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volume 5, issue 3, article 66, 2004.

Received 28 May, 2003;

accepted 12 April, 2004.

Communicated by:C.E.M. Pearce

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Journal of Inequalities in Pure and Applied Mathematics

IMPROVEMENTS OF EULER-TRAPEZOIDAL TYPE INEQUALITIES WITH HIGHER-ORDER CONVEXITY AND APPLICATIONS

DAH-YAN HWANG

Department of General Education Kuang Wu Institute of Technology Peito, Taipei, Taiwan 11271, R.O.C.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 072-03

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Improvements of Euler-Trapezoidal Type Inequalities with Higher-Order

Convexity and Applications Dah-Yan Hwang

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J. Ineq. Pure and Appl. Math. 5(3) Art. 66, 2004

http://jipam.vu.edu.au

Abstract

We improve a bounds relating to Euler’s formula for the case of a function with higher-order convexity properties. These are used to improve estimates of the error involved in the use of the trapezoidal formula for integrating such a function.

2000 Mathematics Subject Classification:26D15, 26D20.

Key words: Hadamard inequality, Euler formula, Convex functions, Integral inequali- ties, Numerical integration.

1. Introduction

Let f : I ⊆ R → R be a convex mapping defined on the interval I of real numbers anda, b∈I witha < b. The following inequality:

(1.1) f

a+b 2

≤ 1

b−a Z b

a

f(x)dx≤ f(a) +f(b) 2

is known in the literature as Hadamard’s inequality for convex mappings. Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the mappings f. Over the last decade this pair of inequalities (1.1) have been improved and extended in a number of ways, including the derivation of estimates of the differences between the two sides of each inequality.

In [4], Dragomir and Agarwal have made use of the latter to derive bounds for the error term in the trapezoidal formula for the numerical integration of an integrable functionf such that|f⁰|^qis convex for someq ≥ 1. Some improvements of their result have been derived in [6] and [7].

Recently, Lj Dedic et al. [3, Theorem 2 forr = 1] establish the following basic result which was obtained for the difference between the two side of the right-hand Hadamard inequality.

Supposef : [a, b]→Ris a real-valued twice differentiable function. If|f⁰⁰|^q is convex for someq≥1, then

(1.2)

Z b a

f(t)dt− b−a

2 [f(a) +f(b)]

≤ (b−a)³ 12

|f⁰⁰(a)|^q+|f⁰⁰(b)|^q 2

¹_q .

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If|f⁰⁰|^qis concave, then

(1.3)

Z b a

f(t)dt−b−a

2 [f(a) +f(b)]

≤ (b−a)³ 12

f⁰⁰

a+b 2

. In this paper, using Euler-trapezoidal formula, we shall generalize and improve the inequalities (1.2) and (1.3). Also, we apply the result to obtain a better estimates of the error in the trapezoidal formula.

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2. The Euler-Trapezoidal Formula and Some Identities

In what follows, let B_n(t), n ≥ 0 be the Bernoulli polynomials and B_n = Bn(0),n≥0, the Bernoulli numbers. The first few Bernoulli polynomials are B₀(t) = 1, B₁(t) =t− 1

2, B₂(t) = t²−t+ 1

6, B₃(t) =t³− 3 2t²+ 1

2t, B₄(t) =t⁴−2t³+t²− 1

30

and the first few Bernoulli numbers are

B₀ = 1, B₁ =−1

2, B₂ = 1

6, B₃ = 0, B₄ =− 1

30, B₅ = 0.

For some details on the Bernoulli polynomials and the Bernoulli numbers, see for example [1,5].

The relevant key properties of the Bernoulli polynomials are B_n⁰(t) =nBn−1(t), (n≥1)

B_n(1 +t)−B_n(t) = ntⁿ⁻¹, (n≥0)

(See for example, [1, Chapter 23]). LetP_2n(t) = [B_n(t)−B_n]/n!. We note that P_2n(t),P_2n(₂^t)andP_2n(1− ^t₂)do not change sign on(0.1).

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By [2, p. 274], we have the following Euler-trapezoidal formula:

(2.1) Z b

a

f(x)dx= b−a

2 [f(a) +f(b)]

−

r−1

X

k=1

(b−a)^2kB2k

(2k)!

f^(2k−1)(b)−f^(2k−1)(a) + (b−a)^2r+1

Z 1 0

P_2r(t)f^(2r)(a+t(b−a))dt.

and by [2, p. 275], we have

(2.2)

Z a+nh a

f(x)dx=T(f;h)

−

r−1

X

k=1

B2kh^2k

(2k)! [f^(2k−1)(a+nh)−f^(2k−1)(a)]

+h^2r+1

n−1

X

k=0

Z 1 0

P2r(t)f^(2r)(a+h(t+k))dt,

where

T(f;h) =h

"

1

2f(a) +

n−1

X

k=1

f(a+nh) + 1

2f(a+nh)

#

is estimates of integralRa+h

a f(x)dx.

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In this article, we adopt the terminology that f is (j + 2)-convex if f^(j) is convex, so ordinary convexity is two-convexity. A corresponding definition applies for(j+ 2)-concavity.

For our results, we need the following identities. By B2j+1 1 2

= 0 for j ≥ 0; B_j = B_j(1) for j ≥ 0 and integrating by parts, we have that the following identities hold:

a_I(r) = Z 1

0

P_2r t

2

dt =− 2B2r+1

(2r+ 1)! − B2r

(2r)!, (I)

a_II(r) = Z 1

0

tP_2r t

2

dt =−4 B_2r+2 ¹₂

−B_2r+2

(2r+ 2)! − B_2r 2(2r)!, (II)

aIII(r) = Z 1

0

(1−t)P2r

t 2

dt (III)

= 4 B_2r+2 ¹₂

−B_2r+2

(2r+ 2)! − 2B_2r+1

(2r+ 1)! − B_2r 2(2r)!, a_IV(r) =

Z 1 0

P_2r

1− t 2

dt= 2B_2r+1

(2r+ 1)! − B_2r (2r)!, (IV)

a_V(r) = Z 1

0

tP_2r

1− t 2

dt =−4 B2r+2 1 2

−B2r+2

(2r+ 2)! − B2r

2(2r)!, (V)

a_VI(r) = Z 1

0

(1−t)P_2r

1− t 2

dt (VI)

=−4 B_2r+2 ¹₂

−B_2r+2

(2r+ 2)! + 2B2r+1

(2r+ 1)! − B2r

2(2r)!.

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3. Results

In the remainder of the paper we shall use the notation

I_r = (−1)^r Z b

a

f(x)dx−(b−a)

2 [f(a) +f(b)]

+

r−1

X

k=1

B_2k(b−a)^2k

(2k)! [f^(2k−1)(b)−f^(2k−1)(a)]

) , As above, the empty sum forr= 1is interpreted as zero.

Theorem 3.1. Supposef : [a, b]→Ris a real-valued(2r)-times differentiable function.

(a) If|f^(2r)|^qis convex forq ≥1, then (3.1) |I_r| ≤(b−a)^2r+1

1 2

¹_q

|B_2r| (2r)!

1−¹

q

(

|a_III(r)| · |f^(2r)(a)|^q

+ (|aII(r)|+|aV(r)|)·

f^(2r)

a+b 2

q

+|aVI(r)| · |f^(2r)(b)|^q )¹_q

. (b) If|f^(2r)|^qis concave forq ≥1, then

(3.2) |I_r| ≤ (b−a)^2r+1 2

× (

|aI(r)| ·

f^(2r) |a_III(r)| ·a+|a_II(r)| ·(^a+b₂ )

|a_I(r)|

!

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+|a_IV(r)| ·

f^(2r) |a_VI(r)| ·b+|a_V(r)| ·(^a+b₂ )

|aIV(r)|

!

) .

Proof. From (2.1) and the definition ofI_r, we have

|I_r| ≤(b−a)^2r+1 Z 1

0

|P_2r(t)f^(2r)(a+t(b−a))|dt

= (b−a)^2r Z b

a

P2r

x−a b−a

· |f^(2r)(x)|dx.

It follows from Hölder’s inequality that

(3.3) |I_r| ≤ Z b

a

P_2r

x−a b−a

dx ¹⁻¹_q

× Z b

a

P_2r

x−a b−a

· |f^(2r)(x)|^qdx ¹q

. Now,

Z b a

P_2r

x−a b−a

dx= (b−a)

Z 1 0

P_2r(t)dt (3.4)

= (b−a)|B_2r| (2r)! ,

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and, by the convexity of|f^(2r)|^q, we have Z b

a

P_2r

x−a b−a

· |f^(2r)(x)|^qdx (3.5)

= Z ^a+b₂

a

P_2r

x−a b−a

· |f^(2r)(x)|^q +

Z b

a+b 2

P_2r

x−a b−a

· |f^(2r)(x)|^qdx

= b−a 2

Z 1 0

P2r

t 2

·

f^(2r)

(1−t)a+t

a+b 2

q

dt +

Z 1 0

P_2r

1− t 2

·

f^(2r)

(1−t)b+t

a+b 2

q

dt

≤ b−a 2

Z 1 0

(1−t)P_2r t

2

dt

· |f^(2r)(a)|^q +

Z 1 0

tP_2r t

2

dt

·

f^(2r)

a+b 2

q

+

Z 1 0

(1−t)P_2r

1− t 2

dt

· |f^(2r)(b)|^q +

Z 1 0

P2r

1− t

2

dt

·

f^(2r)

a+b 2

q

Thus, by (3.3), (3.4) and (3.5) and the identities (II), (III), (V) and (VI), we have the inequality (3.1).

On the other hand, since |f^(2r)|^q is concave implies that |f^(2r)| is concave

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and by Jensen’s integral inequality, we have

|I_r| ≤(b−a)^2r

"

Z ^a+b₂

a

P_2r

x−a b−a

· |f^(2r)(x)|dx

+ Z b

a+b 2

P_2r

x−a b−a

· |f^(2r)(x)|dx

#

= (b−a)^2r+1 Z 1

0

P_2r t

2

·

f^(2r)

(1−t)a+t

a+b 2

dt +

Z 1 0

P_2r

1− t 2

·

f^(2r)

(1−t)b+t

a+b 2

≤ (b−a)^2r+1 2

Z 1 0

P_2r t

2

dt

×

f^(2r)





R1

0 P_2r ₂^t

((1−t)a+t ^a+b₂ )dt

R1 0 P2r t

2

dt



 +

Z 1 0

P_2r

1− t 2

dt

×

f^(2r)





R1

0 P_2r 1− ₂^t

(1−t)b+t ^a+b₂ dt

R1

0 P2r 1− ₂^t dt







. Thus, by identities I, II, III, IV, V and VI, we have the inequality (3.2).

Forr= 1in Theorem3.1, we have the following corollary.

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Corollary 3.2. Under the assumptions of Theorem3.1. Iff is a4-convex func- tion, we have

(3.6)

Z b a

f(x)dx− b−a

2 [f(a) +f(b)]

≤ (b−a)³ 12

"

3|f⁰⁰(a)|^q+ 10

f⁰⁰ ^a+b₂

q+ 3|f⁰⁰(b)|^q 16

#¹_q

and iff is a4-concave function, we have (3.7)

Z b a

f(x)dx− b−a

2 [f(a) +f(b)]

≤ (b−a)³ 12

"

f⁰⁰ ^11a+5b₁₆ +

f⁰⁰ ^5a+11b₁₆ 2

# . Remark 3.1. Using the convexity of|f⁰⁰|^q, we have

f⁰⁰

a+b 2

q

≤ |f⁰⁰(a)|^q+|f⁰⁰(b)|^q

2 ,

Hence inequality (3.6) is an improvement of inequality (1.2).

Since|f⁰⁰|^q is concave implies that|f⁰⁰|is concave, we have 1

2

f⁰⁰

11a+ 5b 16

+

f⁰⁰

5a+ 11b 16

≤

f⁰⁰

a+b 2

. Thus inequality (3.7) is an improvement of inequality (1.3).

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4. Application

To obtain estimates of the error in the trapezoidal formula, we apply the results of the previous section on each interval from the subdivision

[a, a+b], [a+h, a+ 2h], . . . ,[a+ (n−1)h, a+nh].

We define

J_r =

Z a+nh a

f(x)dx−T(f;h) +

r−1

X

k=1

B_2kh^2k

(2k)! [f^(2k−1)(a+nh)−f^(2k−1)(a)].

Theorem 4.1. If f : [a, a +nh] → R is a (2r)-time differentiable function (r ≥1).

(a) If|f^(2r)|^qis convex for someq≥1, then

|J2r| ≤h^2r+1 1

2

¹_q |B_2r| (2r)!

1−¹_q

×

n

X

m=1

(

|a_III(r)| · |f^(2r)(a+ (m−1)h)|^q + (|a_II(r)|+|a_V(r)|)·

f^(2r)

a+

m− 1 2

h

q

+|a_VI(r)| · |f^(2r)(a+mh)|^q )¹_q

.

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(b) If|f^(2r)|^qis concave for someq≥1, then

|J_2r| ≤ h^2r+1 2

n

X

m=1

|a_I(r)|

×

f^(2r)

|a_III(r)|(a+ (m−1·h) +|a_II(r)|(a+ (m− ¹₂)·h)

|a_I(r)|

+|a_IV(r)| ·

f^(2r)

|a_VI(r)|(a+mh) +|a_II(r)| ·(a+ (m− ¹₂)h)

|a_IV(r)|

. Proof. Since

|Jr| ≤

n

X

m=1

Z a+mh a+(m−1)h

f(x)dx− h

2[f(a+ (m−1)h) +f(a+mh)]

+

r−1

X

k=1

B_2kh^2k

(2k)! [f^(2k−1)(a+mh)−f^(2k−1)(a+ (m−1)h)]

) , we have

|J_r| ≤

n

X

m=1

h^2r+1 1

2 ¹_q

|B2r| (2r)!

1−¹_q (

|a_III(r)| · |f^(2r)(a+ (m−1)h)|^q + (|aII(r)|+|aV(r)|)·

f^(2r)

a+

m−1 2

h

q

+|aII(r)| · |f^(2r)(a+mh)|^q ¹_q

,

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by Theorem3.1(a) applied to each interval[a+ (m−1)h, a+mh]. Obviously, the proof (a) is complete.

The proof (b) is similar.

Remark 4.1. As the same discussion in the Section3, Theorem4.1forr = 1is an improvement of Theorem 4 forr= 1in [3].

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References

[1] M. ABRAMOWITZANDI.A. STEGUN, Handbook of Mathematical Func- tions, Dover, New York, 1965.

[2] I.S. BEREZINANDN.P. ZHIDKOV, Computing Methods, Vol. I, Pergamon Press, Oxford, 1965.

[3] LJ. DEDIC, C.E.M. PEARCEANDJ. PE ˇCARI ´C, Hadamard and Dragomir- Agarwal inequalities, Higher-order convexity and the Euler formula, J. Ko- rean Math., 38(6) (2001), 1235–1243.

[4] S.S. DRAGOMIR AND R.P. AGARWAL, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5)(1998), 91–95.

[5] V.I. KRYLOV, Approximate Calculation of Integrals, Macmillan, New York, 1962.

[6] C.E.M. PEARCE AND J. PE ˇCARI ´C, Inequalities for differentiable map- ping with application to special means and quadrature, Appl. Math. Lett., 13 (2000), 51–55.

[7] GAU-SHENG YANG, DAH-YAN HWANG AND KUEI-LIN TSENG, Some inequalities for differentiable convex and concave mappings, Com- puter and Mathematics with Applications, 47 (2004), 207–216.