Hermite Hadamard and Simpson Like Type Inequalities for Differentiable (

International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 809689,12pages

doi:10.1155/2012/809689

Research Article

Hermite-Hadamard and Simpson-Like

Type Inequalities for Differentiable (α, m)-Convex

Mappings

Jaekeun Park

Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si 356-706, Republic of Korea

Correspondence should be addressed to Jaekeun Park,jkpark@hanseo.ac.kr Received 13 July 2011; Revised 13 November 2011; Accepted 28 November 2011 Academic Editor: Jewgeni Dshalalow

Copyrightq2012 Jaekeun Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The author establish several Hermite-Hadamard and Simpson-like type inequalities for mappings whose first derivative in absolute value aroused to theqthq≥1power areα, m-convex. Some applications to special means of positive real numbers are also given.

1. Introduction

Recall that, for some fixedα∈0,1andm∈0,1, a mappingf :I⊆0,∞ → Ris said to beα, m-convex on an intervalIif the inequality

ftxm1−ty≤tα_{fx m1−t}α_{fy 1.1}

holds forx, y∈I, andt∈0,1. Denote byKα

mIthe set of all α, m-convex mappings on I. For recent results and generalizations concerningm-convex andα, m-convex mappings, see1–4.

For the simplicities of notations, forf∈Kα

mI, let us denote

a−mbSba

fα, m, r 1 r

fa r−2f

amb

2

fmb

− 1

mb−a

_mb

a

fxdx.

In 1, 3, Klariˇci´c Bakula and ¨Ozdemir et al., proved the following Hadamard’s inequalities for mappings whose second derivative in absolute value aroused to the q-th

q≥1power areα, m-convex.

Theorem 1.1. Letf : I ⊆ 0, b∗ → Rbe a twice differentiable mapping on the interiorI0 _{of an}

intervalIsuch thatf ∈L1a, b, wherea, b ∈Iwitha < bandb∗ >0. If|f |qisα, m-convex

ona, bforα, m∈0,12andq≥1 with 1/p1/q1, then the following inequality holds:

a Sba

fα, m,2 ≤ mb−a 2

1 6

1/p

μ f a qmν f b q1/q, 1.3

where

μ 1

α2α3, ν

α2_5α

6α2α3, 1.4

b Sba

fα, m,6 ≤ mb−a 2

1 2

1/p

μ0 f a qmν0 f b q

1/q

, 1.5

where

μ0

q qα2

_Γα2Γq

Γqα1,

ν0 1

q1q2 −

q qα2

_Γα2Γq

Γqα1,

1.6

where

Γx

_∞

0

e−ttx−1dt, x >0. 1.7

Theorem 1.2. Under the same notations in Theorem 2.2, if |f |q _{is α, m-convex on a, b for}

α, m∈0,12andq >1 with 1/p1/q1, then the following inequality holds:

_Sb a

fα, m,2 ≤ mb−a 8

Γp1

Γp3/2 1/p

f a q 1

α1m f

_b q α

α1 1/q

. 1.8

Note that forα, m∈ {0,0,α,0,1,0,1, m,1,1,α,1}one obtains the following classes of functions: increasing,α-starshaped, starshaped,m-convex, convex, andα-convex. For the definitions and elementary properties of these classes, see4–8.

For recent years, many authors present some new results about Simpson’s inequality for α, m-convex mappings and have established error estimations for the Simpson’s inequality: for refinements, counterparts, generalizations, and new Simpson’s type inequali-ties, see1–3,6.

Theorem 1.3. Letf : I ⊂ 0,∞ → Rbe an absolutely continuous mapping ona, bsuch that

f ∈Lpa, b, wherea, b∈Iwitha < b. Then the following inequality holds:

_Sb a

f1,1,6 ≤ b−a

−1/p

6

2q1₁

3q1 1/q

_f

p. 1.9

The readers can estimate theerror fin the generalized Simpson’s formula without going through its higher derivatives which may not exist, not be bounded, or may be hard to find.

In this paper, the author establishes some generalizations of Hermite-Hadamard and Simpson-like type inequalities based on differentiableα, m-convex mappings by using the following new identity inLemma 2.1and by using these results, obtain some applications to special means of positive real numbers.

2. Generalizations of Simpson-Like Type Inequalities on

K

m

I

In order to generalize the classical Simpson-like type inequalities and prove them, we need the following lemma6.

Lemma 2.1. Letf :I⊆0, b∗ → Rbe a differentiable mapping on the interiorI0of an intervalI,

wherea, b ∈Iwith 0≤ a < band b∗ > 0. Iff ∈ L1_{a, b, then, forr ≥ 2 andh ∈0,1with}

1/r ≤h≤r−1/r, the following equality holds:

Sba

fα, m, r

1

0

pr, tftam1−tbdt 2.1

forf ∈Kα

ma, band eacht∈0,1, where

pr, t

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

t−1

r t∈

0,1

2

,

t−r−1

r t∈

1 2,1

.

2.2

By the similar way as Theorems1.1–1.3, we obtain the following theorems.

Theorem 2.2. Letf : I⊂ 0, b∗ → Rbe a differentiable mapping onI0_{such thatf ∈La, b,}

wherea, b ∈Iwith 0≤ a < b <∞andb∗ > 0. If|f| ∈Kα

ma, b, for someα, m∈0,12and

mb > a, then, for anyr ≥2, the following inequality holds:

_Sb a

fα, m, r ≤μ1

1μ21μ31μ41 fa

ν1

1ν21ν31ν41

where

μ1₁ 1

α1α2rα2,

μ2₁ 1

α1α2rα2

α1r−2α2 2α2_α1α2r,

μ3₁ r−1

α2

α1α2rα2

2α2−3r

2α2_α1α2r,

μ4₁ r−1

α2

α1α2rα2

α2−r

α1α2r,

ν1

1

1 2r2 −μ

1

1,

ν2

1

r−22 8r2 −μ

2

1,

ν3

1

r−22 8r2 −μ

3

1,

ν₁4 1

2r2 −μ 4

1.

2.4

Proof. FromLemma 2.1and using the properties of the modulus, we have the following:

_Sb a

fα, m, r ≤

_1/r

0

1

r −t ftam1−tb dt

1/2

1/r

t−1

r ftam1−tb dt

r−1/r

1/2

r−1

r −t ftam1−tb dt

₁

r−1/r

t−r−1

r ftam1−tb dt.

2.5

Since|f|isα, m-convex ona, b, we know that for anyt∈0,1

_{ftam1−tb ≤t}α _{fa m1−t}α _{fb . 2.6}

By2.5and2.6, we get the following:

_Sb a

fα, m, r ≤

_1/r

0

1

r −t

tα fa m1−tα fb dt

_1/2

1/r

t− 1 r

_r−1/r

1/2

r−1

r −t

tα fa m1−tα fb dt

₁

r−1/r

t− r−1 r

tα _{fa m1−t}α _{fb dt}

≤

1/2

0

1_r −t tαdt

1

1/2

r−_r1 −t tαdt

_fa

1/2

0

1_r −t 1−tαdt

₁

1/2

r−_r1−t 1−tαdt

m fb

μ1₁μ2₁μ₁3μ4₁ fa ν1₁ν2₁ν3₁ν4₁m fb ,

2.7

which completes the proof.

Corollary 2.3. InTheorem 2.2,iif we chooseα1 andr2, then we have the following:

_mb−aSb a

f1, m,2

fa fmb

2 −

1

mb−a

mb

a

fxdx

≤ mb−a

8 fa m fb ,

2.8

andiiif we chooseα1 andr6, then we have the following

_mb−aSb a

f1, m,6 1 6

fa 4f

amb

2

fmb

− 1

mb−a

_mb

a

fxdx

≤ 5

72mb−a fa m fb .

2.9

Theorem 2.4. Under the same notations inTheorem 2.2, if|f|q _{∈ K}α

ma, b, for someα, m ∈

0,12,mb > aandq >1 with 1/p1/q1, then, for anyr≥2, the following inequality holds:

_Sb a

fα, m, r

≤

1 2r2

1/p

μ1₁ fa qν₁1m fb q1/qμ4₁ fa qν4₁m fb q1/q

1 8

r−2

r

21/p

μ2₁ fa qν₁2m fb q1/qμ₁3 fa qν₁3m fb q1/q

.

Proof. FromLemma 2.1and using the properties of modulus, we have the following:

_Sb a

fα, m, r ≤

_1/r

0

1

r −t ftam1−tb dt

1/2

1/r

t− 1

r ftam1−tb dt

_r−1/r

1/2

r−1

r −t ftam1−tb dt

₁

r−1/r

t− r−1

r ftam1−tb dt.

2.11

Using the power-mean integral inequality and α, m-convexity of |f|q _{for any t ∈}

0,1, we have the following

a

_1/r

0

1

r −t ftam1−tb

q

dt≤μ1₁ fa qν1₁m fb q, 2.12

b

_1/2

1/r

t− 1

r ftam1−tb

q

dt≤μ2₁ fa qν2₁m fb q, 2.13

c

_r−1/r

1/2

r−1

r −t ftam1−tb

q

dt≤μ3₁ fa qν₁3m fb q, 2.14

d

1

r−1/r

t−r−1

r ftam1−tb

q

dt≤μ4₁ fa qν4₁m fb q. 2.15

By the similar way as the above inequalitiesa–d, we have the following:

a

_1/r

0

1

r −t ftam1−tb dt≤

1 2r2

1/p

μ1

1 fa

q

ν1

1m fb

q1/q

b

1/2

1/r

t−1

r ftam1−tb dt≤

1 8

r−2

r

21/p

μ2₁ fa qν2₁m fb 1/q,

2.17

c

_r−1/r

1/2

r−1

r −t ftam1−tb dt≤

1 8

r−2

r

21/p

μ3

1 fa

q

ν3

1m fb

q1/q

,

2.18

d

1

r−1/r

t−r−1

r ftam1−tb dt≤

1 2r2

1/p

μ4₁ fa qν₁4m fb q1/q.

2.19

By2.11and2.16–2.19the assertion2.10holds.

Corollary 2.5. InTheorem 2.4,iif we chooseα1 andr2, then we have that

fa fmb

2 −

1

mb−a

mb

a

fxdx

≤ 6−1/q

8 mb−a fa q

5m fb q1/q5 fa qm fb q1/q,

2.20

andiiif we chooseα1 andr6, then we have that

1

6

fa 4f

amb

2

fmb

− 1

mb−a

mb

a

fxdx

≤mb−a

₁

72

1 18

1/q

fa q17m fb q1/q

17 fa qm fb q1/q

25 288

2 5

2/q

×7 fa q11m fb q1/q11 fa q7m fb q1/q.

Theorem 2.6. Under the same notations inTheorem 2.2, if|f|q _{∈ K}α

ma, b, for someα, m ∈

0,12,mb > aandq >1 with 1/p1/q1, then, for anyr≥2, the following inequality holds: _Sb

a

fα, m, r

≤

1

rp1

1/p

μ1₂ fa qν₂1m fb q1/qμ₂4 fa qν₂4m fb q1/q

r−2p1 2p1_rp1

1/p

μ22 fa

q

ν22m fb

q1/q

μ3₂ fa qν3₂m fb q1/q

,

2.22

where

μ1₂ 1

rα1_α1,

μ22

rα1₋₂α1

2α1_rα1_α1,

μ3₂ 2

α1_r−1α1_−rα1

2α1_rα1_α1 ,

μ4₂ r

α1_−r−1α1

rα1_α1 ,

ν1₂ 1 r −μ

1

2,

ν22

r−2 2r μ

2

2,

ν3₂ r−2

2r μ

3

2,

ν42

1

r −μ

4

2.

2.23

Proof. Suppose thatq >1. FromLemma 2.1, using the H ¨older integral inequality, we get the

following:

_Sb a

fα, m, r ≤

1/r

0

1

r −t

p

dt

1/p_1/r

0

_ftbm1−ta q

dt

1/q

1/2

1/r

t−1 r

p

dt

1/p_1/2

1/r

_ftbm1−ta q

dt

1/q

r−1/r

1/2

r−1

r −t

p

dt

1/p_r−1/r

1/2

_ftbm1−ta q

dt

1/q

1

r−1/r

t−r−1 r

p

dt

1/p₁

r−1/r

_ftbm1−ta q

dt

≤

1

rp1

1/p1/r

0

_ftbm1−ta qdt

1/q

1 2p1

r−2

r

p11/p_1/2

1/r

_ftbm1−ta q

dt

1/q

1 2p1

r−2

r

p11/p_r−1/r

1/2

_ftbm1−ta q

dt

1/q

1

rp1

1/p1

r−1/r

_ftbm1−ta q

dt

1/q

,

2.24

where we have used the fact that 1/2<1/p11/p<1. Since|f|q_∈Km

αa, bfor some fixedα∈0,1andm∈0,1, we have the followings: 1/r

0

_{ftb 1−ta} q

dt≤μ12 fa

q

ν12m fb

q

,

_1/2

1/r

_ftb 1−ta qdt≤μ2₂ fa qν2₂m fb q,

_r−1/r

1/2

_ftb 1−ta qdt≤μ3

2 fa

q

ν3

2m fb

q

,

1

r−1/r

_{ftb 1−ta} q

dt≤μ42 fa

q

ν24m fb

q

.

2.25

Hence, if we combine the inequalities in2.24-2.25, we get the desired result.

Corollary 2.7. InTheorem 2.6,iif we chooseα1 andr2, then we have that

fa fmb

2 −

1

mb−a

_mb

a

fxdx

≤

1 4

₁₁/q

mb−a fa q3m fb q1/q3 fa qm fb q1/q,

2.26

andiiif we chooseα1 andr6, then we have

1

6

fa 4f

amb

2

fmb

− 1

mb−a

mb

a

fxdx

≤mb−a

₁

6 12/q

fa q11m fb q1/q11 fa qm fb q1/q

1 3

21/q

fa q2m fb q1/q2 fa qm fb q1/q,

2.27

3. Applications to Special Means

Now using the results ofSection 2, we give some applications to the following special means of positive real numbersa, b∈Rwithb≥a.

1The arithmetic mean:Aa, b ab/2.

2The geometric mean:Ga, b √ab.

3The logarithmic mean:La, b b−a/lnb−lnafor_{a /}b.

4The harmonic mean:Ha, b 2ab/ab.

5The power mean:Mra, b arbr/21/r,r≥1, a, b∈R.

6The generalized logarithmic mean:

Lna, b

bn1_−an1

b−an1 1/n

, a /b. 3.1

7The identric mean:

Ia, b

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

a ab

1

e

bb

aa

1/b−a

a /b. 3.2

Proposition 3.1. Forn ∈ −∞,0∪1,∞\ {−1}anda, b ∈ 0, b∗withb∗ > 0, we have the

following inequalities:

a|Aan, mnbn−Lnna, mb| ≤

mb−a

8 |n|M n−1

n−1

a, m1/n−1b.

b 1 3Aa

n_{, m}n_bn 2 3A

n_{a, mb−L}n na, mb

≤

5 72

mb−a|n|M_nn−1₋₁a, m1/n−1b.

3.3

Proof. The assertions follow fromCorollary 2.3forfx xn.

Proposition 3.2. Fora, b∈0, b∗, we have the following inequalities:

a H−1a, mb−L−1a, mb ≤

mb−a

8a2_b2

M₂2m1/2a, b,

b 1 3H

−1_{a, mb} 2

3A

−1_{a, mb−L}−1_{a, mb ≤}5mb−a

72a2_b2

M2₂m1/2a, b.

3.4

Proposition 3.3. For n ∈ −∞,0∪1,∞\ {−1} and a, b ∈ 0, b∗, we have the following inequalities:

a|Aan, mnbn−Lnna, mb|

≤

3−1/q

8

mb−a|n| ×

A1/q_an−1q_,5mbn−1q1/q_A1/q_5an−1q_{, mb}n−1q1/q

,

b 1 3Aa

n_{, m}n_bn 2 3A

n_{a, mb−L}n na, mb

≤

1 72

1 9

1/q

mb−a|n| ×A1/q_an−1q_,17mbn−1q_A1/q_17an−1q_{, mb}n−1q

25 288

2 25

1/q

mb−a|n|

×A1/q7an−1q,11mbn−1qA1/q11an−1q,7mbn−1q.

3.5

Proof. The assertions follow fromCorollary 2.5forfx xn_.

Proposition 3.4. For n ∈ −∞,0∪1,∞\ {−1} and a, b ∈ 0, b∗, we have the following

inequalities:

a|Aan, mnbn−Lnna, b| ≤

1 221/q

|n|mb−a

×A1/qan−1q,3mbn−1qA1/q3an−1q, mbn−1q,

b 1 3Aa

n_{, m}n_bn 2 3A

n_{a, mb−L}n na, mb

≤

1 6

₁₁/q₁

3 1/q

|n|mb−a×A1/q_an−1q_,11mbn−1q_A1/q_11an−1q_{, mb}n−1q

1 3

21/q

21/q|n|mb−a×A1/qan−1q,2mbn−1qA1/q2an−1q, mbn−1q.

3.6

Proof. The assertions follow fromCorollary 2.7forfx xn_.

Proposition 3.5. Fora, b∈0, b∗, we have the following inequalities:

a ln Ia, b

Ga, b

≤ b−a

ab

1 411/q

aq3bq1/q 3aqbq1/q,

b ln Ia, b

G1/3_{a, bA}2/3_{a, b}

≤ b−a

ab

1 6

12/q

aq11bq1/q 11aqbq1/q

1 3

₂₁/q

aq2bq1/q 2aqbq1/q.

3.7

Acknowledgment

The author is so indebted to the referee who read carefully through the paper very well and mentioned many scientifically and expressional mistakes.

References

1 M. Klariˇci´c Bakula, M. E. ¨Ozdemir, and J. Peˇcari´c, “Hadamard type inequalities for m andα, m-convex functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 9, no. 4, article 96, p. 12, 2008.

2 M. E. ¨Ozdemir, E. Set, and M. Z. Sarikaya, “Some new Hadamard’s type inequalities for co-ordinated m-convex andα, m-convex functions,”rgmia.org/papers/v13e/oss5.pdf.

3 M. E. ¨Ozdemir, M. Avci, and H. Kavurmaci, “Hermite-Hadamard-type inequalities viaα, m-convexity,”

Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2614–2620, 2011.

4 E. Set, M. E. ¨Ozdemir, and M. Z. Sarikaya, “Inequalities of Hermite-Hadamard’s type for functions whose derivatives absolute values are m-convex,” in Proceedings of the AIP Conference, vol. 1309, pp. 861–873, 2010,rgmia.org/papers/v13e/sos6.pdf.

5 M. Alomari and M. Darus, “On some inequalities of Simpson-type via quasi-convex functions and applications,” Transylvanian Journal of Mathematics and Mechanics, vol. 2, no. 1, pp. 15–24, 2010.

6 J. Park, “Generalizations of simpson-like type inequalities for differentiable m-convex and α, m -convex mappings,” Far East Journal of Mathematical Sciences, vol. 55, no. 1, pp. 97–109, 2011.

7 M. Z. Sarikaya, E. Set, and M. E. ¨Ozdemir, “On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex,” RGMIA Research Report Collection, vol. 13, no. 1, article 1, 2010.

8 G. Toader, “The hierarchy of convexity and some classic inequalities,” Journal of Mathematical

Inequalities, vol. 3, no. 3, pp. 305–313, 2009.

9 S. S. Dragomir, R. P. Agarwal, and P. Cerone, “On Simpson’s inequality and applications,” Journal of

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Probability and Statistics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematical PhysicsAdvances in

Complex Analysis

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Optimization

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Combinatorics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Operations Research

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Decision Sciences

Discrete Mathematics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Stochastic Analysis