Introduction The following inequality is well known in the literature as Simpson’s inequality

FUNCTIONS

MEHMET ZEKI SARIKAYA^?|, ERHAN. SET , AND M. EMIN OZDEMIR

Abstract. In this paper, we introduce some inequalities of Simpson’s type based on convexity. Some applications for special means of real numbers are also given.

1. Introduction

The following inequality is well known in the literature as Simpson’s inequality.

Theorem 1. Letf : [a; b]!Rbe a four times continuously di¤ erentiable mapping on(a; b)and f⁽⁴⁾

1= sup

x2(a;b)

f⁽⁴⁾(x) <1: Then, the following inequality holds:

1 3

f(a) +f(b)

2 + 2f a+b 2

1

b a

Z b a

f(x)dx 1 2880 f⁽⁴⁾

1(b a)⁴:

For recent re…nements, counterparts, generalizations and new Simpson’s type inequalities, see ([1],[2],[6]).

In [2], Dragomir et. al. proved the following some recent developments on Simp- son’s inequality for which the remainder is expressed in terms of lower derivatives than the fourth.

Theorem 2. Supposef : [a; b]!Ris a di¤ erentiable mapping whose derivative is continuous on (a; b) andf⁰ 2L[a; b]. Then the following inequality

(1.1) 1

3

f(a) +f(b)

2 + 2f a+b 2

1

b a

Z b a

f(x)dx b a 3 kf⁰k1

holds, wherekf⁰k1=Rb

ajf⁰(x)jdx:

The bound of (1.1) for L-Lipschitzian mapping was given in [2] by 5

36L(b a): Also, the following inequality was obtained in [2].

Theorem 3. Suppose f : [a; b]!Ris an absolutely continuous mapping on [a; b]

whose derivative belongs toLp[a; b]:Then the following inequality holds, 1

3

f(a) +f(b)

2 + 2f a+b 2

1

b a

Z b a

f(x)dx (1.2)

1 6

2^q+1+ 1 3(q+ 1)

1 q

(b a)^1qkf⁰kp 2000Mathematics Subject Classi…cation. 26D15, 26D10.

Key words and phrases. Simpson’s inequality, convex function.

?corresponding author.

1

where ¹_p+¹_q = 1:

In [4], some inequalities of Hermite-Hadamard’s type for di¤erentiable convex mappings were presented as follows:

Theorem 4. Let f :I R!R be di¤ erentiable mapping on I ; where a; b2I witha < b: If jf⁰j is convex on[a; b]; then the following equality holds,

(1.3) 1

b a

Z b a

f(x)dx f a+b 2

(b a) 4

jf⁰(a)j+jf⁰(b)j

2 :

The main aim of this paper is to establish new Simpson’s type inequalities for the class of functions whose derivatives in absolute value at certain powers are convex functions.

2. Main Results

In order to prove our main theorems, we need the following Lemma.

Lemma 1. Let f : I R! R be an absolutely continuous mapping on I such that f⁰ 2L₁[a; b]; wherea; b2I witha < b; then the following equality holds:

1

6 f(a) + 4f a+b

2 +f(b) 1

b a

Z b a

f(x)dx (2.1)

= b a

2 Z 1

0

t 2

1

3 f⁰ 1 +t

2 b+1 t 2 a + 1

3 t

2 f⁰ 1 +t

2 a+1 t 2 b dt:

Proof. It su¢ ces to note that

I₁ = Z 1

0

t 2

1

3 f⁰ 1 +t

2 b+1 t 2 a dt (2.2)

= t

2 1 3

2

b a f 1 +t

2 b+1 t 2 a

1

0

1

b a

Z 1 0

f 1 +t

2 b+1 t 2 a dt

= 2

6 (b a)f(b) + 2

3 (b a)f(a+b 2 ) 1

b a

Z 1 0

f 1 +t

2 b+1 t 2 a dt:

Settingx= 1 +t

2 b+1 t

2 aanddx= b a

2 dt; which gives (2.3) I1= 2

6 (b a)f(b) + 2

3 (b a)f(a+b

2 ) 2

(b a)² Z b

a+b 2

f(x)dx

Similarly, we can show that

I2 = Z 1

0

1 3

t

2 f⁰ 1 +t

2 a+1 t 2 b dt (2.4)

= 1

3 t 2

2

a b f 1 +t

2 a+1 t 2 b

1

0

+ 1

a b

Z 1 0

f 1 +t

2 a+1 t 2 b dt

= 2

6 (b a)f(a) + 2

3 (b a)f(a+b

2 ) 2

(b a)² Z ^a+b₂

a

f(x)dx:

Adding (2.3) and (2.4), we obtain

b a

2 (I₁+I₂) =

"

f(a) +f(b)

6 +2

3f(a+b

2 ) 1

b a

Z b a

f(x)dx

#

which is required.

The next theorems give a new re…nement of the Simpson inequality for di¤eren- tiable functions:

Theorem 5. Let f : I R ! R be a di¤ erentiable mapping on I such that f⁰ 2 L₁[a; b]; wherea; b 2 I with a < b: If jf⁰j is a convex on [a; b]; then the following inequality holds:

1

6 f(a) + 4f a+b

2 +f(b) 1

b a

Z b a

f(x)dx (2.5)

5 (b a)

72 [jf⁰(a)j+jf⁰(b)j]:

Proof. From Lemma 1 and by used convexity ofjf⁰j;we get 1

6 f(a) + 4f a+b

2 +f(b) 1

b a

Z b a

f(x)dx

b a

2 Z 1

0

t 2

1

3 f⁰ 1 +t

2 b+1 t 2 a + 1

3 t

2 f⁰ 1 +t

2 a+1 t 2 b dt

b a

2 Z 1

0

t 2

1 3 1 +t

2 jf⁰(b)j+1 t

2 jf⁰(a)j+1 +t

2 jf⁰(a)j+1 t

2 jf⁰(b)j dt

= b a

2 Z 1

0

t 2

1

3 [jf⁰(a)j+jf⁰(b)j]dt:

By simple computation, Z 1

0

t 2

1 3 =

Z ²₃

0

1 3

t 2 dt+

Z 1

2 3

t 2

1

3 dt= 5 36: Thus, we get (2.5) which completes the proof.

Corollary 1. In Theorem 5, iff(a) =f(^a+b₂ ) =f(b), then we have

(2.6) 1

b a

Z b a

f(x)dx f a+b 2

5 (b a)

72 [jf⁰⁰(a)j+jf⁰⁰(b)j]:

Remark 1. We note that the obtained midpoint inequality (2.6) is better than the inequality (1.3).

Remark 2. We note that the obtained midpoint inequality (2.5) is better than the inequality (1.1).

Theorem 6. Let f : I R ! R be a di¤ erentiable mapping on I such that f⁰ 2 L1[a; b]; wherea; b2I with a < b: If jf⁰j^q is a convex on [a; b]; q >1; then the following inequality holds:

1

6 f(a) + 4f a+b

2 +f(b) 1

b a

Z b a

f(x)dx (2.7)

(b a) 12

1 + 2^p+1 3(p+ 1)

1 p

( 3jf⁰(b)j^q+jf⁰(a)j^q 4

1 q

+ jf⁰(b)j^q+ 3jf⁰(a)j^q 4

1 q)

where ¹_p+¹_q = 1:

Proof. From Lemma 1 and by Hölder’s integral inequality;we get

1

6 f(a) + 4f a+b

2 +f(b) 1

b a

Z b a

f(x)dx

b a

2 Z 1

0

t 2

1

3 f⁰ 1 +t

2 b+1 t 2 a + 1

3 t

2 f⁰ 1 +t

2 a+1 t 2 b dt

b a

2

( Z 1 0

t 2

1 3

p

dt

1

p Z 1

0

f⁰ 1 +t

2 b+1 t 2 a

q

dt

1 q

+ Z 1

0

1 3

t 2

p ¹p Z 1 0

f⁰ 1 +t

2 a+1 t 2 b

q

dt

1 q)

:

Sincejf⁰j^q is a convex on[a; b];we know that fort2[0;1]

f⁰ 1 +t

2 b+1 t 2 a

q 1 +t

2 jf⁰(b)j^q+1 t

2 jf⁰(a)j^q;

hence

1

6 f(a) + 4f a+b

2 +f(b) 1

b a

Z b a

f(x)dx

b a

2

( Z 1 0

t 2

1 3

p

dt

1

p Z 1

0

1 +t

2 jf⁰(b)j^q+1 t

2 jf⁰(a)j^q dt

1 q

+ Z 1

0

1 3

t 2

p ¹p Z 1 0

1 +t

2 jf⁰(a)j^q+1 t

2 jf⁰(b)j^q dt

1 q)

= b a

2 Z 1

0

t 2

1 3

p

dt

1 p

( 3jf⁰(b)j^q+jf⁰(a)j^q 4

1 q

+ 3jf⁰(a)j^q+jf⁰(b)j^q 4

1 q)

: By simple computation,

Z 1 0

t 2

1 3

p

= Z ²₃

0

1 3

t 2

p

dt+ Z 1

2 3

t 2

1 3

p

dt= 2(1 + 2^p+1) 6^p+1(p+ 1): Thus, we get (2.7) which completes the proof.

Corollary 2. In Theorem 6, iff(a) =f(^a+b₂ ) =f(b), then we have 1

b a

Z b a

f(x)dx f a+b 2 (b a)

12

1 + 2^p+1 3(p+ 1)

1 p

( 3jf⁰(b)j^q+jf⁰(a)j^q 4

1 q

+ jf⁰(b)j^q+ 3jf⁰(a)j^q 4

1 q)

:

Remark 3. We note that the obtained midpoint inequality (2.7) is better than the inequality (1.2).

Theorem 7. Let f : I R ! R be a di¤ erentiable mapping on I such that f⁰ 2 L₁[a; b]; wherea; b2I with a < b: If jf⁰j^q is a convex on [a; b]; q 1; then the following inequality holds:

1

6 f(a) + 4f a+b

2 +f(b) 1

b a

Z b a

f(x)dx (2.8)

b a

72 (5)^{1 1q}

( 61jf⁰(b)j^q+ 29jf⁰(a)j^q 18

1 q

+ 61jf⁰(a)j^q+ 29jf⁰(b)j^q 18

1 q)

:

Proof. From Lemma 1 and by power mean inequality; we get 1

6 f(a) + 4f a+b

2 +f(b) 1

b a

Z b a

f(x)dx

b a

2 Z 1

0

t 2

1

3 f⁰ 1 +t

2 b+1 t 2 a + 1

3 t

2 f⁰ 1 +t

2 a+1 t 2 b dt

b a

2

( Z 1 0

t 2

1 3 dt

1 ¹_q Z 1 0

t 2

1

3 f⁰ 1 +t

2 b+1 t 2 a

q

dt

1 q

+ Z 1

0

1 3

t 2

1 ¹_q Z 1 0

1 3

t

2 f⁰ 1 +t

2 a+1 t 2 b

q

dt

1 q)

: Sincejf⁰j^q is a convex on[a; b];we know that fort2[0;1]

f⁰ 1 +t

2 b+1 t 2 a

q 1 +t

2 jf⁰(b)j^q+1 t

2 jf⁰(a)j^q; hence

1

6 f(a) + 4f a+b

2 +f(b) 1

b a

Z b a

f(x)dx

b a

2

( Z 1 0

t 2

1 3 dt

1 ¹_q

Z 1 0

t 2

1 3

1 +t

2 jf⁰(b)j^q+1 t

2 jf⁰(a)j^q dt

1 q

+ Z 1

0

1 3

t 2

1 ¹_q Z 1 0

1 3

t 2

1 +t

2 jf⁰(a)j^q+1 t

2 jf⁰(b)j^q dt

1 q)

= b a

2 5 36

1 ¹_q

( 61

648jf⁰(b)j^q+ 29

648jf⁰(a)j^q

1 q

+ 61

648jf⁰(a)j^q+ 29

648jf⁰(b)j^q

1 q)

= b a

72 (5)^{1 1q}

( 61jf⁰(b)j^q+ 29jf⁰(a)j^q 18

1 q

+ 61jf⁰(a)j^q+ 29jf⁰(b)j^q 18

1 q)

: By simple computation,

Z 1 0

t 2

1 3

1 +t

2 jf⁰(b)j^q+1 t

2 jf⁰(a)j^q dt= 61

648jf⁰(b)j^q+ 29

648jf⁰(a)j^q Z 1

0

1 3

t 2

1 +t

2 jf⁰(a)j^q+1 t

2 jf⁰(b)j^q dt= 61

648jf⁰(a)j^q+ 29

648jf⁰(b)j^q

Z 1 0

t 2

1 3 = 5

36 Thus, we get (2.8) which completes the proof.

Corollary 3. In Theorem 6, iff(a) =f(^a+b₂ ) =f(b), then we have 1

b a

Z b a

f(x)dx f a+b 2

b a

72 (5)^{1 1q}

( 61jf⁰(b)j^q+ 29jf⁰(a)j^q 18

1 q

+ 61jf⁰(a)j^q+ 29jf⁰(b)j^q 18

1 q)

:

3. Applications to Special Means We shall consider the following special means:

(a) The arithmetic mean: A=A(a; b) :=a+b

2 ; a; b 0;

(b) The harmonic mean:

H=H(a; b) := 2ab

a+b; a; b >0;

(c) The logarithmic mean:

L=L(a; b) :=

8<

:

a if a=b

b a

lnb lna if a6=b

, a; b >0;

(d) Thep logarithmic mean

L_p=L_p(a; b) :=

8>

<

>:

hb^p+1 a^p+1 (p+1)(b a)

i¹_p

if a6=b

a if a=b

, p2R f 1;0g; a; b >0.

It is well known that Lp is monotonic nondecreasing over p2 Rwith L 1 :=L andL0:=I:In particular, we have the following inequalities

H L A:

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

Proposition 1. Let a; b2R,0< a < bandn2N,n 2: Then, we have 1

3A(aⁿ; bⁿ) +2

3Aⁿ(a; b) Lⁿ_n(a; b) n5 (b a)

36 A(a^{n 1}; b^{n 1}):

Proof. The assertion follows from Theorem 5 applied forf(x) =xⁿ; x2[a; b]and n2N:

Proposition 2. Let a; b2R,0< a < b:Then, for all p >1, we have 1

3H ¹(a; b) +2

3A ¹(a; b) L ¹(a; b) (b a)

12a²b²

1 + 2^p+1 3(p+ 1)

1 p(

3a^2q+b^2q 4

1 q

+ a^2q+ 3b^2q 4

1 q)

: Proof. The assertion follows from Theorem 6 applied forf(x) = 1=x; x2[a; b]: Proposition 3. Let a; b2R,0< a < bandn2N,n 2:Then, for allq >1, we have

1

3A(aⁿ; bⁿ) +2

3Aⁿ(a; b) Lⁿ_n(a; b) n5^{1 1q}(b a)

72

( 29a^{(n 1)q}+ 61b^{(n 1)q} 18

1 q

+ 61a^{(n 1)q}+ 29b^{(n 1)q} 18

1 q)

: Proof. The assertion follows from Theorem 7 applied forf(x) =xⁿ; x2[a; b]and n2N:

References

[1] M. Alomari, M. Darus and S.S. Dragomir, New inequalities of Simpson’s type for s- convex functions with applications,RGMIA Res. Rep. Coll., 12 (4) (2009), Article 9. [On- line:http://www.sta¤.vu.edu.au/RGMIA/v12n4.asp]

[2] S.S. Dragomir, R.P. Agarwal and P. Cerone, On Simpson’s inequality and applications,J. of Inequal. Appl., 5(2000), 533-579.

[3] S. Hussain, M.I. Bhatti and M. Iqbal, Hadamard-type inequalities for s-convex functions I, Punjab Univ. Jour. of Math., Vol.41, pp:51-60, (2009).

[4] U.S. K¬rmac¬, Inequalities for di¤erentiable mappings and applications to special means of real numbers and to midpoint formula,Appl. Math. Comp.,147 (2004), 137-146.

[5] D.A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex func- tions,Annals of University of Craiova Math. Comp. Sci. Ser., 34 (2007), 82-87.

[6] B.Z. Liu, An inequality of Simpson type,Proc. R. Soc. A, 461 (2005), 2155-2158.

[7] J. Peµcari´c, F. Proschan and Y.L. Tong, Convex functions, partial ordering and statistical applications,Academic Press, New York, 1991.

|Department of Mathematics,Faculty of Science and Arts, Afyon Kocatepe Univer- sity, Afyon, Turkey

E-mail address: sarikaya@aku.edu.tr, sarikayamz@gmail.com

Atatürk University, K.K. Education Faculty, Department of Mathematics, 25240, Campus, Erzurum, Turkey

E-mail address: erhanset@yahoo.com

Graduate School of Natural and Applied Sciences, A^G¼r¬·Ibrahim Çeçen University, A^G¼r¬, Turkey

E-mail address: emos@atauni.edu.tr