Appl. Math. Inf. Sci.8, No. 2, 607-616 (2014) 607

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.12785/amis/080218

Geometrically Relative Convex FunctionsMuhammad Aslam Noor∗, Khalida Inayat Noor and Muhammad Uzair Awan

Mathematics Department, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan

Received: 22 Mar. 2013, Revised: 23 Jul. 2013, Accepted: 25 Jul. 2013Published online: 1 Mar. 2014

Abstract: In this paper, some new concepts of geometrically relative convex sets and relative convex functions are defined. These newclasses of geometrically relative convex functions unify several known and new classes of relative convex functions such as exponentialconvex functions. New Hermite-Hadamard type integral inequalities are derived for these new classes of geometrically relative convexfunctions and their variant forms. Some special cases, which can be obtained from our results, are discussed. Results proved in thispaper represent significant improvements of the previously known results. We would like to emphasize that the results obtained anddiscussed in this paper may stimulate novel, innovative and potential applications of the geometrically relative convex functions in otherfields.

Keywords: Convex functions, geometrically relative convex functions, Hermite-Hadamard inequality.2010 AMS Subject Classification:26D15, 26A51, 49J40, 90C33

1 Introduction

Recently convexity has seen a dramatic increase in itsapplications for solving a large number of problemswhich arise in various branches of pure and appliedsciences. As a result of these activities, the concept ofconvexity has been extended and generalized in variousdirections using novel and innovative ideas see [1,6,7,9,10,11,12,13,14,15,16,18,19,22,23]. An important andsignificant generalization of the convex functions is theintroduction of relative convex functions by Youness [6].These relative convex functions plays an interesting rolein optimization theory, since they provide a broadersetting to study the optimization and programmingproblems. It is well known [5,6] that the relative convexsets and relative convex functions are nonconvex sets andnonconvex functions respectively. However it has beenshown that the relative convex functions preserve somenice properties that the convex functions have. It has beenshown by Noor [11] that the minimum of a differentiablerelative convex functions on the relative convex set can becharacterized by a class of variational inequalities, whichare known as general variational inequalities. This showsthat the concept of relative convexity plays the same rolefor general variational inequalities as classical convexityplays for variational inequalities. For the applications and

other aspects of the relative convexity, see [10,11,12] andthe references therein.Let f : I ⊆ R → R be a convex function witha < b anda,b ∈ I. Then the following double inequality is known asHermite-Hadamard inequality in the literature.

f

(

a+b2

)

≤ 1b−a

b∫

a

f (x)dx ≤ f (a)+ f (b)2

.

In recent years, much attention has been given to derivethe Hermite-Hadamard type inequalities for various typesof convex functions, see [1,2,4,17,19,20,21,22,23,24].Motivated and inspired by the recent research going on inthis field, we introduce and study a new class of relativeconvex functions, which is called the geometricallyrelative convex functions. We derive severalHermite-Hadamard type integral inequalities for thesenew geometrically relative convex functions. Severalspecial cases are also discussed. The ideas and techniquesof this paper may stimulate further research in thisinteresting field.

2 Preliminaries

In this section, we recall some known concepts and definethe class of geometrically (GG) relative convex functions

∗ Corresponding author e-mail:noormaslam@hotmail.com

c© 2014 NSPNatural Sciences Publishing Cor.

608 M. A. Noor et al: Geometrically Relative Convex Functions

andGA relative convex functions. First of all letRn be thefinite dimensional space, whose inner product and normare denoted by〈., .〉 and||.||, respectively.

Definition 1. Let G ⊆ (0,∞). Then G is said to begeometrically relative convex set, if there exists anarbitrary function g : Rn → R

n such that

(g(x))t(g(y))1−t ∈ G , ∀g(x),g(y) ∈ G , t ∈ [0,1].

UsingAM−GM inequality, we have

g(x),g(y) ∈ G , t ∈ [0,1] ⇒ (g(x))t(g(y))1−t

≤ tg(x)+(1− t)g(y).

Definition 2([6,19]). A set Mg ⊆Rn is said to be a relative

convex (g-convex) set, if there exists a function g :Rn →Rn

such that,

tg(x)+(1− t)g(y) ∈ Mg,

∀x,y ∈ Rn : g(x),g(y) ∈ Mg, t ∈ [0,1]. (1)

Recently it has been shown in [5], that if Mg is a relativeconvex set then it is possible that it may not be a classicalconvex set.

Definition 3. A function f : G → R (on subintervals of(0,∞)) is said to be geometrically relative convex function(GG-relative convex function) if there exists an arbitraryfunction g : Rn → R

n such that,

f ((g(x))t(g(y))1−t)≤ ( f (g(x)))t( f (g(y)))1−t,

∀g(x),g(y) ∈ G , t ∈ [0,1]. (2)

From (2), it follows that

log f ((g(x))t(g(y))1−t)

≤ t log f (g(x))+(1− t) log f (g(y)),

∀g(x),g(y) ∈ G , t ∈ [0,1].

UsingAM−GM inequality, we have

f ((g(x))t(g(y))1−t) ≤ ( f (g(x)))t( f (g(y)))1−t

≤ t f (g(x))+(1− t) f (g(y)).

This implies that every geometrically relative convexfunction (that is GG-relative convex function) is alsoGA-relative convex function, but the converse is not true.

Definition 4. A function f : G → R (on subintervals of(0,∞)) is said to be GA-relative convex function) if thereexists an arbitrary function g : Rn → R

n such that,

f ((g(x))t(g(y))1−t)≤ t f (g(x))+(1− t) f (g(y)),

∀g(x),g(y) ∈ G , t ∈ [0,1]. (3)

We note that forf (x) = ex Definition 4 reduces to one in[8]. That is

f (etx+(1−t)y)≤ t f (ex)+(1− t) f (ey),

∀g(x),g(y) ∈ G , t ∈ [0,1]. (4)

From Definition 3 and Definition 4, it follows thatGG =⇒ GA, but the converse is not true.

Again using theAM −GM inequality from Definition 3,we have the following known concept of relative convexfunctions.

Definition 5([6,19]). A function f is said to be a relativeconvex (g-convex) function (that is AA relative convexfunction) on a relative convex (g-convex) set Mg, if andonly if, there exists a function g : Rn → R

n such that,

f ((1− t)g(x)+ tg(y))≤ (1− t) f (g(x))+ t f (g(y)),

∀x,y ∈ Rn : g(x),g(y) ∈ Mg, t ∈ [0,1]. (5)

It is known [6] that every convex functionf on a convexset is a relative convex function, but the converse is nottrue. There are functions which are relative convexfunction but may not be a convex function in the classicalsense.Noor [11] proved the optimality condition for thedifferentiable relative convex functions on relative convexset can be characterized by a class of variationalinequality which is called as general variationalinequality, for the applications and other aspects ofgeneral variational inequalities, see [9,10,11,12].Now we define the concept of relative log convexfunctions.

Definition 6. A function f : Mg → R (on subintervals of(0,∞)) is said to be relative log convex function (that isAG-relative convex function) if there exists an arbitraryfunction g : Rn → R

n such that,

f (tg(x)+(1− t)g(y))≤ ( f (g(x)))t( f (g(y)))1−t,

∀g(x),g(y) ∈ Mg, t ∈ [0,1]. (6)

From (6) it follows that

log f (tg(x)+(1− t)g(y))

≤ t log f (g(x))+(1− t) log f (g(y)),

∀g(x),g(y) ∈ Mg, t ∈ [0,1].

Definition 7. A function f : Mg → R (on subintervals of(0,∞)) is said to be relative geometrically quasi-convexfunction if there exists an arbitrary function g : Rn → R

n

such that,

f ((g(x))t(g(y))1−t)≤ max{ f (g(x)), f (g(y))},∀g(x),g(y) ∈ Mg, t ∈ [0,1]. (7)

Next we define the concept of geometrically relativeconvex functions on an interval.

c© 2014 NSPNatural Sciences Publishing Cor.

Appl. Math. Inf. Sci.8, No. 2, 607-616 (2014) /www.naturalspublishing.com/Journals.asp 609

Definition 8. Let I be a subinterval of (0,∞). Then f isgeometrically relative convex function, if and only if,∣

∣

∣

∣

∣

∣

1 1 1logg(a) logg(x) logg(b)

log f (g(a)) log f (g(x)) log f (g(b))

∣

∣

∣

∣

∣

∣

≥ 0,

where g(a)≤ g(x)≤ g(a).

One can easily show that the following are equivalent:

1. f is geometrically relative semi-convex function onrelative convex set.

2. f (g(a))log(g(b)) f (g(x))log(g(a)) f (g(b))log(g(x))

≥ f (g(a))log(g(x)) f (g(x))log(g(b)) f (g(b))log(g(a)).

whereg(x) = g(a)tg(b)1−t andt ∈ [0,1].

For g(x) = x the Definition 8 reduces to the definitionfor geometrically convex functions see [3].

3 Main Results

In this section we discuss our main results. For thispurpose we need following lemmas which play a key partin proving our main results.Essentially using the technique of [21] one can prove thefollowing result.

Lemma 1. Let f : I ⊆ R→ R be a differentiable functionon I◦ (the interior of I) and g : R → R be arbitraryfunction. If f ′ ∈ L [g(a),g(b)] for g(a),g(b) ∈ I withg(a)< g(b). Then

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

=g(b)−g(a)

2

1∫

0

(1−2t) f ′(tg(a)+(1− t)g(b))dt.

Lemma 2([17,24]). Let f : I ⊆R→R be a differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. If f ′′ ∈ L [g(a),g(b)] forg(a),g(b) ∈ I with g(a)< g(b). Then

1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)− f

(

g(a)+g(b)2

)

= (g(b)−g(a))

1∫

0

µ(t) f ′(tg(a)+(1− t)g(b))dt

,

where

µ(t) ={

t, [0, 12),

t −1, [ 12 ,1].

Using the technique of [20], one cane prove the followinglemma.

Lemma 3. Let f : I ⊆ R → R be twice differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. If f ′ ∈ L [g(a),g(b)] for g(a),g(b) ∈ Iwith g(a)< g(b). Then

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

=[g(b)−g(a)]2

2

1∫

0

t(1− t) f ′′(tg(a)+(1− t)g(b))dt.

Using the technique of [23] one can prove the followinglemma.

Lemma 4. Let f be a differentiable function on(g(a),g(b)) with g(a) < g(b) where g : Rn → R

n is anyarbitrary function. If f ∈ L [g(a),g(b)]. Then thefollowing identity holds:

[g(b) f (g(b))−g(a) f (g(a))]−g(b)∫

g(a)

f (g(x))dg(x)

= (lng(b)− lng(a))

1∫

0

(g(a))2t(g(b))2(1−t)

× f ′((g(a))t(g(b))1−t)dt.

Now we are in a position to derive our main results. Firstof all, we prove the results for the class of geometricallyrelative convex functions (GG).

Theorem 1. Let f : I ⊆ R → R be a differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. Also f ′ ∈ L [g(a),g(b)] forg(a),g(b) ∈ I with g(a) < g(b). If | f ′| is decreasing andgeometrically relative convex function. Then∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

| f ′(g(b))|Ψ(w),

where w = | f ′(g(a))|| f ′(g(b))| and Ψ(w) = w ln(w)+4

√w−2w−ln(w)−2

ln(w)2.

Proof. Using Lemma 1 and the fact that| f ′| isgeometrically relative convex function, we have∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

1∫

0

|1−2t|| f ′(tg(a)+(1− t)g(b))|dt

≤ g(b)−g(a)2

1∫

0

|1−2t|∣

∣

∣f ′(

g(a)tg(b)1−t)∣

∣

∣dt

c© 2014 NSPNatural Sciences Publishing Cor.

610 M. A. Noor et al: Geometrically Relative Convex Functions

≤ g(b)−g(a)2

| f ′(g(b))|1∫

0

|1−2t|( | f ′(g(a))|| f ′(g(b))|

)t

dt

=g(b)−g(a)

2| f ′(g(b))|

1∫

0

|1−2t|wtdt

=g(b)−g(a)

2| f ′(g(b))|w ln(w)+4

√w−2w− ln(w)−2ln(w)2

.

This completes the proof. �

Theorem 2. Let f : I ⊆ R → R be a differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. Also f ′ ∈ L [g(a),g(b)] forg(a),g(b) ∈ I with g(a) < g(b). If | f ′|q is decreasing andgeometrically relative convex function forp,q > 1, 1

p +1q = 1. Then

∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

| f ′(g(b))|(

1p+1

) 1p(

wq −1q ln(w)

) 1q

,

where w = | f ′(g(a))|| f ′(g(b))| .

Proof. Using Lemma 1, well known Holder’s inequalityand the fact that| f ′|q is geometrically relative convexfunction, we have∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

1∫

0

|1−2t|| f ′(tg(a)+(1− t)g(b))|dt

≤ g(b)−g(a)2

1∫

0

|1−2t|pdt

1p

×

1∫

0

∣

∣

∣f ′(

g(a)tg(b)1−t)∣

∣

∣

qdt

1q

≤ g(b)−g(a)2

| f ′(g(b))|(

1p+1

) 1p

1∫

0

( | f ′(g(a))|| f ′(g(b))|

)qt

dt

1q

=g(b)−g(a)

2| f ′(g(b))|

(

1p+1

) 1p

1∫

0

wqtdt

1q

=g(b)−g(a)

2| f ′(g(b))|

(

1p+1

) 1p(

wq −1q ln(w)

) 1q

.

This completes the proof. �

Theorem 3. Let f : I ⊆ R → R be a differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. Also f ′ ∈ L [g(a),g(b)] for

g(a),g(b) ∈ I with g(a) < g(b). If | f ′|q is decreasing andgeometrically relative convex function for q > 1. Then∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

| f ′(g(b))|(

12

) 1p

(Ψ(w))1q ,

where w =| f ′(g(a))|| f ′(g(b))| and

Ψ(w) = wq ln(w)q−2wq+4w12 q−ln(w)q−2

ln(w)2q2 .

Proof. Using Lemma 1, well known Power meaninequality and the fact that| f ′|q is geometrically relativeconvex function, we have∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

1∫

0

|1−2t|| f ′(tg(a)+(1− t)g(b))|dt

≤ g(b)−g(a)2

1∫

0

|1−2t|dt

1− 1q

×

1∫

0

|1−2t|∣

∣

∣f ′(

g(a)tg(b)1−t)∣

∣

∣

qdt

1q

≤ g(b)−g(a)2

| f ′(g(b))|(

12

) 1p

×

1∫

0

|1−2t|( | f ′(g(a))|| f ′(g(b))|

)qt

dt

1q

=g(b)−g(a)

2| f ′(g(b))|

(

12

) 1p

1∫

0

|1−2t|wqtdt

1q

=g(b)−g(a)

2| f ′(g(b))|

(

12

) 1p

×(

wq ln(w)q−2wq +4w12 q − ln(w)q−2

ln(w)2q2

) 1q

.

This completes the proof. �

Remark. For q = 1 Theorem 3 reduces to Theorem 1.

Theorem 4. Let f : I ⊆ R → R be a differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. Also f ′ ∈ L [g(a),g(b)] forg(a),g(b) ∈ I with g(a) < g(b). If | f ′| is decreasing andgeometrically relative convex function. Then∣

∣

∣

∣

∣

∣

∣

1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)− f

(

g(a)+g(b)2

)

∣

∣

∣

∣

∣

∣

∣

≤ (g(b)−g(a))| f ′(b)| [Ψ1(w)+Ψ2(w)] ,

c© 2014 NSPNatural Sciences Publishing Cor.

Appl. Math. Inf. Sci.8, No. 2, 607-616 (2014) /www.naturalspublishing.com/Journals.asp 611

where w = | f ′(g(a))|| f ′(g(b))| , Ψ1(w) =

−w12 + 1

2w12 ln(w)+1

ln(w)2and

Ψ3(w) =w− 1

2w12 ln(w)−w

12

ln(w)2.

Proof. Using Lemma 2 and the fact that| f ′| isgeometrically relative convex function, we have∣

∣

∣

∣

∣

∣

∣

1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)− f

(

g(a)+g(b)2

)

∣

∣

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∣

(g(b)−g(a))

1∫

0

µ(t) f ′(tg(a)+(1− t)g(b))dt

∣

∣

∣

∣

∣

∣

≤ (g(b)−g(a))

12∫

0

|t|| f ′((g(a))t(g(b))1−t)|dt

+

1∫

12

|(t −1)|| f ′((g(a))t(g(b))1−t)|dt

≤ (g(b)−g(a))| f ′(b)|

×

12∫

0

{

t

( | f ′(a)|| f ′(b)|

)t}

dt +

1∫

12

{

(1− t)

( | f ′(a)|| f ′(b)|

)t}

dt

≤ (g(b)−g(a))| f ′(b)|

12∫

0

twtdt +

1∫

12

(1− t)wtdt

= (g(b)−g(a))| f ′(b)|

×[

−w12 + 1

2w12 ln(w)+1

ln(w)2+

w− 12w

12 ln(w)−w

12

ln(w)2

]

.

This completes the proof. �

Theorem 5. Let f : I ⊆ R → R be a differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. Also f ′ ∈ L [g(a),g(b)] forg(a),g(b) ∈ I with g(a) < g(b). If | f ′|q is decreasing andgeometrically relative convex function forp,q > 1, 1

p +1q = 1. Then

∣

∣

∣

∣

∣

∣

∣

1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)− f

(

g(a)+g(b)2

)

∣

∣

∣

∣

∣

∣

∣

≤ (g(b)−g(a))| f ′(b)|(

1(p+1)2p+1

) 1p

×[(Ψ1(w))1q +(Ψ2(w))

1q ],

where w = | f ′(g(a))|| f ′(g(b))| , Ψ1 =

w12 q−1

q ln(w) and Ψ2(w) = wq−w12 q

q ln(w) .

Proof. Using Lemma 2, well known Holder’s inequalityand the fact that| f ′|q is geometrically relative convexfunction, we have∣

∣

∣

∣

∣

∣

∣

1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)− f

(

g(a)+g(b)2

)

∣

∣

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∣

(g(b)−g(a))

1∫

0

µ(t) f ′(tg(a)+(1− t)g(b))dt

∣

∣

∣

∣

∣

∣

≤ (g(b)−g(a))

12∫

0

|t|| f ′((g(a))t(g(b))1−t)|dt

+

1∫

12

|(t −1)|| f ′((g(a))t(g(b))1−t)|dt

≤ (g(b)−g(a))

12∫

0

t pdt

1p

×

12∫

0

| f ′((g(a))t(g(b))1−t)|qdt

1q

+

1∫

12

|t −1|pdt

1p

1∫

12

| f ′((g(a))t(g(b))1−t)|qdt

1q

≤ (g(b)−g(a))| f ′(b)|(

1(p+1)2p+1

) 1p

×

12∫

0

wqtdt

1q

+

1∫

12

wqtdt

1q

= (g(b)−g(a))| f ′(b)|(

1(p+1)2p+1

) 1p

×

(

w12 q −1

q ln(w)

) 1q

+

(

wq −w12 q

q ln(w)

) 1q

.

This completes the proof. �

Theorem 6. Let f : I ⊆ R → R be a differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. Also f ′ ∈ L [g(a),g(b)] forg(a),g(b) ∈ I with g(a) < g(b). If | f ′|q is decreasing andgeometrically relative convex function for q > 1. Then∣

∣

∣

∣

∣

∣

∣

1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)− f

(

g(a)+g(b)2

)

∣

∣

∣

∣

∣

∣

∣

≤ (g(b)−g(a))| f ′(b)|(

18

)1− 1q

×

(

−w12 q + 1

2w12 qq ln(w)+1

q2 ln(w)2

)1q

+

(

wq − 12w

12 qq ln(w)−w

12 q

q2 ln(w)2

)1q

,

where w = | f ′(g(a))|| f ′(g(b))| .

c© 2014 NSPNatural Sciences Publishing Cor.

612 M. A. Noor et al: Geometrically Relative Convex Functions

Proof. Using Lemma 2, well known Power meaninequality and the fact that| f ′|q is geometrically relativeconvex function, we have∣

∣

∣

∣

∣

∣

∣

1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)− f

(

g(a)+g(b)2

)

∣

∣

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∣

(g(b)−g(a))

1∫

0

µ(t) f ′(tg(a)+(1− t)g(b))dt

∣

∣

∣

∣

∣

∣

≤ (g(b)−g(a))

12∫

0

|t|| f ′((g(a))t(g(b))1−t)|dt

+

1∫

12

|(t −1)|| f ′((g(a))t(g(b))1−t)|dt

≤ (g(b)−g(a))

×

12∫

0

tdt

1− 1q

12∫

0

|t|| f ′((g(a))t(g(b))1−t)|qdt

1q

+

1∫

12

(1− t)dt

1− 1q

1∫

12

|t −1|| f ′((g(a))t(g(b))1−t)|qdt

1q

≤ (g(b)−g(a))| f ′(b)|(

18

)1− 1q

×

12∫

0

twqtdt

1q

+

1∫

12

(1− t)wqtdt

1q

= (g(b)−g(a))| f ′(b)|(

18

)1− 1q

×

(

−w12 q + 1

2w12 qq ln(w)+1

q2 ln(w)2

)1q

+

(

wq − 12w

12 qq ln(w)−w

12 q

q2 ln(w)2

)1q

.

This completes the proof. �

Remark. For q = 1 Theorem 6 reduces to Theorem 4.

Theorem 7. Let f : I ⊆ R → R be twice differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. Also f ′′ ∈ L [g(a),g(b)] forg(a),g(b) ∈ I with g(a) < g(b). If | f ′| is decreasing andgeometrically relative convex function. Then∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

| f ′′(g(b))|Ψ(k),

where k = | f ′′(g(a))|| f ′′(g(b))| and Ψ(k) = k ln(k)−2k+ln(k)+2

ln(k)3.

Proof. Using Lemma 3 and the fact that| f ′′| isgeometrically relative convex function, we have∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

1∫

0

t(1− t)| f ′′(tg(a)+(1− t)g(b))|dt

≤ g(b)−g(a)2

1∫

0

t(1− t)∣

∣

∣f ′′(

g(a)tg(b)1−t)∣

∣

∣dt

≤ g(b)−g(a)2

| f ′(g(b))|1∫

0

t(1− t)

( | f ′′(g(a))|| f ′′(g(b))|

)t

dt

=g(b)−g(a)

2| f ′(g(b))|

1∫

0

t(1− t)ktdt

=g(b)−g(a)

2| f ′′(g(b))| k ln(k)−2k+ ln(k)+2

ln(k)3.

This completes the proof. �

Theorem 8. Let f : I ⊆ R → R be twice differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. Also f ′′ ∈ L [g(a),g(b)] forg(a),g(b) ∈ I with g(a)< g(b). If | f ′′|q is decreasing andgeometrically relative convex function forp,q > 1, 1

p +1q = 1. Then

∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

| f ′(g(b))|(

2(−1−2p)√πΓ (p+1)

Γ(

p+ 32

)

) 1p

×(

kq −1q ln(k)

) 1q

,

where k = | f ′′(g(a))|| f ′′(g(b))| .

Proof. Using Lemma 3, well known Holder’s inequalityand the fact that| f ′′|q is geometrically relative convexfunction, we have∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

1∫

0

t(1− t)| f ′′(tg(a)+(1− t)g(b))|dt

≤ g(b)−g(a)2

1∫

0

(t(1− t))pdt

1p

c© 2014 NSPNatural Sciences Publishing Cor.

Appl. Math. Inf. Sci.8, No. 2, 607-616 (2014) /www.naturalspublishing.com/Journals.asp 613

×

1∫

0

∣

∣

∣f ′′(

g(a)tg(b)1−t)∣

∣

∣

qdt

1q

≤ g(b)−g(a)2

| f ′(g(b))|(

2(−1−2p)√πΓ (p+1)

Γ(

p+ 32

)

) 1p

×

1∫

0

( | f ′′(g(a))|| f ′′(g(b))|

)qt

dt

1q

=g(b)−g(a)

2| f ′(g(b))|

(

2(−1−2p)√πΓ (p+1)

Γ(

p+ 32

)

) 1p

×

1∫

0

kqtdt

1q

=g(b)−g(a)

2| f ′(g(b))|

(

2(−1−2p)√πΓ (p+1)

Γ(

p+ 32

)

) 1p

×(

kq −1q ln(k)

) 1q

.

This completes the proof. �

Theorem 9. Let f : G ⊆ R → R be a differentiablefunction on I◦ (the interior of I) and g : R → R bearbitrary function. Also f ′′ ∈ L [g(a),g(b)] forg(a),g(b) ∈ I with g(a)< g(b). If | f ′′|q is decreasing andgeometrically relative convex function for q > 1. Then∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

| f ′(g(b))|(

16

) 1p

Ψ(k),

where k = | f ′′(g(a))|| f ′′(g(b))| and

Ψ(k) =(

kqq ln(k)−2kq+q ln(k)+2q3 ln(k)3

) 1q.

Proof. Using Lemma 3, well known Power meaninequality and the fact that| f ′′|q is geometrically relativeconvex function, we have∣

∣

∣

∣

∣

∣

∣

f (g(a))+ f (g(b))2

− 1g(b)−g(a)

g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ g(b)−g(a)2

1∫

0

t(1− t)| f ′(tg(a)+(1− t)g(b))|dt

≤ g(b)−g(a)2

1∫

0

t(1− t)dt

1− 1q

×

1∫

0

t(1− t)∣

∣

∣f ′(

g(a)tg(b)1−t)∣

∣

∣

qdt

1q

≤ g(b)−g(a)2

| f ′(g(b))|(

16

) 1p

×

1∫

0

t(1− t)

( | f ′(g(a))|| f ′(g(b))|

)qt

dt

1q

=g(b)−g(a)

2| f ′(g(b))|

(

16

) 1p

1∫

0

t(1− t)kqtdt

1q

=g(b)−g(a)

2| f ′(g(b))|

(

16

) 1p

×(

kqq ln(k)−2kq +q ln(k)+2q3 ln(k)3

) 1q

.

This completes the proof. �

Next we prove the results for the class ofGA-relativeconvex functions.

Theorem 10. Let f : G → R be GA-relative convexfunction such that g(a),g(b) ∈ G with g(a) < g(b). Thenthe following inequality holds:

f√

g(a)g(b) ≤ 1lng(b)− lng(a)

g(b)∫

g(a)

f (g(x))g(x)

dg(x)

≤ f (g(a))+ f (g(b))2

. (8)

Proof. Since f is GA-relative convex function. Thus

f (√

g(x)g(y))≤ f (g(x))+ f (g(y))2

.

Let g(x) = (g(a))1−t(g(b))t andg(y) = (g(a))t(g(b))1−t . Thenthis implies

f (√

g(a)g(b))

=

t∫

o

f (√

g(a)g(b))dt

≤t∫

o

f ((g(a))1−t(g(b))t)+ f ((g(a))t(g(b))1−t)

2dt

=1

lng(b)− lng(a)

g(b)∫

g(a)

f (g(x))g(x)

dg(x)

=

1∫

0

f ((g(a))t(g(b))1−t)dt

≤ f (g(a))+ f (g(b))2

.

This completes the proof. �

Theorem 11.Let f : G → R be differentiable function on(g(a),g(b)) with g(a) < g(b) and f ′ ∈ L [g(a),g(b)]. If| f ′|q is GA-relative convex function for q ≥ 1, thenfollowing inequality holds:∣

∣

∣

∣

∣

∣

∣

[g(b) f (g(b))−g(a) f (g(a))]−g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

c© 2014 NSPNatural Sciences Publishing Cor.

614 M. A. Noor et al: Geometrically Relative Convex Functions

≤ (g(b))2(lng(b)− lng(a))

(

12

)1+ 1q[

h2−1ln(h)

]1− 1q

×[

2h2 ln(h)−h2+1ln(h)2

| f ′(a)|q + h2−2ln(h)−1ln(h)2

| f ′(b)|q]

1q

.

Proof. Since | f ′|q is GA-relative convex function. Thenfrom lemma 4 and well known power mean inequality, wehave∣

∣

∣

∣

∣

∣

∣

[g(b) f (g(b))−g(a) f (g(a))]−g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ (g(b))2(lng(b)− lng(a))

1∫

0

(

g(a)g(b)

)2t

×| f ′((g(a))t(g(b))1−t)|dt

≤ (g(b))2(lng(b)− lng(a))

1∫

0

(

g(a)g(b)

)2t

dt

1− 1q

×

1∫

0

(

g(a)g(b)

)2t

{t| f ′(a)|q +(1− t)| f ′(b)|q}dt

1q

= (g(b))2(lng(b)− lng(a))

1∫

0

h2tdt

1− 1q

×

1∫

0

h2t{t| f ′(a)|q +(1− t)| f ′(b)|q}dt

1q

= (g(b))2(lng(b)− lng(a))

[

12

h2−1ln(h)

]1− 1q

×[

14

2h2 ln(h)−h2+1ln(h)2

| f ′(a)|q + 14

h2−2ln(h)−1ln(h)2

| f ′(b)|q]

1q

= (g(b))2(lng(b)− lng(a))

(

12

)1+ 1q[

h2−1ln(h)

]1− 1q

×[

2h2 ln(h)−h2+1ln(h)2

| f ′(a)|q + h2−2ln(h)−1ln(h)2

| f ′(b)|q]

1q

.

This completes the proof. �

Corollary 1. Under the assumptions of Theorem 11, if q =1, we have∣

∣

∣

∣

∣

∣

∣

[g(b) f (g(b))−g(a) f (g(a))]−g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ (g(b))2(lng(b)− lng(a))

×14

[

2h2 ln(h)−h2+1ln(h)2

| f ′(a)|+ h2−2ln(h)−1ln(h)2

| f ′(b)|]

.

Theorem 12.Let f : G → R be differentiable function on(g(a),g(b)) with g(a) < g(b) and f ′ ∈ L [g(a),g(b)]. If| f ′|q is GA-relative convex function for q > 1, then

following inequality holds:∣

∣

∣

∣

∣

∣

∣

[g(b) f (g(b))−g(a) f (g(a))]−g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ (lng(b)− lng(a))[

L((g(a))2q

q−1 ,(g(b))2q

q−1 )]1− 1

q

×[

A(| f ′(a)|q, | f ′(b)|q)] 1

q .

Proof. Since | f ′|q is GA-relative convex function. Thenfrom lemma 4 and well known Holder’s inequality, wehave∣

∣

∣

∣

∣

∣

∣

[g(b) f (g(b))−g(a) f (g(a))]−g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ (g(b))2(lng(b)− lng(a))

1∫

0

(

g(a)g(b)

)2t

×| f ′((g(a))t(g(b))1−t)|dt

≤ (g(b))2(lng(b)− lng(a))

1∫

0

(

g(a)g(b)

)2tqq−1

dt

1− 1q

×

1∫

0

{t| f ′(a)|q +(1− t)| f ′(b)|q}dt

1q

= (lng(b)− lng(a))

(q−1)(

(g(b))2q

q−1 − (g(a))2q

q−1

)

2q(lng(b)− lng(a))

1− 1q

×[ | f ′(a)|q + | f ′(b)|q

2

]1q

= (lng(b)− lng(a))[

L((g(a))2q

q−1 ,(g(b))2q

q−1 )]1− 1

q

×[

A(| f ′(a)|q, | f ′(b)|q)] 1

q .

This completes the proof. �

Theorem 13.Let f : G → R be differentiable function on(g(a),g(b)) with g(a) < g(b) and f ′ ∈ L [g(a),g(b)]. If| f ′|q is GA-relative convex function for q ≥ 1, thenfollowing inequality holds:∣

∣

∣

∣

∣

∣

∣

[g(b) f (g(b))−g(a) f (g(a))]−g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ (lng(b)− lng(a))1−1q

(2q)1q

×{

[

(g(b))2q −L((g(a))2q,(g(b))2q)| f ′(g(a))|q

]

+[

L((g(a))2q,(g(b))2q)− (g(a))2q

]

| f ′(b)|q} 1

q

.

c© 2014 NSPNatural Sciences Publishing Cor.

Appl. Math. Inf. Sci.8, No. 2, 607-616 (2014) /www.naturalspublishing.com/Journals.asp 615

Proof. Since | f ′|q is GA-relative convex function. Thenfrom lemma 4 and well known Holder’s inequality, wehave∣

∣

∣

∣

∣

∣

∣

[g(b) f (g(b))−g(a) f (g(a))]−g(b)∫

g(a)

f (g(x))dg(x)

∣

∣

∣

∣

∣

∣

∣

≤ (g(b))2(lng(b)− lng(a))

1∫

0

(

g(a)g(b)

)2t

×| f ′((g(a))t(g(b))1−t)|dt

≤ (g(b))2(lng(b)− lng(a))

1∫

0

1dt

1− 1q

×

1∫

0

(

g(a)g(b)

)2qt

{t| f ′(a)|q +(1− t)| f ′(b)|q}dt

1q

= (lng(b)− lng(a))

×[

(g(b))2q(ln(g(b))2q − ln(g(a))2q)− (g(b))2q +(g(a))2q

(ln(g(b))2q − ln(g(a))2q)2

×| f ′(g(a))|q

+(g(b))2q − (g(a))2q(ln(g(b))2q − ln(g(a))2q)− (g(a))2q

(ln(g(b))2q − ln(g(a))2q)2

×| f ′(g(b))|q] 1

q

≤ (lng(b)− lng(a))1−1q

(2q)1q

×{

[

(g(b))2q −L((g(a))2q,(g(b))2q)| f ′(g(a))|q

]

+[

L((g(a))2q,(g(b))2q)− (g(a))2q

]

| f ′(b)|q} 1

q

.

This completes the proof. �

Acknowledgement

The authors are grateful to Dr. S. M. Junaid Zaidi, Rector,COMSATS Institute of Information Technology, Pakistanfor providing excellent research and academicenvironment.

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MuhammadAslam Noor earnedhis PhD degree from BrunelUniversity, London, UK(1975) in the field of AppliedMathematics(NumericalAnalysis and Optimization).He has vast experienceof teaching and researchat university levels in various

countries including Pakistan, Iran, Canada, Saudi Arabiaand United Arab Emirates. His field of interest andspecialization is versatile in nature. It covers many areasof Mathematical and Engineering sciences such asVariational Inequalities, Operations Research andNumerical Analysis. He has been awarded by thePresident of Pakistan: President’s Award for pride ofperformance on August 14, 2008, in recognition of hiscontributions in the field of Mathematical Sciences. Hewas awarded HEC Best Research paper award in 2009.He has supervised successfully several Ph.D andMS/M.Phil students. He is currently member of theEditorial Board of several reputed international journalsof Mathematics and Engineering sciences. He has morethan 750 research papers to his credit which werepublished in leading world class journals.

Khalida Inayat Nooris a leading world-knownfigure in mathematics andis presently employed as HECForeign Professor at CIIT,Islamabad. She obtained herPhD from Wales University(UK). She has a vastexperience of teaching andresearch at university levelsin various countries including

Iran, Pakistan, Saudi Arabia, Canada and United ArabEmirates. She was awarded HEC best research paperaward in 2009 and CIIT Medal for innovation in 2009.She has been awarded by the President of Pakistan:Presidents Award for pride of performance on August 14,2010 for her outstanding contributions in mathematicalsciences and other fields. Her field of interest andspecialization is Complex analysis, Geometric functiontheory, Functional and Convex analysis. She introduced anew technique, now called as Noor Integral Operatorwhich proved to be an innovation in the field of geometricfunction theory and has brought new dimensions in therealm of research in this area. She has been personallyinstrumental in establishing PhD/MS programs at CIIT.Dr. Khalida Inayat Noor has supervised successfullyseveral Ph.D and MS/M.Phil students. She has been aninvited speaker of number of conferences and haspublished more than 400 (Four hundred ) research articlesin reputed international journals of mathematical andengineering sciences. She is member of editorial boardsof several international journals of mathematical andengineering sciences.

Muhammad Uzair Awanhas earned his MS degreefrom COMSATS Instituteof Information Technology,Islamabad, Pakistan. His fieldof interest is Convex Analysisand Numerical Optimization.

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