Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011Β (2011), Article IDΒ 594758, 9 pages
http://dx.doi.org/10.5402/2011/594758
Research Article

A Family of Mappings Associated with Hadamard's Inequality on a Hypercube

School of Mathematics and Physics, Changzhou University, Changzhou 213164, China

Received 24 December 2010; Accepted 16 January 2011

Academic Editors: V.Β Kravchenko and S.Β Pilipovic

Copyright Β© 2011 FuLi Wang. 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.

Abstract

This paper provides a method to generalize and refine the Hadamard's inequality on a hypercube in 𝑛-dimensional Euclidean space by establishing a family of mappings.

1. Introduction

For a convex function 𝑓 defined on an open interval 𝐼 of real numbers and π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, then the following inequality: π‘“ξ‚€π‘Ž+𝑏2≀1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“(π‘₯)𝑑π‘₯≀𝑓(π‘Ž)+𝑓(𝑏)2(1.1) is known as the classical Hadamard’s inequality for the convex function.

In [1], Dragomir established the following mapping:[]1𝐻∢0,1βŸΆπ‘,𝐻(𝑑)=ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“ξ‚€π‘‘π‘₯+(1βˆ’π‘‘)π‘Ž+𝑏2𝑑π‘₯,(1.2) then the mapping 𝐻 is convex, increasing on [0,1], and for all π‘‘βˆˆ[0,1], we have π‘“ξ‚€π‘Ž+𝑏21=𝐻(0)≀𝐻(𝑑)≀𝐻(1)=ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“(π‘₯)𝑑π‘₯,(1.3) which is a refinement to the left side of (1.1).

In [2], Yang and Hong established the following mapping:[]1𝐺∢0,1βŸΆπ‘,𝐺(𝑑)=ξ€œ2(π‘βˆ’π‘Ž)π‘π‘Žξ‚ƒπ‘“ξ‚€1+𝑑2π‘Ž+1βˆ’π‘‘2π‘₯+𝑓1+𝑑2𝑏+1βˆ’π‘‘2π‘₯𝑑π‘₯,(1.4) then the mapping 𝐺 is convex, increasing on [0,1], and for all π‘‘βˆˆ[0,1], we have 1ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“(π‘₯)𝑑π‘₯=𝐺(0)≀𝐺(𝑑)≀𝐺(1)=𝑓(π‘Ž)+𝑓(𝑏)2,(1.5) which is a refinement to the right side of (1.1).

In this paper, we first establish a mapping in Section 2, defined by 𝐴1∢[]0,1βŸΆπ‘,𝐴11(𝑑)=2𝑓1+𝑑2π‘Ž+1βˆ’π‘‘2𝑏+𝑓1βˆ’π‘‘2π‘Ž+1+𝑑2𝑏,(1.6) then the mapping 𝐴1 is convex, increasing on [0,1], and for all π‘‘βˆˆ[0,1] we have π‘“ξ‚€π‘Ž+𝑏2=𝐴1(0)≀𝐴1(𝑑)≀𝐴1(1)=𝑓(π‘Ž)+𝑓(𝑏)2.(1.7) Furthermore, there is a point πœ‰βˆˆ(0,1) so that 𝐴1∫(πœ‰)=(1/(π‘βˆ’π‘Ž))π‘π‘Žπ‘“(π‘₯). Thus, (1.1) has been proved based on the properties of 𝐴1, and the inequality (1.7) is a refinement to the both sides of (1.1).

In Section 3, we also use an analogue of (1.6) to obtain the following Hadamard’s type inequality on a square: 1𝑓(π‘Ž,𝑏)β‰€β„Ž2|π‘₯βˆ’π‘Ž|β‰€β„Ž/2|π‘¦βˆ’π‘|β‰€β„Ž/21𝑓(π‘₯,𝑦)𝑑π‘₯π‘‘π‘¦β‰€ξ€Ÿ4β„Ž|π‘₯βˆ’π‘Ž|=β„Ž/2|π‘¦βˆ’π‘|=β„Ž/2𝑓(π‘₯,𝑦)𝑑𝑠,(1.8) which is a generalization of (1.1) to two-dimensional Euclidean space.

In Section 4, by establishing another analogue of (1.6) on a hypercube, the following Hadamard’s inequality is also pointed out: 1𝑓(𝑐)β‰€β„Žπ‘›ξ€œβ€–π‘₯βˆ’π‘β€–βˆžβ‰€β„Ž/21𝑓(π‘₯)𝑑π‘₯≀2π‘›β„Žπ‘›βˆ’1ξ€Ÿ||π‘₯{π‘₯βˆΆπ‘–βˆ’π‘π‘–||=β„Ž/2}𝑓(π‘₯)𝑑𝑠,(1.9) which is a generalization of (1.1) to 𝑛-dimensional Euclidean space.

All the above Hadamard’s inequalities (1.1), (1.8), and (1.9) have a uniform format 1𝑓(𝑐)β‰€ξ€œVol(Ξ©)Ξ©1𝑓(π‘₯)𝑑π‘₯β‰€ξ€ŸArea(Ξ©)πœ•Ξ©π‘“(π‘₯)𝑑𝑠.(1.10)

Thus, by establishing a family of the mappings, we have provided a method to get Hadamard’s inequality of the convex function on a hypercube in 𝑛-dimensional Euclidean space. The motivation for the present work is from many generalizations and refinements of the classical inequality (1.1) in [1–8].

2. Properties of the Mapping

The main properties of the mapping 𝐴1 defined by (1.6) are embodied in the following theorem.

Theorem 2.1. If 𝑓 is a convex function defined on an open interval 𝐼 of real numbers, π‘Ž,π‘βˆˆπΌ with π‘Ž<𝑏, and the mapping 𝐴1 is defined by (1.6), then, one has (i)the mapping 𝐴1 is convex on [0,1],(ii)one has the bounds: inf[]π‘‘βˆˆ0,1𝐴1(𝑑)=𝐴1ξ‚€(0)=π‘“π‘Ž+𝑏2,sup[]π‘‘βˆˆ0,1𝐴1(𝑑)=𝐴1(1)=𝑓(π‘Ž)+𝑓(𝑏)2,(2.1)(iii)the mapping 𝐴1 is increasing on [0,1],(iv)there is a point πœ‰βˆˆ(0,1) so that 𝐴11(πœ‰)=ξ€œπ‘βˆ’π‘Žπ‘π‘Žπ‘“(π‘₯)𝑑π‘₯,(2.2)(v)one obtains, the classical inequality (1.1) and has a refinement of (1.1): 𝐴1(0)≀𝐴1(πœ‚)≀𝐴1(πœ‰)≀𝐴1(πœ‡)≀𝐴1(1),βˆ€πœ‚βˆˆ(0,πœ‰),πœ‡βˆˆ(πœ‰,1).(2.3)

Proof. (i) Let 𝑑1,𝑑2∈[0,1] and 𝛼,𝛽β‰₯0 with 𝛼+𝛽=1, then we have 𝐴1𝛼𝑑1+𝛽𝑑2ξ€Έ=12𝑓1+𝛼𝑑1+𝛽𝑑22π‘Ž+1βˆ’π›Όπ‘‘1βˆ’π›½π‘‘22𝑏+𝑓1βˆ’π›Όπ‘‘1βˆ’π›½π‘‘22π‘Ž+1+𝛼𝑑1+𝛽𝑑22𝑏=1ξ‚Άξ‚Ή2𝑓𝛼1+𝑑12π‘Ž+1βˆ’π‘‘12𝑏+𝛽1+𝑑22π‘Ž+1βˆ’π‘‘22𝑏𝛼+𝑓1βˆ’π‘‘12π‘Ž+1+𝑑12𝑏+𝛽1βˆ’π‘‘22π‘Ž+1+𝑑22𝑏≀1ξ‚Άξ‚Ήξ‚Ό2𝛼𝑓1+𝑑12π‘Ž+1βˆ’π‘‘12𝑏+𝛽𝑓1+𝑑22π‘Ž+1βˆ’π‘‘22𝑏+𝛼𝑓1βˆ’π‘‘12π‘Ž+1+𝑑12𝑏+𝛽𝑓1βˆ’π‘‘22π‘Ž+1+𝑑22𝑏=1ξ‚Άξ‚Ή2𝛼𝑓1+𝑑12π‘Ž+1βˆ’π‘‘12𝑏+𝑓1βˆ’π‘‘12π‘Ž+1+𝑑12𝑏+1ξ‚Άξ‚Ή2𝛽𝑓1+𝑑22π‘Ž+1βˆ’π‘‘22𝑏+𝑓1βˆ’π‘‘22π‘Ž+1+𝑑22𝑏=𝛼𝐴1𝑑1ξ€Έ+𝛽𝐴1𝑑2ξ€Έ,(2.4) which proves the convexity of 𝐴1 on [0,1].
(ii) By the convexity of 𝑓 on the interval [π‘Ž,𝑏], we have𝐴1(1𝑑)≀21+𝑑2𝑓(π‘Ž)+1βˆ’π‘‘2𝑓(𝑏)+1βˆ’π‘‘2𝑓(π‘Ž)+1+𝑑2ξ‚„=𝑓(𝑏)𝑓(π‘Ž)+𝑓(𝑏)2=𝐴1(𝐴1),11(𝑑)β‰₯𝑓2ξ‚€1+𝑑2π‘Ž+1βˆ’π‘‘2𝑏+1βˆ’π‘‘2π‘Ž+1+𝑑2𝑏=π‘“π‘Ž+𝑏2=𝐴1(0),(2.5) the bounds (2.1) hold.
(iii) Let 0<𝑑1<𝑑2≀1. By the convexity of the mapping 𝐴1, we have𝐴1𝑑2ξ€Έβˆ’π΄1𝑑1𝑑2βˆ’π‘‘1β‰₯𝐴1𝑑1ξ€Έβˆ’π΄1(0)𝑑1β‰₯0.(2.6) Since we have proved that 𝐴1(𝑑1)β‰₯𝐴1(0) for all 𝑑1∈(0,1] in (ii), the monotonicity of 𝐴1 is established.
(iv) We define the function πœ™ on [0,1] byξ€œπœ™(𝑑)=((1βˆ’π‘‘)/2)π‘Ž+((1+𝑑)/2)𝑏((1+𝑑)/2)π‘Ž+((1βˆ’π‘‘)/2)π‘ξ€œπ‘“(π‘₯)𝑑π‘₯βˆ’π‘‘π‘π‘Žπ‘“(π‘₯)𝑑π‘₯,(2.7) then πœ™πœ™(0)=πœ™(1)=0,ξ…ž(𝑑)=(π‘βˆ’π‘Ž)𝐴1ξ€œ(𝑑)βˆ’π‘π‘Žπ‘“(π‘₯)𝑑π‘₯.(2.8) According to the Rolle’s mean-value theorem, there is a point πœ‰βˆˆ(0,1) so that 0=πœ™ξ…ž(πœ‰)=(π‘βˆ’π‘Ž)𝐴1ξ€œ(πœ‰)βˆ’π‘π‘Žπ‘“(π‘₯)𝑑π‘₯.(2.9) Thus, (2.2) holds.
(v) Considering the properties (ii), (iii), and (iv) above, we have proved the Hadamard’s inequality (1.1) and the inequality (2.3).

3. The Hadamard’s Inequality on a Square

We consider a convex function with two variables defined on a square in two-dimensional Euclidean space.

Theorem 3.1. If 𝑓 is a convex function defined on an open region DβŠ†π‘2, the square ξ‚†β„Ž(π‘₯,𝑦)∢|π‘₯βˆ’π‘Ž|≀2,||||β‰€β„Žπ‘¦βˆ’π‘2ξ‚‡βŠ†π·,(3.1) and the mapping A2 is defined by 𝐴2∢[]0,1βŸΆπ‘,𝐴2(1𝑑)=ξ€Ÿ4β„Ž|𝑒|=β„Ž/2|𝑣|=β„Ž/2𝑓(π‘Ž+𝑑𝑒,𝑏+𝑑𝑣)𝑑𝑠,(3.2) which is a parameter curvilinear integral with respect to arc length, then one has (i)the mapping 𝐴2 is convex on [0,1],(ii)one has the bounds infπ‘‘βˆˆ[0,1]𝐴2(𝑑)=𝐴2(0)=𝑓(π‘Ž,𝑏),sup[]π‘‘βˆˆ0,1𝐴2(𝑑)=𝐴21(1)=ξ€Ÿ4β„Ž|π‘₯βˆ’π‘Ž|=β„Ž/2|π‘¦βˆ’π‘|=β„Ž/2𝑓(π‘₯,𝑦)𝑑𝑠,(3.3)(iii)the mapping 𝐴2 is increasing on [0,1],(iv)there is a point πœ‰βˆˆ(0,1) so that 𝐴2(1πœ‰)=β„Ž2|π‘₯βˆ’π‘Ž|β‰€β„Ž/2|π‘¦βˆ’π‘|β‰€β„Ž/2𝑓(π‘₯,𝑦)𝑑π‘₯𝑑𝑦,(3.4)(v)one has the Hadamard’s type inequality (1.8) and a refinement 𝐴2(0)≀𝐴2(πœ‚)≀𝐴2(πœ‰)≀𝐴2(πœ‡)≀𝐴2(1),βˆ€πœ‚βˆˆ(0,πœ‰),πœ‡βˆˆ(πœ‰,1).(3.5)

Proof. For a fixed π‘‘βˆˆ[0,1], let 𝑒=𝑧 if |𝑣|=β„Ž/2, and 𝑣=𝑧 if |𝑒|=β„Ž/2, we have 𝐴2(1𝑑)=ξ€Ÿ4β„Ž|𝑒|=β„Ž/2|𝑣|=β„Ž/2=1𝑓(π‘Ž+𝑑𝑒,𝑏+𝑑𝑣)π‘‘π‘ ξ€œ4β„Žβ„Ž/2βˆ’β„Ž/2ξ‚ƒπ‘“ξ‚€π‘Ž+𝑑𝑧,π‘βˆ’π‘‘β„Ž2+π‘“π‘Ž+𝑑𝑧,𝑏+π‘‘β„Ž2+π‘“π‘Žβˆ’π‘‘β„Ž2,𝑏+𝑑𝑧+π‘“π‘Ž+π‘‘β„Ž2,𝑏+𝑑𝑧𝑑𝑧,(3.6) which indicates the mapping 𝐴2(𝑑) is continuous on [0,1].
(i) Let 𝑑1,𝑑2∈[0,1] and 𝛼,𝛽β‰₯0 with 𝛼+𝛽=1. By the convexity of 𝑓, we haveπ‘“ξƒ¬ξ€·π‘Ž+𝛼𝑑1+𝛽𝑑2𝑧,π‘βˆ’π›Όπ‘‘1+𝛽𝑑2ξ€Έβ„Ž2𝛼=π‘“π‘Ž+𝑑1𝑧+π›½π‘Ž+𝑑2𝑧𝑑,π›Όπ‘βˆ’1β„Ž2𝑑+π›½π‘βˆ’2β„Ž2ξ‚΅ξ‚Άξ‚Ήβ‰€π›Όπ‘“π‘Ž+𝑑1𝑑𝑧,π‘βˆ’1β„Ž2ξ‚Άξ‚΅+π›½π‘“π‘Ž+𝑑2𝑑𝑧,π‘βˆ’2β„Ž2ξ‚Ά.(3.7) Similarly, we have π‘“ξƒ¬ξ€·π‘Ž+𝛼𝑑1+𝛽𝑑2𝑧,𝑏+𝛼𝑑1+𝛽𝑑2ξ€Έβ„Ž2ξƒ­ξ‚΅β‰€π›Όπ‘“π‘Ž+𝑑1𝑑𝑧,𝑏+1β„Ž2ξ‚Άξ‚΅+π›½π‘“π‘Ž+𝑑2𝑑𝑧,𝑏+2β„Ž2ξ‚Ά,π‘“ξƒ¬ξ€·π‘Žβˆ’π›Όπ‘‘1+𝛽𝑑2ξ€Έβ„Ž2ξ€·,𝑏+𝛼𝑑1+𝛽𝑑2ξ€Έπ‘§ξƒ­ξ‚΅π‘‘β‰€π›Όπ‘“π‘Žβˆ’1β„Ž2,𝑏+𝑑1𝑧𝑑+π›½π‘“π‘Žβˆ’2β„Ž2,𝑏+𝑑2𝑧,π‘“ξƒ¬ξ€·π‘Ž+𝛼𝑑1+𝛽𝑑2ξ€Έβ„Ž2ξ€·,𝑏+𝛼𝑑1+𝛽𝑑2ξ€Έπ‘§ξƒ­ξ‚΅π‘‘β‰€π›Όπ‘“π‘Ž+1β„Ž2,𝑏+𝑑1𝑧𝑑+π›½π‘“π‘Ž+2β„Ž2,𝑏+𝑑2𝑧.(3.8) Thus, 𝐴2𝛼𝑑1+𝛽𝑑2≀𝛼𝐴2𝑑1ξ€Έ+𝛽𝐴2𝑑2ξ€Έ,(3.9) which proves the convexity of 𝐴2 on [0,1].
(ii) By the convexity of 𝑓 and (3.6), we have𝐴21(𝑑)β‰₯ξ‚΅ξ€œ2π‘‘β„Žπ‘Ž+π‘‘β„Ž/2π‘Žβˆ’π‘‘β„Ž/2π‘“ξ€œ(π‘₯,𝑏)𝑑π‘₯+𝑏+π‘‘β„Ž/2π‘βˆ’π‘‘β„Ž/2𝑓(π‘Ž,𝑦)𝑑𝑦β‰₯𝑓(π‘Ž,𝑏)=𝐴2(0),(3.10) the last inequality holds owing to (1.1); as the convexity of 𝐴2 we have 𝐴2(𝑑)≀𝑑𝐴2(1)+(1βˆ’π‘‘)𝐴2(0)≀𝐴2(1),(3.11) and note that 𝐴2(11)=ξ€Ÿ4β„Ž|𝑒|=β„Ž/2|𝑣|=β„Ž/21𝑓(π‘Ž+𝑒,𝑏+𝑣)𝑑𝑠=ξ€Ÿ4β„Ž|π‘₯βˆ’π‘Ž|=β„Ž/2|π‘¦βˆ’π‘|=β„Ž/2𝑓(π‘₯,𝑦)𝑑𝑠,(3.12) the bounds (3.3) hold.
(iii) The monotonicity of 𝐴2 follows as in the proof of Theorem 2.1, and we omit the detail.
(iv) We define the function πœ™ on [0,1] byξ€πœ™(𝑑)=|π‘₯βˆ’π‘Ž|β‰€π‘‘β„Ž/2|π‘¦βˆ’π‘|β‰€π‘‘β„Ž/2𝑓(π‘₯,𝑦)𝑑π‘₯π‘‘π‘¦βˆ’π‘‘2|π‘₯βˆ’π‘Ž|β‰€β„Ž/2|π‘¦βˆ’π‘|β‰€β„Ž/2𝑓(π‘₯,𝑦)𝑑π‘₯𝑑𝑦,(3.13) then πœ™πœ™(0)=πœ™(1)=0,ξ…žβ„Ž(𝑑)=2ξ‚΅ξ€œπ‘Ž+π‘‘β„Ž/2π‘Žβˆ’π‘‘β„Ž/2𝑓π‘₯,𝑏+π‘‘β„Ž2+𝑓π‘₯,π‘βˆ’π‘‘β„Ž2+ξ€œξ‚ξ‚„π‘‘π‘₯𝑏+π‘‘β„Ž/2π‘βˆ’π‘‘β„Ž/2ξ‚ƒπ‘“ξ‚€π‘Ž+π‘‘β„Ž2,𝑦+π‘“π‘Žβˆ’π‘‘β„Ž2,π‘¦ξ‚ξ‚„π‘‘π‘¦βˆ’2𝑑|π‘₯βˆ’π‘Ž|β‰€β„Ž/2|π‘¦βˆ’π‘|β‰€β„Ž/2=β„Žπ‘“(π‘₯,𝑦)𝑑π‘₯𝑑𝑦,2ξ€Ÿ|π‘₯βˆ’π‘Ž|=π‘‘β„Ž|π‘¦βˆ’π‘|=π‘‘β„Žξ€π‘“(π‘₯,𝑦)π‘‘π‘ βˆ’2𝑑|π‘₯βˆ’π‘Ž|β‰€β„Ž/2|π‘¦βˆ’π‘|β‰€β„Ž/2𝑓(π‘₯,𝑦)𝑑π‘₯𝑑𝑦.(3.14) Let π‘₯=π‘Ž+𝑑𝑒 and 𝑦=𝑏+𝑑𝑣, we have πœ™ξ…ž(𝑑)=π‘‘β„Ž2ξ€Ÿ|𝑒|=β„Ž/2|𝑣|=β„Ž/2𝑓(π‘Ž+𝑑𝑒,𝑏+𝑑𝑣)π‘‘π‘ βˆ’2𝑑|π‘₯βˆ’π‘Ž|β‰€β„Ž/2|π‘¦βˆ’π‘|β‰€β„Ž/2𝑓(π‘₯,𝑦)𝑑π‘₯𝑑𝑦=2π‘‘β„Ž2𝐴2(𝑑)βˆ’2𝑑|π‘₯βˆ’π‘Ž|β‰€β„Ž/2|π‘¦βˆ’π‘|β‰€β„Ž/2𝑓(π‘₯,𝑦)𝑑π‘₯𝑑𝑦.(3.15) According to the Rolle’s mean-value theorem, there is a point πœ‰βˆˆ(0,1) so that 0=πœ™ξ…ž(πœ‰)=2β„Ž2πœ‰π΄2(ξ€πœ‰)βˆ’2πœ‰|π‘₯βˆ’π‘Ž|β‰€β„Ž/2|π‘¦βˆ’π‘|β‰€β„Ž/2𝑓(π‘₯,𝑦)𝑑π‘₯𝑑𝑦,(3.16) thus, (3.4) holds.
(v) Considering the properties (ii), (iii), and (iv) above, we have proved the Hadamard’s type inequality (1.8) and the inequality (3.5).

4. Remark

In recent years, several authors have discussed the problem of extending the classical inequality (1.1) to a convex function 𝑓 on a general convex body Ξ© of 𝑛-dimensional space [6–9]. For the left side of (1.1) (i.e., Jensen’s inequality), the following inequality was showed in [6, 9] (also see [7]): 1𝑓(𝑐)β‰€ξ€œπœ‡(Ξ©)Ω𝑓(π‘₯)𝑑π‘₯,(4.1) where 𝑐 is the barycenter of Ξ©, and πœ‡(Ξ©) denotes 𝑛-dimensional Lebesgue’s measure of Ξ©.

It is a much more delicate problem to extend the right side of (1.1) on a general convex body of 𝑛-dimensional space [7]. On a closed hyperball of the 𝑛-dimensional space, de la Cal and CΓ‘rcamo [7] obtained the inequality 1ξ€œVol(𝐾)𝐾1𝑓(π‘₯)𝑑π‘₯≀Area(πΎβˆ—)ξ€ŸπΎβˆ—π‘“(π‘₯)𝑑𝑠,(4.2) where Vol(𝐾) denotes the volumes of 𝐾, and Area(πΎβˆ—) denotes the areas of the hypersphere πΎβˆ—, for 𝑛=2,3 which was early obtained by Dragomir [4, 5] based on Calculus.

When the convex function 𝑓 was supposed to be differentiable on a closed convex body Ξ©, the following inequality was proved in [9] based on Stokes’ Formula 1ξ€œVol(Ξ©)Ξ©1𝑓(π‘₯)𝑑π‘₯≀1𝑛+1ξ€ŸArea(πœ•Ξ©)πœ•Ξ©ξ‚΅1+Area(πœ•Ξ©)ξ‚ΆVol(Ξ©)(π‘₯βˆ’πœƒ)πœ‹(π‘₯)𝑓(π‘₯)𝑑𝑠,(4.3) where πœ•Ξ© denotes the piecewisely smooth boundary of Ξ©, πœƒ denotes the barycenter of πœ•Ξ©, and πœ‹(π‘₯) denotes the exterior unit normal vector in the point π‘₯.

For a closed hyperball Ξ©, the inequality (4.2) is able to be deduced from (4.3); and for a hypercube Ξ©, the inequality (1.9) is also able to be deduced from (4.3); however, these are on the condition that 𝑓 is differentiable.

Now, let 𝑓 be a convex function defined on an open region π·βŠ†π‘π‘›, the hypercube {π‘₯βˆΆβ€–π‘₯βˆ’π‘β€–βˆžβ‰€β„Ž/2}βŠ†π·, the mapping 𝐴𝑛 is defined by π΄π‘›βˆΆ[]0,1βŸΆπ‘,𝐴𝑛(1𝑑)=2π‘›β„Žπ‘›βˆ’1ξ€Ÿβ€–π‘’β€–βˆž=β„Ž/2𝑓(𝑐+𝑑𝑒)𝑑𝑠,(4.4) which is a parameter surface integral with respect to area of π‘›βˆ’1 dimension, and letξ€œπœ™(𝑑)=β€–π‘₯βˆ’π‘β€–βˆžβ‰€π‘‘β„Ž/2𝑓(π‘₯)𝑑π‘₯βˆ’π‘‘π‘›ξ€œβ€–π‘₯βˆ’π‘β€–βˆžβ‰€β„Ž/2𝑓(π‘₯)𝑑π‘₯,(4.5) in the same way as the proof of Theorem 3.1, by the induction we will be able to obtain the inequality (1.9).

Some mappings associated to Hadamard’s inequality have been established in [1–5], which refine the classical Hadamard’s inequality. These mappings are based on Hadamard’s inequality. However, our method in this paper to establish mappings is different from theirs: we first proved (1.1) by the properties of 𝐴1 and subsequently established 𝐴2 and proved (1.8) based on (1.1). Thus, using the induction, we will be able to establish the mapping 𝐴𝑛 and prove Hadamard’s inequality on a hypercube.

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