On an implicit convexity concept and some integral inequalities

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R E S E A R C H

Open Access

On an implicit convexity concept and

some integral inequalities

Bessem Samet

*

*_{Correspondence:}

[email protected] Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

Abstract

We introduce a new concept of convexity that depends on a function F:R×R×R×(0, 1)→Rsatisfying certain axioms. The presented concept generalizes many kinds of convexity including

ε

-convex functions,

α

-convex functions, andh-convex functions. Moreover, some integral inequalities are provided via our notion of convexity.

MSC: 26A51; 26D15; 35A23

Keywords: F-convex; integral inequalities

1 Introduction

Convexity is an important concept in many branches of mathematics, pure and applied. In particular, many important integral inequalities are based on a convexity assumption of a certain function, such as Jensen’s inequality, the Hermite-Hadamard inequality, the Hardy-Littlewood-Pólya majoration inequality, Petrović’s inequality, Popoviciui’s convex function inequality, and many others. For more details as regards inequalities via con-vex functions, we refer the reader to the monograph []. However, for many encountered problems, convexity is not satisﬁed. This leads to the necessity to extend this concept.

In the last  years, great attention has been focused on the generalization of the no-tion of convexity. Let us cite some references in this direcno-tion. In [], Deﬁnetti introduced the class of quasi-convex functions. In [], Mangasarian introduced the notion of pseudo-convex functions. Polyak [] deﬁned the concept of strongly pseudo-convex functions. The class ofε-convex functions was introduced by Hyers and Ulam []. In [], Varosanec intro-duced the notion ofh-convexity that includes the class ofs-convex functions (see Hudzik []). For other work in this direction, we refer the reader to [–] and the references therein.

In this paper, we present a new concept of convexity that depends on a certain function satisfying some axioms. This new notion generalizes diﬀerent types of convexity, including ε-convex functions,α-convex functions,h-convex functions, and many others. Moreover,

some integral inequalities are established via this new notion of convexity. As particular cases, we retrieve several existing inequalities from the literature.

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2 An implicit convexity concept

We denote byFthe family of mappingsF:R×R×R×(, )→Rsatisfying the following axioms:

(A) Ifui∈L(, ),i= , , , then for everyλ∈(, ), we have





Fu(t),u(t),u(t),λ

dt=F





u(t)dt,





u(t)dt,





u(t)dt,λ

.

(A) For everyu∈L(, ),w∈L∞(, ), and(z,z)∈R, we have





Fw(t)u(t),w(t)z,w(t)z,t

dt=TF,w





w(t)u(t)dt,z,z

,

whereTF,w:R×R×R→Ris a function that depends on(F,w), and it is

nondecreasing with respect to the ﬁrst variable. (A) For any(w,u,u,u)∈R,u∈(, ), we have

wF(u,u,u,u) =F(wu,wu,wu,u) +Lw,

whereLw∈Ris a constant that depends only onw.

We introduce the new concept of convexity as follows.

Deﬁnition . Letf : [a,b]→R, (a,b)∈R,a<b, be a given function. We say thatf is a convex function with respect to someF∈F(orF-convex function) iﬀ

Fftx+ ( –t)y,f(x),f(y),t≤, (x,y,t)∈[a,b]×[a,b]×(, ). (.)

The following property follows immediately from (.).

Proposition . Let f : [a,b]→R, (a,b)∈R,a<b,be an F-convex function,for some F∈F.Then

Ff(x),f(x),f(x),t≤, (x,t)∈[a,b]×(, ).

Proof Takingx=yin (.), the desired inequality follows.

Now, we give some examples ofF-convex functions.

Example . Letε≥, and letf : [a,b]→R, (a,b)∈R,a<b, be anε-convex function, that is (see []),

ftx+ ( –t)y≤tf(x) + ( –t)f(y) +ε, (x,y,t)∈[a,b]×[a,b]×[, ].

Deﬁne the functionF:R×R×R×[, ]→Rby

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Letui∈L(, ),i= , , , and letλ∈[, ]. We have





dt=





u(t) –λu(t) – ( –λ)u(t) –ε

dt

=





u(t)dt–λ





u(t)dt– ( –λ)





u(t)dt–ε

=F





u(t)dt,





u(t)dt,





u(t)dt,λ

.

Therefore, the function F satisﬁes axiom (A). Now, let u∈L_{(, ),}_w_∈_L∞_{(, ), and}

(z,z)∈R. We have





dt

=





w(t)u(t) –tw(t)z– ( –t)w(t)z–ε

dt

=





w(t)u(t)dt–z





tw(t)dt–z





( –t)w(t)dt–ε

=TF,w





,

whereTF,w:R×R×R→Ris deﬁned by

TF,w(u,u,u) =u–





tw(t)dt

u–





( –t)w(t)dt

u–ε,

(u,u,u)∈R×R×R. (.)

Then the functionFsatisﬁes axiom (A). Now, let (w,u,u,u)∈Randu∈(, ). We

have

wF(u,u,u,u) =w

u–uu– ( –u)u–ε

=wu–u(wu) – ( –u)(wu) –wε

=wu–u(wu) – ( –u)(wu) –ε

+ ( –w)ε

=F(wu,wu,wu,u) + ( –w)ε.

Therefore, axiom (A) is satisﬁed with

Lw= ( –w)ε. (.)

Thus we proved thatF∈F. On the other hand, sincef isε-convex, for all (x,y,t)∈[a,b]× [a,b]×(, ), we have

Fftx+ ( –t)y,f(x),f(y),t=ftx+ ( –t)y–tf(x) – ( –t)f(y) –ε≤.

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Remark . Takingε=  in the above example, we observe that any convex function f : [a,b]→R, (a,b)∈R_,_a_<_b_{, is an}_F_{-convex function with respect to the function}_F_: R×R×R×(, )→Rdeﬁned by

F(u,u,u,u) =u–uu– ( –u)u, (u,u,u,u)∈R×R×R×(, ).

Example . Letf: [a,b]→R, (a,b)∈R_,_a_<_b_{, be an}_α_{-convex function,  <}_α_≤_{, that}

is,

ftx+ ( –t)y≤tαf(x) + –tαf(y), (x,y,t)∈[a,b]×[a,b]×[, ].

Deﬁne the functionF:R×R×R×[, ]→Rby

F(u,u,u,u) =u–uαu–

 –uα 

u, (u,u,u,u)∈R×R×R×[, ]. (.)





dt=





u(t) –λαu(t) –

 –λαu(t)

dt

=





u(t)dt–λα





u(t)dt–

 –λα





u(t)dt

=F





u(t)dt,





u(t)dt,





u(t)dt,λ

.

Therefore, the function F satisﬁes axiom (A). Now, let u∈L_{(, ),}_w_∈_L∞_{(, ), and}

(z,z)∈R. We have





dt

=





w(t)u(t) –tα_w₍_t₎_z –

 –tα_w₍_t₎_z 

dt

=





w(t)u(t)dt–z





tαw(t)dt–z





 –tαw(t)dt

=TF,w





,

TF,w(u,u,u) =u–





tαw(t)dt

u–





 –tαw(t)dt

u,

(u,u,u)∈R×R×R. (.)

have

wF(u,u,u,u) =w

u–uαu–

 –uα_u

=wu–uα(wu) –

 –uα_(wu)

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Lw= . (.)

Thus we proved thatF∈F. On the other hand, sincef isα-convex, for all (x,y,t)∈[a,b]× [a,b]×(, ), we have

Fftx+ ( –t)y,f(x),f(y),t=ftx+ ( –t)y–tαf(x) – –tαf(y)

≤.

As a consequence,f is anF-convex function.

Example . Leth:J→[,∞) be a given function which is not identical to , whereJis an interval inRsuch that (, )⊆J. Letf : [a,b]→[,∞), (a,b)∈R_,_a_<_b_{, be a}_h_-convex

function, that is (see []),

ftx+ ( –t)y≤h(t)f(x) +h( –t)f(y), (x,y,t)∈[a,b]×[a,b]×(, ).

We suppose thath∈L(, ). Deﬁne the functionF:R×R×R×(, )→Rby

F(u,u,u,u) =u–h(u)u–h( –u)u,

(u,u,u,u)∈R×R×R×(, ). (.)





dt=





u(t) –h(λ)u(t) –h( –λ)u(t)

dt

=





u(t)dt–h(λ)





u(t)dt–h( –λ)





u(t)dt

=F





u(t)dt,





u(t)dt,





u(t)dt,λ

.

Therefore, the function F satisﬁes axiom (A). Now, let u∈L(, ),w∈L∞(, ), and (z,z)∈R. We have





dt

=





w(t)u(t) –h(t)w(t)z–h( –t)w(t)z

dt

=





w(t)u(t)dt–z





h(t)w(t)dt–z





h( –t)w(t)dt

=TF,w





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TF,w(u,u,u) =u–





h(t)w(t)dt

u–





h( –t)w(t)dt

u,

(u,u,u)∈R×R×R. (.)

have

wF(u,u,u,u) =w

u–h(u)u–h( –u)u

=wu–h(u)(wu) –h( –u)(wu)

=F(wu,wu,wu,u).

Lw= . (.)

Thus we proved thatF∈F. On the other hand, sincefish-convex, for all (x,y,t)∈[a,b]× [a,b]×(, ), we have

Fftx+ ( –t)y,f(x),f(y),t=ftx+ ( –t)y–h(t)f(x) –h( –t)f(y)

≤.

As a consequence,f is anF-convex function.

3 Integral inequalities involvingF-convex functions

Some integral inequalities viaF-convex functions are presented in this section.

Theorem . Let f: [a,b]→R, (a,b)∈R_,_a_<_b_,_{be an F-convex function}_,_{for some F}_∈_F_.

Suppose that f ∈L₍_a_,_b_)._Then

F

f

a+b 

,  b–a

b

a

f(x)dx,  b–a

b

a

f(x)dx, 

≤, (.)

TF,

 b–a

b

a

f(x)dx,f(a),f(b)

≤. (.)

Proof Sincef is anF-convex function, for everyu,v∈[a,b], we have

F

f

 u+

 – 

v

,f(u),f(v), 

≤.

Taking

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we obtain

F

f

a+b 

,fta+ ( –t)b,ftb+ ( –t)a, 

≤.

Using axiom (A), we get

F





f

a+b 

dt,





fta+ ( –t)bdt,





ftb+ ( –t)adt, 

≤.

On the other hand, we have





fta+ ( –t)bdt=





ftb+ ( –t)adt=  b–a

b

a

f(x)dx.

Therefore,

F





f

a+b 

dt,  b–a

b

a

f(x)dx,  b–a

b

a

f(x)dx, 

≤,

which proves (.).

Again, sincef is anF-convex function, for everyt∈(, ), we have

Ffta+ ( –t)b,f(a),f(b),t≤.

Using axiom (A) withw≡, and integrating over (, ) with respect to the variablet, we obtain

TF,





fta+ ( –t)bdt,f(a),f(b)

≤,

that is,

TF,

 b–a

b

a

f(x)dx,f(a),f(b)

≤,

which proves (.).

Remark . Note that in the proof of Theorem ., we used only the axioms (A) and (A). So, Theorem . holds true for any functionFsatisfying (A) and (A).

The following lemma will be useful later (see []).

Lemma . Let f :I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b. Then

f(a) +f(b)

 –

 b–a

b

a

f(x)dx=b–a 





( – t)fta+ ( –t)bdt.

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Theorem . Let f:I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b. Suppose that

(i) |f|isF-convex on[a,b],for someF∈F.

(ii) The functiont∈(, )→Lw(t)belongs toL(, ),wherew(t) =| – t|.

Then

TF,w

 b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx,f(a),f(b)

+





Lw(t)dt≤. (.)

Proof Since|f|isF-convex, we have

Ffta+ ( –t)b,f(a),f(b),t≤, t∈(, ).

Multiplying this inequality byw(t) and using axiom (A), we get

Fw(t)fta+ ( –t)b,w(t)f(a),w(t)f(b),t+Lw(t)≤, t∈(, ).

Integration over (, ) with respect to the variabletand using axiom (A), we obtain

TF,w





w(t)fta+ ( –t)bdt,f(a),f(b)

+





Lw(t)dt≤.

On the other hand, from Lemma ., we have

 b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx≤





w(t)fta+ ( –t)bdt.

SinceTF,wis nondecreasing with respect to the ﬁrst variable, we deduce that

TF,w

 b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx,f(a),f(b)

+





Lw(t)dt≤,

which proves (.).

Another similar result is given by the following theorem.

Theorem . Let f :I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a< b,and let p> .Suppose that|f|p/(p–) is F-convex on[a,b],for some F ∈F,and|f| ∈ Lp/(p–)(a,b).Then

TF,

A(p,f),f(a) p p–_,_f₍_b₎

p

p–≤_, _(.)

where

A(p,f) =

 b–a

p p–

(p+ )p–f(a) +f(b)

 –

 b–a

b

a

f(x)dx p p–

.

Proof Since|f|pp– _is_F_{-convex, we have}

Ffta+ ( –t)b p p–_,_f₍_a₎

p p–_,_f₍_b₎

p

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Using axiom (A) withw≡, and integrating over (, ) with respect to the variablet, we get

TF,





fta+ ( –t)b p

p–_dt_,_f₍_a₎

p p–_,_f₍_b₎

p p–

≤.

On the other hand, using Lemma . and Hölder’s inequality, we obtain

 b–a

p p–

(p+ )p–f(a) +f(b)

 –

 b–a

b

a

f(x)dx p p–

≤





fta+ ( –t)b p p–_dt_,

that is,

A(p,f)≤





fta+ ( –t)b p p–_dt_.

SinceTF,is nondecreasing with respect to the ﬁrst variable, we obtain

TF,

A(p,f),f(a) p p–_,_f₍_b₎

p p–≤_,

which proves (.).

4 Particular cases

As consequences of the presented theorems, we obtain in this section some integral in-equalities for diﬀerent (and independent) kinds of convexity.

4.1 The case of

ε

-convexity

We have the following Hermite-Hadamard inequalities forε-convex functions.

Corollary . Let f: [a,b]→R, (a,b)∈R_,_a_<_b_,_{be an}_{ε-convex function}_,_ε_≥_._Suppose

that f ∈L(a,b).Then

f

a+b 

–ε≤  b–a

b

a

f(x)dx≤f(a) +f(b)  +ε.

Proof From Example ., we know that anε-convex function is anF-convex. Using (.) and (.) withw≡, we have

F(u,u,u,u) =u–uu– ( –u)u–ε, (u,u,u,u)∈R×R×R×[, ],

and

TF,(u,u,u) =u–

u+u

 –ε, (u,u,u)∈R×R×R.

So, applying Theorem ., we obtain the desired result.

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Corollary . Let f : [a,b]→R, (a,b)∈R_,_a_<_b_,_{be a convex function}_._{Suppose that}

f ∈L₍_a_,_b_)._Then

f

a+b 

≤  b–a

b

a

f(x)dx≤f(a) +f(b)

 .

Corollary . Let f :I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b. Suppose that the function|f|isε-convex on[a,b],ε≥.Then

f(a) +_f(b)–  b–a

b

a

f(x)dx≤(b–a)

_|

f(a)|+|f(b)|

 +

ε  .

Proof Using (.) withw(t) =| – t|, we obtain





Lw(t)dt=ε





 –w(t)dt=ε

 –





| – t|dt

=ε .

Using (.) withw(t) =| – t|, we obtain

TF,w(u,u,u) =u–





t| – t|dt

u–





( –t)| – t|dt

u–ε,

for all (u,u,u)∈R×R×R. On the other hand, simple computations yield





t| – t|dt=





( –t)| – t|dt= .

Therefore, we have

TF,w(u,u,u) =u–

u+u

 –ε, (u,u,u)∈R×R×R.

Then

TF,w

 b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx,f(a),f(b)

+





Lw(t)dt

=  b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx–|f

₍_a_)|₊_|_f₍_b_)|

 –

ε .

Now, by Theorem ., we have

TF,w

 b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx,f(a),f(b)

+





Lw(t)dt≤,

that is,

f(a) +_f(b)–  b–a

b

a

f(x)dx≤(b–a)

|f(a)|+|f(b)|

 +

ε  ,

which proves the desired inequality.

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Corollary . Let f :I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b. Suppose that the function|f|is convex on[a,b].Then

f(a) +_f(b)–  b–a

b

a

f(x)dx≤(b–a)(|f

₍_a₎_|₊_|_f₍_b₎_|₎

 .

Corollary . Let f :I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b, and let p> .Suppose that|f|p/(p–)_is_{ε-convex on}_[_a_,_b_],_ε_≥_,_and_|_f_{| ∈}_Lp/(p–)₍_a_,_b_).

Then

f(a) +_f(b)–  b–a

b

a

f(x)dx≤ (b–a) (p+ )p

_|

f(a)| p

p–₊|_f₍_b)|

p p–

 +ε

p–

p .

Proof Using (.) withw≡, we obtain

TF,(u,u,u) =u–

u+u

 –ε, (u,u,u)∈R×R×R.

Then

TF,

A(p,f),f(a) p p–_,_f₍_b₎

p p–

=A(p,f) –|f

₍_a₎_|pp–₊|_f₍_b₎|

p p–

 –ε,

where

A(p,f) =

 b–a

p p–

(p+ )p–f(a) +f(b)

 –

 b–a

b

a

f(x)dx p p–

.

By Theorem ., we have

TF,

A(p,f),f(a) p p–_,_f₍_b₎

p p–≤_,

that is,

f(a) +_f(b)–  b–a

b

a

f(x)dx≤ (b–a) (p+ )p

|f(a)|p–p ₊|_f₍_b₎|

p p–

 +ε

p–

p ,

which proves the desired inequality.

Takingε=  in Corollary ., we obtain the following result (see []).

Corollary . Let f :I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b, and let p> .Suppose that|f|p/(p–)is convex on[a,b],and|f| ∈Lp/(p–)(a,b).Then

f(a) +_f(b)–  b–a

b

a

f(x)dx≤ (b–a) (p+ )p

_|

f(a)| p

p–₊|_f₍_b)|

p p– 

p–

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4.2 The case of

α

-convexity

We have the following Hermite-Hadamard inequalities forα-convex functions.

Corollary . Let f : [a,b]→R, (a,b)∈R_, _a_<_b_, _{be an}_{α-convex function}_,_α_∈_{(, ].}

Suppose that f ∈L₍_a_,_b_)._Then

f

a+b 

≤  b–a

b

a

f(x)dx≤f(a) +αf(b) α+  .

Proof From Example ., we know that anα-convex function isF-convex. Using (.) and (.) withw≡, we have

F(u,u,u,u) =u–uαu–

 –uα_u, (u,u,u,u)∈R×R×R×[, ],

and

TF,(u,u,u) =u–

u

α+ – α

α+ u, (u,u,u)∈R×R×R.

Remark . Takingα=  in Corollary ., we obtain the standard Hermite-Hadamard inequalities for convex functions (see Corollary .).

Corollary . Let f :I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b. Suppose that the function|f|isα-convex on[a,b],α∈(, ].Then

f(a) +_f(b)–  b–a

b

a

f(x)dx

≤ (b–a) (α+ )(α+ )

–α₊_α_f₍_a₎₊[α(α+ ) + ( – – α_)]

 f

₍_b₎_.

Proof Using (.), we have





Lw(t)dt= .

Using (.) withw(t) =| – t|, we obtain

TF,w(u,u,u) =u–





tα| – t|dt

u–





 –tα| – t|dt

u,

for all (u,u,u)∈R×R×R. Simple computations yield





tα_|_{ – }_t_|_dt₌ 

(α+ )(α+ )

–α₊_α

and





 –tα| – t|dt=α(α+ ) + ( – 

–α₎

(13)

Therefore,

TF,w(u,u,u) =u–

(–α₊_α₎_u 

(α+ )(α+ )–

[α(α+ ) + ( – –α_)]_u 

(α+ )(α+ ) ,

for all (u,u,u)∈R×R×R. Now, we have

TF,w

 b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx,f(a),f(b)

+





Lw(t)dt

=TF,w

 b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx,f(a),f(b)

=  b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx

–(

–α₊_α_)|_f₍_a_)|

(α+ )(α+ ) –

[α(α+ ) + ( – –α_)]|_f₍_b_)|

(α+ )(α+ ) .

So, applying Theorem ., we obtain the desired result.

Remark . Takingα=  in Corollary ., we obtain the result given by Corollary ..

Corollary . Let f:I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b, and let p> .Suppose that|f|p/(p–)_is_{α-convex on}_[_a_,_b_],_α_∈_{(, ],}_and_|_f_{| ∈}_Lp/(p–)₍_a_,_b_).

Then

f(a) +f(b)

 –

 b–a

b

a

f(x)dx≤ (b–a) (p+ )p

_|

f(a)|pp–₊_α|_f₍_b₎|

p p–

α+ 

p–

p .

Proof Using (.) withw≡, we obtain

TF,(u,u,u) =u–

u

α+ – α

α+ u, (u,u,u)∈R×R×R.

Then

TF,

A(p,f),f(a) p p–_,_f₍_b₎

p p–

=A(p,f) –|f

₍_a_)|pp–₊_α|_f₍_b)|

p p–

α+  ,

where

A(p,f) =

 b–a

p p–

(p+ )p–f(a) +f(b)

 –

 b–a

b

a

f(x)dx p p–

.

TF,

A(p,f),f(a) p p–_,_f₍_b₎

(14)

that is,

f(a) +_f(b)–  b–a

b

a

f(x)dx≤ (b–a) (p+ )p

|f(a)|p–p ₊_α|_f₍_b₎|

p p–

α+ 

p–

p ,

which is the desired inequality.

Remark . Takingα=  in Corollary ., we obtain the result given by Corollary ..

4.3 The case ofh-convex functions

Leth:J→[,∞) be a given function which is not identical to , whereJis an interval in

Rsuch that (, )⊆J. We suppose thath∈L_{(, ) and}_h₍ )= .

We have the following Hermite-Hadamard inequalities forh-convex functions (obtained by Sarikayaet al.[]).

Corollary . Let f : [a,b]→[,∞), (a,b)∈R_,_a_<_b_,_{be a h-convex function}_._Suppose

that f ∈L₍_a_,_b_)._Then

 h(

)

f

a+b 

≤  b–a

b

a

f(x)dx≤





h(t)dtf(a) +f(b).

Proof From Example ., we know that ah-convex function isF-convex. Using (.) and (.) withw≡, we have

F(u,u,u,u) =u–h(u)u–h( –u)u, (u,u,u,u)∈R×R×R×(, ).

and

TF,(u,u,u) =u–





h(t)dt

(u+u), (u,u,u)∈R×R×R.

Corollary . Let f:I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b. Suppose that the function|f|is h-convex on[a,b].Then

f(a) +_f(b)–  b–a

b

a

f(x)dx≤(b–a)





h(t)| – t|dt

_|

f(a)|+|f(b)| 

.

Proof Using (.), we have





Lw(t)dt= .

Using (.) withw(t) =| – t|, we obtain

TF,w(u,u,u) =u–





h(t)| – t|dt

(15)

Now, we have

TF,w

 b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx,f(a),f(b)

+





Lw(t)dt

=TF,w

 b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx,f(a),f(b)

=  b–a

f(a) +_f(b)–  b–a

b

a

f(x)dx

–





h(t)| – t|dtf(a)+f(b).

So, applying Theorem ., we obtain the desired result.

Corollary . Let f:I◦⊆R→Rbe a diﬀerentiable mapping on I◦, (a,b)∈I◦×I◦,a<b, and let p> .Suppose that|f|p/(p–)_{is h-convex on}_[_a_,_b_],_and_|_f_{| ∈}_Lp/(p–)₍_a_,_b_)._Then

f(a) +_f(b)–  b–a

b

a

f(x)dx

≤ (b–a) (p+ )p





h(t)dt

p–

p f(a)

p

p–₊_f₍_b₎

p p–

p–

p _.

Proof Using (.) withw≡, we obtain

TF,(u,u,u) =u–





h(t)dt

(u+u), (u,u,u)∈R×R×R.

Then

TF,

A(p,f),f(a) p p–_,_f₍_b₎

p

p–₌_A₍_p_,_f_{) –}





h(t)dtf(a) p

p– ₊_f₍_b₎

p p–_,

where

A(p,f) =

 b–a

p p–

(p+ )p–f(a) +f(b)

 –

 b–a

b

a

f(x)dx p p–

.

TF,

A(p,f),f(a) p p–_,_f₍_b₎

p p–≤_,

that is,

f(a) +_f(b)–  b–a

b

a

f(x)dx

≤ (b–a) (p+ )p





h(t)dt

p–

p f(a)

p

p–₊_f₍_b₎

p p–

p–

p _,

(16)

Competing interests

The author declares to have no competing interests.

Acknowledgements

The author extends his appreciation to the Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).

Received: 4 September 2016 Accepted: 18 November 2016

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