ f ( a ) ] 2 + [ f ( b ) ] 2 + [ w ( a ) ] 2 + [ w ( b ) ] 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq47_HTML.gif.

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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equak_HTML.gif

 □

Theorem 3.18Let f , w : I R R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq43_HTML.gifbe 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equal_HTML.gif
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 ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equam_HTML.gif

Integrating both sides of the above inequality with respect to t on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq44_HTML.gif, we have the required result. □

Theorem 3.19Let f , w : I R R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq43_HTML.gifbe 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equan_HTML.gif

where Θ ( a , b ) = [ f ( a ) ] 2 + [ f ( b ) ] 2 + [ w ( a ) ] 2 + [ w ( b ) ] 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq47_HTML.gif.

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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equao_HTML.gif

which is the required result. □

Theorem 3.20Let f , w : I ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq60_HTML.gifbe increasing and relative logarithmic semi-convex functions onIwith g ( a ) , g ( b ) I https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq61_HTML.gif. 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 ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equap_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equaq_HTML.gif
Now, using x 1 x 2 , x 3 x 4 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq62_HTML.gif ( x 1 , x 2 , x 3 , x 4 R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq63_HTML.gif) and x 1 < x 2 < x 3 < x 4 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq64_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equar_HTML.gif
Integrating the above inequalities with respect to t on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_IEq44_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equas_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equat_HTML.gif
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 ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-332/MediaObjects/13660_2013_Article_778_Equau_HTML.gif

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.

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© Noor et al.; licensee Springer 2013

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