Volume 2009, Article ID 283147,13pages doi:10.1155/2009/283147
Research Article
On The Hadamard’s Inequality for Log-Convex
Functions on the Coordinates
Mohammad Alomari and Maslina Darus
School of Mathematical Sciences, Universiti Kebangsaan Malaysia, UKM, Bangi, 43600 Selangor, Malaysia
Correspondence should be addressed to Maslina Darus,maslina@ukm.my
Received 15 January 2009; Revised 31 May 2009; Accepted 20 July 2009
Recommended by Sever Silvestru Dragomir
Inequalities of the Hadamard and Jensen types for coordinated log-convex functions defined in a rectangle from the plane and other related results are given.
Copyrightq2009 M. Alomari and M. Darus. 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.
1. Introduction
Letf :I ⊆R → Rbe a convex mapping defined on the intervalIof real numbers anda, b∈I, witha < b, then
f
ab
2
≤
b
a
fxdx≤ fa fb
2 1.1
holds, this inequality is known as the Hermite-Hadamard inequality. For refinements, counterparts, generalizations and new Hadamard-type inequalities, see1–8.
A positive functionfis called log-convex on a real intervalI a, b, if for allx, y ∈
a, bandλ∈0,1,
fλx 1−λy≤fλxf1−λy. 1.2
Iffis a positive log-concave function, then the inequality is reversed. Equivalently, a function
The logarithmic mean of the positive real numbersa, b,a /b, is defined as
La, b b−a
logb−loga. 1.3
A version of Hadamard’s inequality for log-convexconcavefunctions was given in9, as follows.
Theorem 1.1. Suppose thatfis a positive log-convex function ona, b, then
1
b−a b
a
fxdx≤Lfa, fb. 1.4
Iffis a positive log-concave function, then the inequality is reversed.
For refinements, counterparts and generalizations of log-convexity see9–13.
A convex function on the coordinates was introduced by Dragomir in8. A function
f : Δ → Rwhich is convex inΔis called coordinated convex onΔif the partial mapping
fy : a, b → R,fyu fu, yand fx : c, d → R,fxv fx, v, are convex for all y∈c, dandx∈a, b.
An inequality of Hadamard’s type for coordinated convex mapping on a rectangle from the planeR2established by Dragomir in8, is as follows.
Theorem 1.2. Suppose thatf:Δ → Ris coordinated convex onΔ, then
f
ab
2 ,
cd
2
≤ 1
b−ad−c
b
a d
c
fx, ydy dx
≤ fa, c fa, d fb, c fb, d
4 .
1.5
The above inequalities are sharp.
The maximum modulus principle in complex analysis states that iffis a holomorphic function, then the modulus|f|cannot exhibit a true local maximum that is properly within the domain off. Characterizations of the maximum principle for subsuperharmonic functions are considered in14, as follows.
Theorem 1.3. LetG⊆R2be a region and letf :G → Rbe a sub(super)harmonic function. If there
is a pointλ∈Gwithfλ≥fx, for allx∈Gthenfxis a constant function.
Theorem 1.4. LetG⊆R2be a region and letfandgbe bounded real-valued functions defined onG
such thatfis subharmonic andgis superharmonic. If for each pointa∈∂∞G
lim
x→asupfx≤xlim→ainfgx, 1.6
In this paper, a new version of the maximum minimum principle in terms of convexity, and some inequalities of the Hadamard type are obtained.
2. On Coordinated Convexity and Sub(Super)Harmonic Functions
Consider the 2-dimensional intervalΔ: a, b×c, dinR2. A functionf :Δ → Ris called convex inΔif
fλx 1−λy≤λfx 1−λfy 2.1
holds for allx,y∈Δandλ∈0,1.
As in8, we define a log-convex function on the coordinates as follows: a function
f:Δ → Rwill be calledcoordinated log-convexonΔif the partial mappingsfy:a, b → R, fyu fu, yandfx : c, d → R,fxv fx, v, are log-convex for ally ∈ c, dand x∈a, b. A formal definition of a coordinated log-convex function may be stated as follows.
Definition 2.1. A functionf :Δ → Rwill be calledcoordinated log-convexonΔ, for allt, s ∈
0,1andx, y,u, v∈Δ, if the following inequality holds,
ftx 1−ty, su 1−sw
≤ftsx, ufs1−ty, uft1−sx, wf1−t1−sy, w.
2.2
Equivalently, we can determine whether or not the function f is coordinated log-convex by using the following lemma.
Lemma 2.2. Letf :Δ → R. Iffis twice differentiable thenfis coordinated log-convex onΔif and
only if for the functionsfy :a, b → R, defined byfyu fu, yandfx :c, d → R, defined
byfxv fx, v, we have
fx·fx−
fx2≥0, fy·fy−
fy2≥0. 2.3
Proof. The proof is straight forward using the elementary properties of log-convexity in one
variable.
Proposition 2.3. Suppose thatg :a, b → R is twice differentiable ona, band log-convex on
a, band h : c, d → R is twice differentiable onc, dand log-convex onc, d. Letf : Δ
a, b×c, d → R be a twice differentiable function defined byfx, y gxhy, thenf is
coordinated log-convex onΔ.
Proof. This follows directly usingLemma 2.2.
The following result holds.
Proposition 2.4. Every log-convex functionf : Δ a, b×c, d → R is log-convex on the
Proof. Suppose thatf : Δ → Ris convex inΔ. Consider the functionfx : c, d → R, fxv fx, v, then forλ∈0,1, andv, w∈c, d, we have
fxλv 1−λw fx, λv 1−λw
fλx 1−λx, λv 1−λw ≤fλx, vf1−λx, w
fxλvfx1−λw,
2.4
which shows the log-convexity of fx. The proof that fy : a, b → R, fyu fu, y,
is also log-convex ona, bfor all y ∈ c, dfollows likewise. Now, consider the mapping
f0:0,12 → Rgiven byf0x, y exy. It is obvious thatf0is log-convex on the coordinates
but not log-convex on0,12. Indeed, ifu,0,0, w∈0,12andλ∈0,1, we have: logf0λu,0 1−λ0, w logf0λu,1−λw λ1−λuw,
λ logf0u,0 1−λlogf00, w 0.
2.5
Thus, for allλ∈0,1andu, w∈0,1, we have
logf0λu,0 1−λ0, w> λlogf0u,0 1−λlogf00, w 2.6
which shows thatf0is not log-convex on0,12.
In the following, a Jensen-type inequality for coordinated log-convex functions is considered.
Proposition 2.5. Letfbe a positive coordinated log-convex function on the open seta, b×c, d
and letxi∈a, b,yj ∈c, d. Ifαi, βj>0and ni0αi1, mj0βj1, then
logf n
i1
αixi, m
i1
βjyj
≤n i1
m
j1
αiβj logf
xi, yj
. 2.7
Proof. Letxi ∈a, b,αi >0 be such that mj0αi 1, and letyi ∈c, d,βj >0 be such that m
j0βj 1, then we have,
f ⎛ ⎝n
i1
αixi, m
j1
βjyj ⎞ ⎠≤n
i1
fαi ⎛ ⎝xi,m
j1
βjyj ⎞ ⎠≤n
i1
m
j1
fαiβjxi, yj
, 2.8
and, sincefis positive,
logf ⎛ ⎝n
i1
αixi, m
j1
βjyj ⎞ ⎠≤n
i1
m
j1
αiβj logf
xi, yj
, 2.9
Remark 2.6. Letfx, y xy, then the following inequality holds: log ⎡ ⎣ n
i1
αixi ⎛
⎝m j1
βjyj ⎞ ⎠ ⎤ ⎦≤n
i1
m
j1
αiβj logxiyj. 2.10
The above result may be generalized to the integral form as follows.
Proposition 2.7. Letfbe a positive coordinated log-convex function on theΔ◦: a, b×c, d,and letxt : r1, r2 → Rbe integrable witha < xt< b, and letyt :s1, s2 → Rbe integrable
withc < yt< d. Ifα:r1, r2 → Ris positive,
r2
r1αtdt1, andαxtis integrable onr1, r2
andβ:s1, s2 → Ris positive,
s2
s1βtdt1, andβytis integrable ons1, s2,then
logf r2
r1
αtxtdt, s2
s1
βuyudu ≤ r2 r1 s2 s1
αtβulogfxt, yudu dt.
2.11
Proof. Applying Jensen’s integral inequality in one variable on thex-coordinate and on the
y-coordinate we get the required result. The details are omitted.
Theorem 2.8. Let f : Δ → R be a positive coordinated log-convex function in Δ, then for all
distinctx1, x2, x3 ∈ a, b, such thatx1 < x2 < x3 and distincty1, y2, y3 ∈c, dsuch that y1 <
y2< y3, the following inequality holds:
fx2y2y3x3x 1, y1
·fy1x2y2x3x 1, y3
·fx1y2x2y3x 3, y1
·fx1y1y2x2x 3, y3
·fx1y3x3y1x 2, y2
≥fx2y3y2x3x 1, y1
·fy1x3x2y2x 1, y3
·fx1y3x2y2x 3, y1
·fx1y2y1x2x 3, y3
·fx1y1x3y3x 2, y2
.
2.12
Proof. Let x1, x2, x3 be distinct points in a, band let y1, y2, y3 be distinct points in c, d.
Settingα x3−x2/x3−x1,x2 αx1 1−αx3 and letβ y3−y2/y3−y1,y2
βy1 1−βy3, we have
logfx2, y2
logfαx1 1−αx3, βy1
1−βy3
≤αβlogfx1, y1
α1−βlogfx1, y3
β1−αlogfx3, y1
1−α1−βlogfx3, y3
x3−x2
x3−x1
y3−y2
y3−y1
logfx1, y1
x3−x2
x3−x1
y2−y1
y3−y1
logfx1, y3
x2−x1
x3−x1
y3−y2
y3−y1
logfx3, y1
x2−x1
x3−x1
y2−y1
y3−y1
logfx3, y3
,
and we can write
logf
x2y2y3x3x 1, y1
fy1x2y2x3x 1, y3
fx1y2x2y3x 3, y1
fx1y1y2x2x 3, y3
fx1y3x3y1x 2, y2
fx2y3y2x3x1, y1fy1x3x2y2x1, y3fx1y3x2y2x3, y1fx1y2y1x2x3, y3fx1y1x3y3x2, y2 ≥0.
2.14
From this inequality it is easy to deduce the required result2.12.
The subharmonic functions exhibit many properties of convex functions. Next, we give some results for the coordinated convexity and subsuperharmonic functions.
Proposition 2.9. Letf : Δ ⊆ R2 → Rbe coordinated convex (concave) on Δ. If f is a twice
differentiable onΔ◦, thenfis sub(super)harmonic onΔ◦.
Proof. Sincefis coordinated convex onΔthen the partial mappingsfy:a, b → R,fyu
fu, yandfx : c, d → R,fxv fx, v, are convex for ally ∈ c, dand x ∈ a, b.
Equivalently, sincefis differentiable we can write
0≤fx ∂
2f
∂2y 2.15
for ally∈c, d, and
0≤fy ∂
2f
∂2x 2.16
for allx∈a, b, which imply that
fxfy ∂
2f
∂2x
∂2f
∂2y ≥0 2.17
which shows thatf is subharmonic. Iff is coordinated concave onΔ, replace “≤” by “≥” above, we get thatfis superharmonic onΔ◦.
We now give two versions of the MaximumMinimum Principle theorem using convexity on the coordinates.
Theorem 2.10. Let f : Δ ⊆ R2 → R be a coordinated convex (concave) function onΔ. If f is
twice differentiable inΔ◦and there is a pointa a1, a2∈Δ◦withfa1, a2≥≤fx, y, for all
x, y∈Δthenfis a constant function.
Proof. By Proposition 2.9, we get thatf is subsuperharmonic. Therefore, by Theorem 1.3
Theorem 2.11. Letf and g be two twice differentiable functions inΔ◦. Assume thatf and g are
bounded real-valued functions defined onΔsuch thatf is coordinated convex andg is coordinated
concave. If for each pointa a1, a2∈∂∞Δ
lim
x,y→a1,a2
supfx, y≤ lim
x,y→a1,a2
infgx, y, 2.18
thenfx, y< gx, yfor allx, y∈Δorfgandfis harmonic.
Proof. ByProposition 2.9, we get thatf is subharmonic and g is superharmonic. Therefore,
byTheorem 1.4and using the maximum principal the required result holds,see14, page 264.
Remark 2.12. The above two results hold for log-convex functions on the coordinates, simply,
replacingfby logf, to obtain the results.
3. Some Inequalities and Applications
In the following we develop a Hadamard-type inequality for coordinated log-convex functions.
Corollary 3.1. Suppose thatf : Δ a, b×c, d → R is log-convex on the coordinates ofΔ, then
logf
ab
2 ,
cd
2
≤ 1 b−ad−c
b
a d
c
logfx, ydy dx
≤log4fa, cfa, dfb, cfb, d.
3.1
For a positive coordinated log-concave functionf, the inequalities are reversed.
Proof. InTheorem 1.2, replacefby log fand we get the required result.
Lemma 3.2. ForA, B, C∈RwithA, B, C >1, the function
ψβCβAβB−1
lnAβB, 0≤β≤1 3.2
is convex for allβ∈0,1. Moreover,
1
0
ψβdβ≤ ψ0 ψ1
2 , 3.3
for allA, B, C >1.
is equivalent to saying thatψ is increasing and henceψ is convex. Now, using inequality
1.1, we get
1
0
CβA βB−1
lnAβBdβ 1
0
ψβdβ≤ ψ0 ψ1
2
1 2
B−1 lnBC·
AB−1 lnAB
, 3.4
which completes the proof.
Theorem 3.3. Suppose thatf:Δ a, b×c, d → Ris log-convex on the coordinates ofΔ. Let
A fa, c fb, c
fb, d
fa, d, B
fa, d
fb, d, C fb, c
fb, d, 3.5
then the inequalities
I 1
b−ad−c
d
c b
a
fx, ydx dy
≤fb, d× ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1, ABC1,
B−1
lnB
C−1 lnC
, A1,
HC, B1,
HB, C1, C−1
lnC, AB1,
B−1
lnB, AC1,
−γln−lnA Ei1,−lnA
lnA , BC1,
1 2
B−1 lnBC·
AB−1 lnAB
, A, B, C >1, 1
0
CβAβB−1
lnAβBdβ, otherwise
3.6
hold, whereγis the Euler constant,
Hx Ei1,−lnx lnln x−Ei1,−lnAx−lnlnAx
lnA ⎧ ⎪ ⎨ ⎪ ⎩
2 lnlnA−ln−lnA
lnA ,
lnx
lnA <0, −
lnx
lnA <1,
0, otherwise,
Eix V.P. ∞
−x e−t
t dt
is the exponential integral function. For a coordinated log-concave function f, the inequalities are reversed.
Proof. Sincef :Δ a, b×c, d → Ris log-convex on the coordinates ofΔ, then
fαa 1−αb, βc1−βd
≤fαβa, cfβ1−αb, cfα1−βa, df1−α1−βb, d
fαβa, cfβb, cf−αβb, cfαa, df−αβa, d
×fb, df−βb, df−αb, dfαβb, d
fa, c
fb, c fb, d fa, d
αβfa, d
fb, d
αfb, c
fb, d β
fb, d.
3.8
Integrating the previous inequality with respect toαandβon0,12, we have,
1
0
1
0
fαa 1−αb, βc1−βddα dβ
≤fb, d 1
0
1
0
fa, c
fb, c fb, d fa, d
αβfa, d
fb, d
αfb, c
fb, d β
dα dβ.
3.9
Therefore, by3.9and for nonzero, positiveA, B, C, we have the following cases.
1IfABC1, the result is trivial.
2IfA1, then
1
0
1
0
fαa 1−αb, βc1−βddα dβ
≤fb, d 1
0
1
0
fa, d fb, d
αfb, c
fb, d β
dα dβ
fb, d
1
0
Bαdα
1
0
Cβdβ
fb, d
B−1
lnB
C−1 lnC
.
3IfB1, then
1
0
1
0
fαa 1−αb, βc1−βddα dβ
≤fb, d 1
0
1
0
AαβCβdα dβ
fb, d
1
0
AαC−1
lnAαCdα
fb, d
⎡ ⎢ ⎣
⎛ ⎜ ⎝
⎧ ⎪ ⎨ ⎪ ⎩
2 lnlnA−ln−lnA
lnA ,
lnC
lnA <0, −
lnC
lnA <1
0, otherwise
⎞ ⎟ ⎠
Ei1,−lnC lnlnC−Ei1,−lnAC−lnlnAC
lnA
⎤ ⎥ ⎦.
3.11
4IfC1, then
1
0
1
0
fαa 1−αb, βc1−βddα dβ
≤fb, d 1
0
1
0
AαβBαdα dβ
fb, d
1
0
AβB−1 lnAβBdβ
fb, d
⎡ ⎢ ⎣
⎛ ⎜ ⎝
⎧ ⎪ ⎨ ⎪ ⎩
2 lnlnA−ln−lnA
ln A ,
lnB
lnA <0, −
lnB
lnA <1
0, otherwise
⎞ ⎟ ⎠
Ei1,−lnB lnlnC−Ei1,−lnAB−lnlnAB
lnA
⎤ ⎥ ⎦.
3.12
5IfAB1, then
1
0
1
0
fαa 1−αb, βc1−βddα dβ
≤fb, d 1
0
1
0
AαβBαCβdα dβfb, d 1
0
Cβdβfb, dC−1
lnC.
6IfAC1, then
1
0
1
0
fαa 1−αb, βc1−βddα dβ
≤fb, d 1
0
1
0
AαβBαCβdα dβfb, d 1
0
Bαdαfb, dB−1
lnB.
3.14
7IfBC1, then
1
0
1
0
fαa 1−αb, βc1−βddα dβ
≤fb, d 1
0
1
0
AαβBαCβdα dβ
fb, d
1
0
1
0
Aβαdα dβ
fb, d
1
0
Aα−1
lnAαdα
−fb, dγln−lnA Ei1,−lnA
lnA .
3.15
8IfA, B, C >1,then
fb, d 1
0
1
0
AαβBαCβdα dβfb, d 1
0
Cβ "
AβB−1
lnAβB #
dβ. 3.16
Therefore, byLemma 3.2, we deduce that
fb, d 1
0
1
0
AαβBαCβdα dβ≤ fb, d
2
B−1 lnBC·
AB−1 lnAB
. 3.17
9IfA, B, C /1, we have
fb, d 1
0
1
0
AαβBαCβdα dβfb, d 1
0
Cβ "
AβB−1
lnAβB #
dβ, 3.18
which is difficult to evaluate because it depends on the values ofA, B,andC.
Remark 3.4. The integrals in3,4, and7in the proof ofTheorem 2.11are evaluated using
Corollary 3.5. InTheorem 3.3, if
1fx, y fx, then
1
b−a b
a
fxdx≤Lfa, fb, 3.19
and for instance, iff1x ex
p
,p≥1we deduce
1
b−a b
a
expdx≤Leap, ebp. 3.20
2fx, y f1xf2y, then
I≤Lf1a, f1b
Lf2c, f2d
, 3.21
and for instance, iff1x, y ex
pyq
, p, q≥1, we deduce
1
b−ad−c
b
a d
c
expyqdx dy≤Leap, ebpLecp, edp. 3.22
Proof. Follows directly by applying inequality1.4.
Acknowledgment
The authors acknowledge the financial support of the Faculty of Science and Technology, Universiti Kebangsaan MalaysiaUKM–GUP–TMK–07–02–107.
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