On new general integral inequalities for s-convex functions

Applied Mathematics and Computation 246 (2014) 306–315

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

On new general integral inequalities for s-convex functions

http://dx.doi.org/10.1016/j.amc.2014.08.0410096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected], [email protected] (_I. _Is�can), [email protected] (E. Set), [email protected] (M.E. Özdemir)

_Imdat _Is�can a, Erhan Set b,⇑, M. Emin Özdemir c

a Department of Mathematics, Faculty of Arts and Sciences, Giresun University, 28100 Giresun, Turkeyb Department of Mathematics, Faculty of Arts and Sciences, Ordu University, 52200 Ordu, Turkeyc Atatürk University, K.K. Education Faculty, Department of Mathematics, 25240 Campus, Erzurum, Turkey

a r t i c l e i n f o

Keywords:Convex functions-Convex functionSimpson’s inequalityHermite–Hadamard’s inequality

a b s t r a c t

In this paper, the authors establish some new estimates for the remainder term of the mid-point, trapezoid, and Simpson formula using functions whose derivatives in absolute valueat certain power are s-convex. Some applications to special means of real numbers are pro-vided as well.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Let f : I # R! R be a convex function defined on the interval I of real numbers and a; b 2 I with a < b. The followinginequality

faþ b

2

� �6

1b� a

Z b

af ðxÞdx 6

f ðaÞ þ f ðbÞ2

ð1:1Þ

holds. This double inequality is known in the literature as Hermite–Hadamard integral inequality for convex functions. See[2,4,6–12,15] for the results of the generalization, improvement and extension of the famous integral inequality (1.1).

In 1978, Breckner introduced s-convex functions as a generalization of convex functions as follows [3]:

Definition 1. Let s 2 ð0;1� be a fixed real number. A function f : ½0;1Þ ! ½0;1Þ is said to be s-convex (in the second sense),orthat f belongs to the class K2

s , if

f axþ ð1� aÞyð Þ 6 asf ðxÞ þ ð1� aÞsf ðyÞ

for all x; y 2 ½0;1Þ and a 2 ½0;1�.Of course, s-convexity means just convexity when s ¼ 1. For other recent results concerning s-convex functions see [1–

19].The following inequality is well known in the literature as Simpson’s inequality:Let f : a; b½ � ! R be a four times continuously differentiable mapping on a; bð Þ and f ð4Þ

�� 1¼ supx2 a;bð Þ f ð4ÞðxÞ

�� <1. Thenthe following inequality holds:

13

f ðaÞ þ f ðbÞ2

þ 2faþ b

2

� �� 1

b� a

Z b

af ðxÞdx

�� 6 1

2880f ð4Þ��

1b� að Þ4:

.

http://crossmark.crossref.org/dialog/?doi=10.1016/j.amc.2014.08.041&domain=pdf

http://dx.doi.org/10.1016/j.amc.2014.08.041

mailto:[email protected]




http://dx.doi.org/10.1016/j.amc.2014.08.041

http://www.sciencedirect.com/science/journal/00963003

http://www.elsevier.com/locate/amc

_I. _Is�can et al. / Applied Mathematics and Computation 246 (2014) 306–315 307

In recent years many authors have studied error estimations for Simpson’s inequality. For refinements, counterparts, gen-eralizations of the Simpson’s inequality and new Simpson’s type inequalities, see [1,6–8,13,14,19].

In [4], Dragomir and Fitzpatrick proved a variant of Hermite–Hadamard inequality which holds for the s-convexfunctions.

Theorem 1. Suppose that f : 0;1½ Þ ! 0;1½ Þ is an s-convex function in the second sense, where s 2 ð0;1� and leta; b 2 0;1½ Þ; a < b. If f 2 L a; b½ �, then the following inequalities hold

2s�1faþ b

2

� �6

1b� a

Z b

af ðxÞdx 6

f ðaÞ þ f ðbÞsþ 1

ð1:2Þ

the constant k ¼ 1sþ1 is the best possible in the second inequality in (1.2). The above inequalities are sharp.

In [6], Iscan obtained a new generalization of some integral inequalities for differentiable convex mapping which are con-nected Simpson and Hadamard type inequalities, and he used the following lemma to prove this.

Lemma 1. Let f : I # R! R be a differentiable mapping on I� such that f 0 2 L½a; b�, where a; b 2 I with a < b and a; k 2 0;1½ �. Thenthe following equality holds:

k af ðaÞ þ 1� að Þf ðbÞð Þ þ 1� kð Þf ðaaþ 1� að ÞbÞ � 1b� a

Z b

af ðxÞdx

¼ b� að ÞZ 1�a

0t � akð Þf 0 tbþ ð1� tÞað Þdt þ

Z 1

1�at � 1þ k 1� að Þð Þf 0 tbþ ð1� tÞað Þdt

� �:

The main inequality in [6], pointed out, is as follows.

Theorem 2. Let f : I # R! R be a differentiable mapping on I� such that f 0 2 L½a; b�, where a; b 2 I� with a < b and a; k 2 0;1½ �. Iff 0�� q is convex on ½a; b�; q P 1, then the following inequality holds:

k af ðaÞþ 1�að Þf ðbÞð Þþ 1�kð Þf ðaaþ 1�að ÞbÞ� 1b�a

Z b

af ðxÞdx

��

6

b�að Þ c1�1

q2 l1 f 0ðbÞ

�� qþl2 f 0ðaÞ�� q� 1

qþt1�1q

2 g3 f 0ðbÞ�� qþg4 f 0ðaÞ

�� q� 1q

�; ak61�a61�k 1�að Þ

b�að Þ c1�1

q2 l1 f 0ðbÞ

�� qþl2 f 0ðaÞ�� q� 1

qþt1�1q


�� q� 1q

�; ak61�k 1�að Þ61�a

b�að Þ c1�1

q1 l3 f 0ðbÞ

�� qþl4 f 0ðaÞ�� q� 1

qþt1�1q


�� q� 1q

�; 1�a6ak61�k 1�að Þ

8>>>>>>><>>>>>>>:

ð1:3Þ

where

c1 ¼ 1� að Þ ak� 1� að Þ2

� �; c2 ¼ akð Þ2 � c1;

t1 ¼1� 1� að Þ2

2� a 1� k 1� að Þ½ �;

t2 ¼1þ 1� að Þ2

2� kþ 1ð Þ 1� að Þ 1� k 1� að Þ½ �;

l1 ¼akð Þ3 þ 1� að Þ3

3� ak

1� að Þ2

2;

l2 ¼1þ a3 þ 1� akð Þ3

3� 1� akð Þ

21þ a2�

;

l3 ¼ ak1� að Þ2

2� 1� að Þ3

3;

l4 ¼ak� 1ð Þ 1� a2

� 2

þ 1� a3

3;

g1 ¼1� 1� að Þ3

3� 1� k 1� að Þ½ �

2a 2� að Þ;

g2 ¼k 1� að Þa2

2� a3

3;

308 _I. _Is�can et al. / Applied Mathematics and Computation 246 (2014) 306–315

g3 ¼1� k 1� að Þ½ �3

3� 1� k 1� að Þ½ �

21þ 1� að Þ2�

þ 1þ 1� að Þ3

3;

g4 ¼k 1� að Þ½ �3

3� k 1� að Þa2

2þ a3

3:

In [2] Alomari et al. obtained the following inequalities of the left-hand side of Hermite–Hadamard’s inequality for s-con-vex mappings.

Theorem 3. Let f : I # ½0;1Þ ! R be a differentiable mapping on I�, such that f 0 2 L½a; b�, where a; b 2 I with a < b. If jf 0jq; q P 1,is s-convex on ½a; b�, for some fixed s 2 ð0;1�, then the following inequality holds:

faþ b

2

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

82

ðsþ 1Þðsþ 2Þ

� �1q

21�s þ 1�

f 0ðbÞ�� q þ 21�s f 0ðaÞ

�� qn o1q

�

þ 21�s þ 1�

f 0ðaÞ�� q þ 21�s f 0ðbÞ

�� qn o1q�: ð1:4Þ

Theorem 4. Let f : I # ½0;1Þ ! R be a differentiable mapping on I�, such that f 0 2 L½a; b�, where a; b 2 I with a < b. If jf 0jp

p�1; p > 1,is s -convex on ½a; b�, for some fixed s 2 ð0;1�, then the following inequality holds:

faþ b

2

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

4

� �1

pþ 1

� �1p 1

sþ 1

� �2q

� 21�s þ sþ 1�

f 0 að Þ�� q þ 21�s f 0 bð Þ

�� q� 1q þ 21�s þ sþ 1

� f 0 bð Þ�� q þ 21�s f 0 að Þ

�� q� 1q

� �;

ð1:5Þ

where p is the conjugate of q; q ¼ p=ðp� 1Þ.In [14], Sarikaya et al. obtained a new upper bound for the right-hand side of Simpson’s inequality for s-convex mapping

as follows:

Theorem 5. Let f : I # ½0;1Þ ! R be a differentiable mapping on I�, such that f 0 2 L½a; b�, where a; b 2 I� with a < b. If jf 0jq; iss-convex on ½a; b�, for some fixed s 2 ð0;1� and q > 1, then the following inequality holds:

16

f ðaÞ þ 4faþ b

2

� �þ f ðbÞ

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

121þ 2pþ1

3 pþ 1ð Þ

!1p

�f 0 aþb

2

� �� q þ f 0 að Þ�� q

sþ 1

!1q

þf 0 aþb

2

� �� q þ f 0 bð Þ�� q

sþ 1

!1q

8<:

9=;; ð1:6Þ

where 1p þ 1

q ¼ 1.In [10], Kirmaci et al. proved the following trapezoid inequality:

Theorem 6. Let f : I # ½0;1Þ ! R be a differentiable mapping on I�, such that f 0 2 L½a; b�, where a; b 2 I�; a < b. If jf 0jq;is s-convexon ½a; b�, for some fixed s 2 ð0;1Þ and q > 1, then

f að Þþ f bð Þ2

� 1b�a

Z b

af ðxÞdx

��6 b�a

2q�1

2 2q�1ð Þ

� �q�1q 1

sþ1

� �1q

� f 0aþb

2

� ��q

þ f 0 að Þ�� q� �1

q

þ f 0aþb

2

� ��q

þ f 0 bð Þ�� q� �1

q( )

:

ð1:7Þ

2. Main results

Let f : I # R! R be a differentiable function on I�, the interior of I, throughout this section we will take

If k;a; a; bð Þ ¼ k af ðaÞ þ 1� að Þf ðbÞð Þ þ 1� kð Þf ðaaþ 1� að ÞbÞ � 1b� a

Z b

af ðxÞdx

where a; b 2 I� with a < b and a; k 2 0;1½ �.

Theorem 7. Let f : I # R! R be a differentiable mapping on I� such that f 0 2 L½a; b�, where a; b 2 I� with a < b and a; k 2 0;1½ �. Iff 0�� q is s-convex on ½a; b�, for some fixed s 2 ð0;1� and q P 1, then


(i) for ak 6 1� a 6 1� k 1� að Þ we have

If k;a; a; bð Þ�� 6 b� að Þ c

1�1q

2 ða; kÞ c1ða; k; sÞ f 0ðbÞ�� q þ c2ða; k; sÞ f 0ðaÞ

�� q� 1q

�

þc1�1

q2 ð1� a; kÞ c2ð1� a; k; sÞ f 0ðbÞ

�� q þ c1ð1� a; k; sÞ f 0ðaÞ�� q� 1

q�;

(ii) for ak 6 1� k 1� að Þ 6 1� a we have


1�1q


�� q� 1q

�

þc1�1


�� q þ c3ð1� a; k; sÞ f 0ðaÞ�� q� 1

q�;

(iii) for 1� a 6 ak 6 1� k 1� að Þ we have


1�1q


�� q� 1q

�

þc1�1


�� q þ c1ð1� a; k; sÞ f 0ðaÞ�� q� 1

q�

where

c1ða; kÞ ¼ 1� að Þ ak� 1� að Þ2

� �;

c2ða; kÞ ¼ akð Þ2 � c1ða; kÞ;

c1ða; k; sÞ ¼ akð Þsþ2 2sþ 1ð Þ sþ 2ð Þ � akð Þ 1� að Þsþ1

sþ 1þ 1� að Þsþ2

sþ 2;

c2ða; k; sÞ ¼ 1� akð Þsþ2 2sþ 1ð Þ sþ 2ð Þ �

1� akð Þ 1þ asþ1�

sþ 1þ 1þ asþ2

sþ 2;

c3ða; k; sÞ ¼ akð Þ 1� að Þsþ1

sþ 1� 1� að Þsþ2

sþ 2;

c4ða; k; sÞ ¼ak� 1ð Þ 1� asþ1

� sþ 1

þ 1� asþ2

sþ 2:

Proof. Suppose that q P 1. From Lemma 1 and using the well known power mean inequality, we have

If k;a; a; bð Þ�� 6 b� að Þ

Z 1�a

0t � akj j f 0 tbþ ð1� tÞað Þ

�� dt þZ 1

1�at � 1þ k 1� að Þj j f 0 tbþ ð1� tÞað Þ

�� dt� �

6 b� að ÞZ 1�a

0t � akj jdt

� �1�1q Z 1�a


�� qdt� �1

q

8<:

þZ 1

1�at � 1þ k 1� að Þj jdt

� �1�1q Z 1


�� qdt� �1

q

9=; ð2:1Þ

Consider

I1 ¼Z 1�a


�� qdt; I2 ¼Z 1


�� qdt

Since f 0�� q is s-convex on ½a; b�,

I1 6 f 0ðbÞ�� q Z 1�a

0t � akj jtsdt þ f 0ðaÞ

�� q Z 1�a

0t � akj jð1� tÞsdt: ð2:2Þ

Similarly

I2 6 f 0ðbÞ�� q Z 1

1�at � 1þ k 1� að Þj jtsdt þ f 0ðaÞ

�� q Z 1

1�at � 1þ k 1� að Þj jð1� tÞsdt: ð2:3Þ

Additionally, by simple computation
Z 1�a
0t � akj jdt ¼

c2ða; kÞ; ak 6 1� ac1ða; kÞ; ak P 1� a

; ð2:4Þ


c1ða; kÞ ¼ 1� að Þ ak� 1� að Þ2

� �; c2ða; kÞ ¼ akð Þ2 � c1ða; kÞ;

Z 1

1�at � 1þ k 1� að Þj jdt ¼

Z a

0t � 1� að Þkj jdt ¼

c1ð1� a; kÞ; 1� k 1� að Þ 6 1� ac2ð1� a; kÞ; 1� k 1� að ÞP 1� a

;

Z 1�a

0t � akj jtsdt ¼

c1ða; k; sÞ; ak 6 1� ac3ða; k; sÞ; ak P 1� a

Z 1�a

0t � akj jð1� tÞsdt ¼

c2ða; k; sÞ; ak 6 1� ac4ða; k; sÞ; ak P 1� a

Z 1

1�at � 1þ k 1� að Þj jts ¼

Z a

0t � 1� að Þkj jð1� tÞsdt ¼

c4ð1� a; k; sÞ; 1� k 1� að Þ 6 1� ac2ð1� a; k; sÞ; 1� k 1� að ÞP 1� a

Z 1

1�at � 1þ k 1� að Þj jð1� tÞsdt ¼

Z a

0t � 1� að Þkj jtsdt ¼

c3ð1� a; k; sÞ; 1� k 1� að Þ 6 1� ac1ð1� a; k; sÞ; 1� k 1� að ÞP 1� a

: ð2:5Þ

Thus, using (2.2)–(2.5) in (2.1), we obtain desired results. This completes the proof. h

Corollary 1. Under the assumptions of Theorem 7 with q ¼ 1,

(i) if ak 6 1� a 6 1� k 1� að Þ, then we have

If k;a; a; bð Þ�� 6 b� að Þ c1ða; k; sÞ þ c2ð1� a; k; sÞð Þ f 0ðbÞ

�� þ c2ða; k; sÞ þ c1ð1� a; k; sÞð Þ f 0ðaÞ��

;

(ii) if ak 6 1� k 1� að Þ 6 1� a, then we have



;

(iii) if 1� a 6 ak 6 1� k 1� að Þ, then we have



Remark 1. In Theorem 7, if we take s ¼ 1, then we obtain the inequality (1.3).

Remark 2. In Theorem 7, if we take a ¼ 12 and k ¼ 1

3, then we have the following Simpson type inequality

16

f ðaÞ þ 4faþ b

2

� �þ f ðbÞ

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

25

36

� �1�1q

� ð2sþ 1Þ3sþ1 þ 2

3� 6sþ1ðsþ 1Þðsþ 2Þf 0ðbÞ�� q þ 2� 5sþ2 þ ðs� 4Þ6sþ1 � ð2sþ 7Þ3sþ1

3� 6sþ1ðsþ 1Þðsþ 2Þf 0ðaÞ�� q !1

q

8<:

þ 2� 5sþ2 þ ðs� 4Þ6sþ1 � ð2sþ 7Þ3sþ1

3:6sþ1ðsþ 1Þðsþ 2Þf 0ðbÞ�� q þ ð2sþ 1Þ3sþ1 þ 2

3� 6sþ1ðsþ 1Þðsþ 2Þf 0ðaÞ�� q !1

q

9=;; ð2:6Þ

which is the same of the inequality in [14, Theorem 10].

Remark 3. In Theorem 7, if we take a ¼ 12 and k ¼ 0, then we have following midpoint inequality

faþ b

2

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

82

ðsþ 1Þðsþ 2Þ

� �1q

�21�s sþ 1ð Þ f 0ðbÞ

�� q2

þ21�s 2sþ2 � s� 3

� f 0ðaÞ�� q

2

0@

1A

1q

8><>:

þ21�s sþ 1ð Þ f 0ðaÞ

�� q2

þ21�s 2sþ2 � s� 3

� f 0ðbÞ�� q

2

0@

1A

1q9>=>;: ð2:7Þ

We note that the obtained midpoint inequality (2.7) is better than the inequality (1.4). Because sþ12 6 1 and 2sþ2�s�3

2 621�sþ1

21�s .


Remark 4. In Theorem 7, if we take a ¼ 12 , and k ¼ 1, then we get the following trapezoid inequality

f að Þ þ f bð Þ2

� 1b� a

Z b

af ðxÞdx

�� 6 b� a

821�s

ðsþ 1Þðsþ 2Þ

!1q

� f 0ðbÞ�� q þ f 0ðaÞ

�� q 2sþ1 þ 1� � 1

q þ f 0ðaÞ�� q þ f 0ðbÞ

�� q 2sþ1 þ 1� � 1

q �

Using Lemma 1 we shall give another result for convex functions as follows.

Theorem 8. Let f : I # R! R be a differentiable mapping on I� such that f 0 2 L½a; b�, where a; b 2 I� with a < b and a; k 2 0;1½ �. Iff 0�� q is s-convex on ½a; b�, for some fixed s 2 ð0;1� and q > 1, then

If k;a;a;bð Þ�� 6 b�að Þ 1

pþ1

� �1p 1

sþ1

� �1q

�

e1=p1 ða;k;pÞC

1=qf ða;qÞþe1=p

1 ð1�a;k;pÞD1=qf ða;qÞ

h i; ak61�a61�k 1�að Þ

e1=p1 ða;k;pÞC

1=qf ða;qÞþe1=p


h i; ak61�k 1�að Þ61�a

e1=p2 ða;k;pÞC

1=qf ða;qÞþe1=p


h i; 1�a6ak61�k 1�að Þ

8>>>><>>>>:

;

ð2:8Þ

where

Cf ða; qÞ ¼ 1� að Þ f 0 1� að Þbþ aað Þ�� q þ f 0 að Þ

�� qh i; ð2:9Þ

Df ða; qÞ ¼ a f 0 1� að Þbþ aað Þ�� q þ f 0 bð Þ

�� qh i;

e1ða; k;pÞ ¼ akð Þpþ1 þ 1� a� akð Þpþ1; ð2:10Þ

e2ða; k;pÞ ¼ akð Þpþ1 � ak� 1þ að Þpþ1;

and 1p þ 1

q ¼ 1.

Proof. From Lemma 1 and by Hölder’s integral inequality, we have

If k;a; a; bð Þ�� 6 b� að Þ

Z 1�a


�� dt þZ 1


�� dt� �

6 b� að ÞZ 1�a

0t � akj jpdt

� �1p Z 1�a

0f 0 tbþ ð1� tÞað Þ�� qdt

� �1q

8<:

þZ 1

1�at � 1þ k 1� að Þj jpdt

� �1p Z 1

1�af 0 tbþ ð1� tÞað Þ�� qdt

� �1q

9=;: ð2:11Þ

Since f 0�� q is s-convex on ½a; b�, for a 2 0;1½ Þ by the inequality (1.2), we get
Z 1�a
0f 0 tbþ ð1� tÞað Þ�� qdt ¼ 1� að Þ 1

1� að Þ b� að Þ

Z ð1�aÞbþaa

af 0 xð Þ�� qdx

" #

6 1� að Þf 0 1� að Þbþ aað Þ�� q þ f 0 að Þ

�� qsþ 1

" #: ð2:12Þ

The inequality (2.12) also holds for a ¼ 1. Similarly, for a 2 0;1ð � by the inequality (1.2), we have
Z 1
1�af 0 tbþ ð1� tÞað Þ�� qdt ¼ a

1a b� að Þ

Z b

ð1�aÞbþaaf 0 xð Þ�� qdx

" #6 a

f 0ðð1� aÞbþ aaÞ�� q þ f 0 bð Þ

�� qsþ 1

" #: ð2:13Þ

The inequality (2.13) also holds for a ¼ 0. By simple computation
Z 1�a
0t � akj jpdt ¼

akð Þpþ1þ 1�a�akð Þpþ1

pþ1 ; ak 6 1� aakð Þpþ1� ak�1það Þpþ1

pþ1 ; ak P 1� a

8<: ; ð2:14Þ

and

Z 1

1�at � 1þ k 1� að Þj jpdt ¼

k 1�að Þ½ �pþ1þ a�k 1�að Þ½ �pþ1

pþ1 ; 1� a 6 1� k 1� að Þk 1�að Þ½ �pþ1� k 1�að Þ�a½ �pþ1

pþ1 ; 1� a P 1� k 1� að Þ

8<: ; ð2:15Þ

thus, using (2.12)–(2.15) in (2.11), we obtain the inequality (2.8). This completes the proof. h


Corollary 2. Under the assumptions of Theorem 8 with s ¼ 1, we have

If k;a;a;bð Þ�� 6 b�að Þ 1

pþ1

� �1p 1

2

� �1q

�

e1=p1 ða;k;pÞC

1=qf ða;qÞþe1=p


h i; ak61�a61�k 1�að Þ

e1=p1 ða;k;pÞC

1=qf ða;qÞþe1=p


h i; ak61�k 1�að Þ61�a

e1=p2 ða;k;pÞC

1=qf ða;qÞþe1=p


h i; 1�a6ak61�k 1�að Þ

8>>>><>>>>:

;

where e1; e2; Cf and Df are defined as in (2.9).

Remark 5. In Theorem 8, if we take a ¼ 12 and k ¼ 1

3, then we have the following Simpson type inequality

16

f ðaÞþ4faþb

2

� �þ f ðbÞ

� �� 1

b�a

Z b

af ðxÞdx

��6 b�a

121þ2pþ1

3 pþ1ð Þ

!1p f 0 aþb

2

� �� qþ f 0 að Þ�� q

sþ1

!1q

þf 0 aþb

2

� �� qþ f 0 bð Þ�� q

sþ1

!1q

8<:

9=;; ð2:16Þ

which is the same of the inequality (1.6).

Remark 6. In Theorem 8, if we take a ¼ 12 and k ¼ 0, then we have the following midpoint inequality

faþ b

2

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

41

pþ 1

� �1p f 0 aþb

2

� �� q þ f 0 að Þ�� q

sþ 1

!1q

þf 0 aþb

2

� �� q þ f 0 bð Þ�� q

sþ 1

!1q

8<:

9=;:

We note that by inequality

2s�1 f 0aþ b

2

� ��q

6f 0 að Þ�� q þ f 0 bð Þ

�� qsþ 1

we have

faþ b

2

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

4

� �1

pþ 1

� �1p 1

sþ 1

� �2q

� 21�s þ sþ 1�

f 0 að Þ�� q þ 21�s f 0 bð Þ

�� q� 1q þ 21�s þ sþ 1

� f 0 bð Þ�� q þ 21�s f 0 að Þ

�� q� 1q

� �;

which is the same of the inequality (1.5).

Remark 7. In Theorem 8, if we take a ¼ 12 and k ¼ 1, then we have the following trapezoid inequality


� 1b� a

Z b

af ðxÞdx

�� 6 b� a

41

pþ 1

� �1p

�f 0 aþb

2

� �� q þ f 0 að Þ�� q

sþ 1

!1q

þf 0 aþb

2

� �� q þ f 0 bð Þ�� q

sþ 1

!1q

8<:

9=;: ð2:17Þ

We note that the obtained midpoint inequality (2.17) is better than the inequality (1.7).

Theorem 9. Let f : I # R! R be a differentiable mapping on I� such that f 0 2 L½a; b�, where a; b 2 I� with a < b and a; k 2 0;1½ �. Iff 0�� q is s-concave on ½a; b�, for some fixed s 2 ð0;1� and q > 1, then the following inequality holds:

If k;a;a;bð Þ�� 6 b� að Þ2

s�1q

1pþ 1

� �1p

�

e1=p1 ða;k;pÞE

1=qf ða;qÞ þ e1=p

1 ð1�a;k;pÞF1=qf ða;qÞ

h i; ak6 1�a6 1� k 1�að Þ

e1=p1 ða;k;pÞE



h i; ak6 1� k 1�að Þ6 1� a

e1=p2 ða;k;pÞE



h i; 1�a6 ak6 1� k 1�að Þ

8>>>><>>>>:

;

ð2:18Þ

where

Ef ða; qÞ ¼ 1� að Þ f 01� að Þbþ 1þ að Þa

2

� ��q

; Ff ða; qÞ ¼ a f 02� að Þbþ aa

2

� ��

q

;

and e1; e2 are defined as in (2.9).


Proof. We proceed similarly as in the proof Theorem 8. Since f 0�� q is s-concave on ½a; b�, for a 2 0;1½ Þ by the inequality (1.2),

we get
Z 1�a
0f 0 tbþ ð1� tÞað Þ�� qdt ¼ 1� að Þ 1

1� að Þ b� að Þ

Z ð1�aÞbþaa

af 0 xð Þ�� qdx

" #

6 2s�1 1� að Þ f 01� að Þbþ 1þ að Þa

2

� ��q

ð2:19Þ

The inequality (2.19) also holds for a ¼ 1. Similarly, for a 2 0;1ð � by the inequality (1.2), we have
Z 1
1�af 0 tbþ ð1� tÞað Þ�� qdt ¼ a

1a b� að Þ

Z bþaab

1�að Þf 0 xð Þ�� qdx

" #6 2s�1a f 0

2� að Þbþ aa2

� ��q

ð2:20Þ

The inequality (2.20) also holds for a ¼ 0. Thus, using (2.14),(2.15), (2.19) and (2.20) in (2.11), we obtain the inequality(2.18). This completes the proof. h

Corollary 3. Under the assumptions of Theorem 9 with s ¼ 1, we have

If k;a; a; bð Þ�� 6 b� að Þ 1

pþ 1

� �1p

�

e1=p1 ða; k;pÞE

1=qf ða; qÞ þ e1=p

1 ð1� a; k;pÞF1=qf ða; qÞ

h i; ak 6 1� a 6 1� k 1� að Þ

e1=p1 ða; k;pÞE



h i; ak 6 1� k 1� að Þ 6 1� a

e1=p2 ða; k;pÞE



h i; 1� a 6 ak 6 1� k 1� að Þ

8>>>><>>>>:

;

where e1; e2; Ef and Ff are defined as in Theorem 9.



� 1b� a

Z b

af ðxÞdx

�� 6 b� a

41

pþ 1

� �1p

� 12

� �1�sq

f 03bþ a

4

� ��þ f 0

3aþ b4

� ��

� �

which is the same of the inequality in [12, Theorem 8 (i)].

Remark 9. In Theorem 9, if we take a ¼ 12 and k ¼ 0, then we have the following midpoint inequality

faþ b

2

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

41

pþ 1

� �1p

� 12

� �1�sq

f 03bþ a

4

� ��þ f 0

3aþ b4

� ��

� �

which is the same of the inequality in [12, Theorem 8 (ii)].



� 1b� a

Z b

af ðxÞdx

�� 6 b� a

41

pþ 1

� �1p

f 03bþ a

4

� ��þ f 0

3aþ b4

� ��

� �ð2:21Þ

which is the same of the inequality in [10, Theorem 2].


faþ b

2

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

41

pþ 1

� �1p

f 03bþ a

4

� ��þ f 0

3aþ b4

� ��

� �ð2:22Þ

which is the same of the inequality in [2, Theorem 2.5].

Remark 12. In Theorem 9, since f 0�� q; q > 1, is concave on a; b½ �, using the power mean inequality, we have

f 0 kxþ 1� kð Þyð Þ�� q P k f 0 xð Þ

�� q þ 1� kð Þ f 0 yð Þ�� q P k f 0 xð Þ

�� þ 1� kð Þ f 0 yð Þ�� q

;

8x; y 2 a; b½ � and k 2 0;1½ �. Hence

f 0 kxþ 1� kð Þyð Þ�� P k f 0 xð Þ

�� þ 1� kð Þ f 0 yð Þ��


so f 0�� is also concave. Then by the inequality (1.1), we have

f 03bþ a

4

� ��þ f 0

3aþ b4

� �� 6 2 f 0

aþ b2

� ��: ð2:23Þ

Thus, using the inequality (2.23) in (2.21) and (2.22) we get


� 1b� a

Z b

af ðxÞdx

�� 6 b� a

21

pþ 1

� �1p

f 0aþ b

2

� ��;

faþ b

2

� �� 1

b� a

Z b

af ðxÞdx

�� 6 b� a

21

pþ 1

� �1p

f 0aþ b

2

� ��:

3. Some applications for special means

Let us recall the following special means of arbitrary real numbers a; b with a – b and a 2 0;1½ �:

(1) The weighted arithmetic mean

Aa a; bð Þ :¼ aaþ ð1� aÞb; a; b 2 R:

(2) The unweighted arithmetic mean

A a; bð Þ :¼ aþ b2

; a; b 2 R:

(3) Then p-Logarithmic mean

Lp a; bð Þ :¼ bpþ1 � apþ1

ðpþ 1Þðb� aÞ

!1p

; p 2 R n �1; 0f g; a; b > 0:

From known Example 1 in [5], we may find that for any s 2 0;1ð Þ and b > 0; f : 0;1½ Þ ! 0;1½ Þ; f ðtÞ ¼ bts; f 2 K2s .

Now, using the results of Section 2, some new inequalities are derived for the above means.

Proposition 1. Let a; b 2 Rwith 0 < a < b; q P 1 and s 2 0; 1q

� . Then

Theorem 10

(i) for ak 6 1� a 6 1� k 1� að Þ we have

kAa asþ1; bsþ1�

þ 1� kð ÞAsþ1a a; bð Þ � Lsþ1

sþ1 a; bð Þ�� 6 b� að Þ sþ 1ð Þ c

1�1q

2 ða; kÞ c1ða; k; sÞbsq þ c2ða; k; sÞasq� 1

q

�

þc1�1

q2 ð1� a; kÞ c2ð1� a; k; sÞbsq þ c1ð1� a; k; sÞasq

� 1q

�;

(ii) for ak 6 1� k 1� að Þ 6 1� a we have

kAa asþ1; bsþ1�



1�1q


q

�

þc1�1


� 1q

�;

(iii) for 1� a 6 ak 6 1� k 1� að Þ we have

kAa asþ1; bsþ1�



1�1q


q

�

þc1�1


� 1q

�

where c1; c2; c1; c2; c3; c4 numbers are defined as in Theorem 7.

Proof. The assertion follows from applied the inequalities in Theorem 7 to the function f ðtÞ ¼ tsþ1; t 2 a; b½ � and s 2 0; 1q

� ,

which implies that f 0ðtÞ ¼ ðsþ 1Þts; t 2 a; b½ � and f 0ðtÞ�� q ¼ ðsþ 1Þqtqs; t 2 a; b½ � is a s-convex function in the second sense

since qs 2 0;1ð Þ and ðsþ 1Þq > 0. h


Proposition 2. Let a; b 2 R with 0 < a < b; p; q > 1; 1pþ 1

q ¼ 1 and s 2 0; 1q

� we have the following inequality:

kAa asþ1; bsþ1�


sþ1 a; bð Þ�� 6 b� að Þ 1

pþ 1

� �1p

sþ 1ð Þ1�1q

�

e1=p1 ða; k;pÞC

1=qs ða; qÞ þ e1=p

1 ð1� a; k;pÞD1=qs ða; qÞ

h i; ak 6 1� a 6 1� k 1� að Þ

e1=p1 ða; k;pÞC



h i; ak 6 1� k 1� að Þ 6 1� a

e1=p2 ða; k;pÞC



h i; 1� a 6 ak 6 1� k 1� að Þ

8>>>><>>>>:

;

where

Csða; qÞ ¼ 1� að Þ Asqa a; bð Þ þ asq

� �; Dsða; qÞ ¼ a Asq

a a; bð Þ þ bsq� �;

and e1 and e2 numbers are defined as in (2.10).

Proof. The assertion follows from applied the inequality (2.8) to the function f ðtÞ ¼ tsþ1; t 2 a; b½ � and s 2 0; 1q

� , which

implies that f 0ðtÞ ¼ ðsþ 1Þts; t 2 a; b½ � and f 0ðtÞ�� q ¼ ðsþ 1Þqtqs; t 2 a; b½ � is a s-convex function in the second sense since

qs 2 0;1ð Þ and ðsþ 1Þq > 0. h

References

[1] M. Alomari, M. Darus, S.S. Dragomir, New inequalities of Simpson’s Type for s-convex functions with applications, RGMIA Res. Rep. Coll. 12 (4) (2009).Article 9. <http://ajmaa.org/RGMIA/v12n4.php>.

[2] M.W. Alomari, M. Darus, U.S. Kirmaci, Some inequalities of Hermite–Hadamard type for s-convex functions, Acta Math. Sci. 31B (4) (2011) 1643–1652.[3] W.W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Räumen, Publ. Inst. Math. 23

(1978) 13–20.[4] S.S. Dragomir, S. Fitzpatrik, The Hadamard’s inequality for s-convex functions in the second sense, Demonstr. Math. 32 (4) (1999) 687–696.[5] H. Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48 (1) (1998) 100–111.[6] I. Iscan, A new generalization of some integral inequalities and their applications, Int. J. Eng. Appl. Sci. 3 (3) (2013) 17–27.[7] I. Iscan, Generalization of different type integral inequalities for s-convex functions via fractional integrals, Appl. Anal. (2013), http://dx.doi.org/

10.1080/00036811.2013.851785. <http://dx.doi.org/10.1080/00036811.2013.851785>.[8] I. Iscan, New estimates on generalization of some integral inequalities for s-convex functions and their applications, Int. J. Pure Appl. Math. 86 (4)

(2013) 727–746.[9] U.S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput.

147 (2004) 137–146.[10] U.S. Kirmaci, M.K. Bakula, M.E. Özdemir, J. Pecari c, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput. 193 (2007) 26–35.[11] Z. Liu, A note on Ostrowski type inequalities related to some s-convex functions in the second sense, Bull. Korean Math. Soc. 49 (4) (2012) 775–785.[12] J. Park, Hermite–Hadamard-type inequalities for real a-star s-convex mappings, J. Appl. Math. Inf. 28 (5–6) (2010) 1507–1518.[13] M.Z. Sarikaya, N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Modell. 54 (2011) 2175–2182.[14] M.Z. Sarikaya, E. Set, M.E. Özdemir, On new inequalities of Simpson’s type for s-convex functions, Comput. Math. Appl. 60 (2010) 2191–2199.[15] M.Z. Sarkaya, E. Set, H. Ögülmüs�, Some new inequalities of Hermite–Hadamard type for mappings whose derivatives are s-convex in the second sense,

Int. J. Mod. Math. Sci. 8 (3) (2013) 212–218.[16] E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Comput. Math.

Appl. 63 (2012) 1147–1154.[17] E. Set, M.E. Özdemir, M.Z. Sarkaya, New inequalities of Ostrowski’s type for s-convex functions in the second sense with applications, Facta

Universitatis Ser. Math. Inf. 27 (1) (2012) 67–82.[18] M.E. Özdemir, Ç. Yldz, A.O. Akdemir, E. Set, On some inequalities for s-convex functions and applications, J. Inequal. Appl. (1) (2013) 333. 11 pages.[19] M.Z. Sarkaya, E. Set, M.E. Özdemir, On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex, J. Appl.

Math. Stat. Inf. 9 (1) (2013) 37–45.

http://ajmaa.org/RGMIA/v12n4.php

http://refhub.elsevier.com/S0096-3003(14)01150-3/h0010






http://dx.doi.org/10.1080/00036811.2013.851785

http://dx.doi.org/10.1080/00036811.2013.851785

http://dx.doi.org/10.1080/00036811.2013.851785





















Documents

On new general integral inequalities for s-convex functions