TWO NEW MAPPINGS ASSOCIATED WITH INEQUALITIES OF HADAMARD-TYPE FOR CONVEX FUNCTIONS

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Mappings Associated with Inequalities of Hadamard-type

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TWO NEW MAPPINGS ASSOCIATED WITH INEQUALITIES OF HADAMARD-TYPE FOR

CONVEX FUNCTIONS

LAN HE

Department of Mathematics and Physics Chongqing Institute of Science and Technology Xingsheng Lu 4, YangjiaPing 400050

Chongqing City, China.

EMail:helan0505@163.com

Received: 18 April, 2008

Accepted: 23 April, 2009

Communicated by: S.S. Dragomir

2000 AMS Sub. Class.: Primary 26D07; Secondary 26B25, 26D15.

Key words: Convex function, Monotonicity, Integral inequality, Refinement.

Abstract: In this paper, we define two mappings associated with the Hadamard inequality, investigate their main properties and give some refinements.

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1. Introduction

Letf,−g : [a, b]→ Rboth be continuous functions. Iff is a convex function, then we have

(1.1) f

a+b 2

≤ 1 b−a

Z b

a

f(t)dt.

The inequality (1.1) is well known as the Hadamard inequality (see [1] – [6]). For some recent results which generalize, improve, and extend this classical inequality, see the references of [3].

Whenf,−g both are convex functions satisfyingRb

a g(x)dx >0andf(^a+b₂ )≥0, S.-J. Yang in [7] generalized (1.1) as

(1.2) f ^a+b₂

g ^a+b₂ ≤ Rb

a f(t)dt Rb

a g(t)dt.

To go further in exploring (1.2), we define two mappingsLandF byL: [a, b]× [a, b]7→R,

L(x, y;f, g) = Z y

x

f(t)dt−(y−x)f

x+y

2 (y−x)g

x+y 2

− Z y

x

g(t)dt

andF : [a, b]×[a, b]7→R, F(x, y;f, g) = g

x+y 2

Z y

x

f(t)dt−f

x+y 2

Z y

x

g(t)dt.

The aim of this paper is to study the properties ofLandF and obtain some new refinements of (1.2).

To prove the theorems of this paper we need the following lemma.

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Lemma 1.1. Letf be a convex function on[a, b]. The mappingHis defined as H(x, y;f) =

Z y

x

f(t)dt−(y−x)f

x+y 2

.

Then H(a, y;f) is nonnegative and monotonically increasing with y on [a, b] (see [8]), H(x, b;f) is nonnegative and monotonically decreasing with x on [a, b] (see [9]).

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2. Main Results

The properties ofLare embodied in the following theorem.

Theorem 2.1. Letf and−gboth be convex functions on[a, b]. Then we have:

1. L(a, y;f, g)is nonnegative increasing withyon[a, b],L(x, b;f, g)is nonnega- tive decreasing withxon[a, b].

2. When Rb

a g(x)dx > 0and f ^a+b₂

≥ 0, for anyx, y ∈ (a, b)andα ≥ 0 and β ≥0such thatα+β = 1, we have the following refinement of (1.2)

f ^a+b₂

g ^a+b₂ ≤ (b−a)f ^a+b₂ 2Rb

ag(t)dt + Rb

af(t)dt 2(b−a)g ^a+b₂ (2.1)

≤ (b−a)f ^a+b₂ 2Rb

ag(t)dt + Rb

af(t)dt 2(b−a)g ^a+b₂ + αL(a, y;f, g) +βL(x, b;f, g)

2(b−a)g ^a+b₂ Rb a g(t)dt

≤ Rb

af(t)dt 2Rb

a g(t)dt + 2f ^a+b₂ 2g ^a+b₂ ≤

Rb a f(t)dt Rb

a g(t)dt. The main properties ofF are given in the following theorem.

Theorem 2.2. Letfand−gboth be nonnegative convex functions on[a, b]satisfying Rb

a g(x)dx >0. Then we have the following two results:

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1. If f and −g both are increasing, then F(a, y;f, g) is nonnegative increasing withyon[a, b], and we have the following refinement of (1.2)

(2.2) f ^a+b₂

g ^a+b₂ ≤ f ^a+b₂

g ^a+b₂ + F(a, y;f, g) g ^a+b₂ Rb

a g(t)dt

≤ Rb

a f(t)dt Rb

a g(t)dt ,

wherey∈(a, b).

2. If f and −g both are decreasing, thenF(x, b;f, g)is nonnegative decreasing withxon[a, b], and we have the following refinement of (1.2)

(2.3) f ^a+b₂

g ^a+b₂ ≤ f ^a+b₂

g ^a+b₂ + F(x, b;f, g) g ^a+b₂ Rb

a g(t)dt ≤ Rb

a f(t)dt Rb

a g(t)dt, wherex∈(a, b).

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3. Proof of Theorems

Proof of Theorem2.1.

(1) By Lemma1.1 and the convexity off and−g, it is obvious thatH(a, y;f)and H(a, y;−g)both are nonnegative increasing with yon [a, b]. Then L(a, y;f, g) = H(a, y;f)H(a, y;−g)is nonnegative increasing withyon[a, b]. By the same argu- ments of proof for L(a, y;f, g), we can also prove that L(x, b;f, g) is nonnegative decreasing withxon[a, b].

(2) SinceH(a, y;f)is monotonically increasing withy on[a, b], for anyy ∈ (a, b) andα≥0, we have

(3.1) 0 = αL(a, a;f, g)≤αL(a, y;f, g)≤αL(a, b;f, g).

AsH(x, b;f) is monotonically decreasing with x on[a, b], for anyx ∈ (a, b) and β ≥0, we have

(3.2) 0 = βL(a, a;f, g)≤βL(x, b;f, g)≤βL(a, b;f, g).

Whenα+β = 1, expression (3.1) plus (3.2) yields

(3.3) 0 =L(a, a;f, g)≤αL(a, y;f, g) +βL(x, b;f, g)≤L(a, b;f, g).

Expression (3.3) plus (b−a)²f

a+b 2

g

a+b 2

+

Z b

a

f(t)dt Z b

a

g(t)dt yields

(b−a)²f

a+b 2

g

a+b 2

+

Z b

a

f(t)dt Z b

a

g(t)dt (3.4)

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≤(b−a)²f

a+b 2

g

a+b 2

+

Z b

a

f(t)dt Z b

a

g(t)dt +αL(a, y;f, g) +βL(x, b;f, g)

≤(b−a)g

a+b 2

Z b

a

f(t)dt+ (b−a)f

a+b 2

Z b

a

g(t)dt.

By the convexity off andg,Rb

a g(x)dx >0,f ^a+b₂

≥0and (1.1), we get (3.5) (b−a)g

a+b 2

≥ Z b

a

g(t)dt >0, Z b

a

f(t)dt≥(b−a)f

a+b 2

≥0.

Using (3.5), we obtain (b−a)²f

a+b 2

g

a+b 2

+

Z b

a

f(t)dt Z b

a

g(t)dt (3.6)

≥(b−a)f

a+b 2

Z b

a

g(t)dt+ (b−a)f

a+b 2

Z b

a

g(t)dt

= 2(b−a)f

a+b 2

Z b

g(t)dt

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Combining (3.4), (3.6) and (3.7), and dividing the combined formula by 2(b−a)g

a+b 2

Z b

a

g(t)dt yields (2.1).

This completes the proof of Theorem2.1.

Proof of Theorem2.2.

(1) By Lemma1.1 and the convexity of f and −g, we can see thatH(a, y;f) and H(a, y;−g)both are nonnegative increasing withy on[a, b]. From the nonnegative increasing properties off andg, we get that

F(a, y;f, g) =g

a+y 2

Z y

a

f(t)dt−f

a+y 2

Z y

a

g(t)dt

=g

a+y 2

Z y

a

f(t)dt−(y−a)f

a+y 2

+f

a+y 2

Z y

a

g(t)dt−(y−a)g

a+y 2

=g

a+y 2

·H(a, y;f) +f

a+y 2

·H(a, y;−g) is nonnegative increasing withyon[a, b].

SinceF(a, y;f, g)is monotonically increasing withyon[a, b], for anyy∈(a, b), we have

(3.8) 0 = F(a, a;f, g)≤F(a, y;f, g)≤F(a, b;f, g).

Expression (3.8) plus

f

a+b 2

Z b

a

g(t)dt

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yields f

a+b 2

Z b

a

g(t)dt≤f

a+b 2

Z b

a

g(t)dt+F(a, y;f, g) (3.9)

≤f

a+b 2

Z b

a

g(t)dt+F(a, b;f, g)

=g

a+b 2

Z b

a

f(t)dt.

Expression (3.9) divided by g

a+b 2

Z b

a

g(t)dt yields (2.2).

(2) By Lemma1.1 and the convexity of f and−g, we can see that H(x, b;f) and H(x, b;−g)are both nonnegative decreasing withxon[a, b]. Further, from the nonnegative decreasing properties off andg, we obtain that

F(x, b;f, g) =g

x+b 2

·H(x, b;f) +f

x+b 2

·H(x, b;−g)

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References

[1] J. HADAMARD, Etude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann, J. Math. Pures Appl., 58 (1893), 171–

215.

[2] L.-C. WANG, Three mapping related to Hermite-Hadamard inequalities, J.

Sichuan Univ., 39 (2002), 652–656. (In Chinese).

[3] S.S. DRAGOMIR, Y.J. CHOANDS.S. KIM, Inequalities of Hadamard’s type for Lipschitzian mappings and their applications, J. Math. Anal. Appl., 245 (2000), 489–501.

[4] G.-S. YANG AND K.-L. TSENG, Inequalities of Hadamard’s type for Lips- chitzian mappings, J. Math. Anal. Appl., 260 (2001), 230–238.

[5] M. MATICANDJ. PE ˇCARI ´C, Note on inequalities of Hadamard’s type for Lip- schitzian mappings, Tamkang J. Math., 32(2) (2001), 127–130.

[6] L.-C. WANG, Convex Functions and Their Inequalities, Sichuan University Press, Chengdu, China, 2001. (Chinese).

[7] S.-J. YANG, A direct proof and extensions of an inequality, J. Math. Res. Ex- posit., 24(4) (2004), 649–652.

[8] S.S. DRAGOMIR AND R.P. AGARWAL, Two new mappings associated with Hadamard’s inequalities for convex functions, Appl. Math. Lett., 11(3) (1998), 33–38.

[9] L.-C. WANG, Some refinements of Hermite-Hadamard inequalities for convex functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 15 (2004), 40–45.