On The Hadamard’s Inequality for Log-Convex Functions on the Coordinates

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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,[email protected] 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 → R be 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,

f

λx 1−λy

≤f^λxf^1−λ y

. 1.2

Iffis a positive log-concave function, then the inequality is reversed. Equivalently, a function fis log-convex onIiffis positive and logfis convex onI. Also, iff >0 andfexists onI, thenfis log-convex if and only iff·f−f²≥0.

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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≤L

fa, 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 → R which 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 → R is coordinated convex onΔ, then

f ab

2 ,cd 2

≤ 1

b−ad−c _b

a

_d

c

f x, y

dy 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

xlim→asupfx≤ lim

x→ainfgx, 1.6

thenfx< gxfor allx∈Gorfgandfis harmonic.

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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 → R is 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 called coordinated log-convex onΔif the partial mappingsf_y:a, b → R, f_yu fu, yandf_x : c, d → R,f_xv 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 called coordinated log-convex onΔ, for allt, s ∈ 0,1andx, y,u, v∈Δ, if the following inequality holds,

f

tx 1−ty, su 1−sw

≤f^tsx, uf^s1−t y, u

f^t1−sx, wf^1−t1−s y, 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 functionsf_y :a, b → R, defined byf_yu fu, yandf_x :c, d → R, defined byfxv fx, v, we have

fx·f_x− f_x₂

≥0, fy·f_y− f_y₂

≥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 coordinates, but the converse is not generally true.

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Proof. Suppose thatf : Δ → R is convex inΔ. Consider the functionfx : c, d → R, f_xv fx, v, then forλ∈0,1, andv, w∈c, d, we have

f_xλv 1−λw fx, λv 1−λw

fλx 1−λx, λv 1−λw

≤f^λx, vf^1−λx, w f_x^λvf_x^1−λw,

2.4

which shows the log-convexity of f_x. The proof that f_y : a, b → R, f_yu fu, y, is also log-convex ona, bfor all y ∈ c, dfollows likewise. Now, consider the mapping f0:0,1² → Rgiven byf0x, y e^xy. 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 letx_i∈a, b,y_j ∈c, d. Ifα_i, β_j>0 and ⁿ_i0α_i1, ^m_j0β_j1, 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 ^m_j0αi 1, and letyi ∈c, d,βj >0 be such that

mj0β_j 1, then we have,

f

⎛

⎝ⁿ

i1

α_ix_i, m j1

β_jy_j

⎞

⎠≤ⁿ

i1

f^αⁱ

⎛

⎝x_i, m j1

β_jy_j

⎞

⎠≤ⁿ

i1

m j1

f^αⁱ^β^j x_i, y_j

, 2.8

and, sincefis positive,

logf

⎛

⎝ⁿ

i1

αixi, m j1

βjyj

⎞

⎠≤ⁿ

i1

m j1

αiβj logf xi, yj

, 2.9

which is as required.

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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 → R be integrable witha < xt< b, and letyt :s1, s2 → R be integrable withc < yt< d. Ifα:r1, r2 → R is positive,_r₂

r1αtdt1, andαxtis integrable onr1, r2 andβ:s1, s₂ → R is positive,_s₂

s1βtdt1, andβytis integrable ons1, s₂,then logf

_r₂

r1

αtxtdt, _s₂

s1

βuyudu

≤ _r₂

r1

_s₂

s1

αtβulogf

xt, yu du 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:

f^x²^y²^y³^x³ x1, y1

·f^y¹^x²^y²^x³ x1, y3

·f^x¹^y²^x²^y³ x3, y1

·f^x¹^y¹^y²^x² x3, y3

·f^x¹^y³^x³^y¹ x2, y2

≥f^x²^y³^y²^x³ x₁, y₁

·f^y¹^x³^x²^y² x₁, y₃

·f^x¹^y³^x²^y² x₃, y₁

·f^x¹^y²^y¹^x² x3, y3

·f^x¹^y¹^x³^y³ x2, 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

logf x2, y2

logf

αx1 1−αx3, βy1 1−β

y3

≤αβlogf x₁, y₁

α 1−β

logf x₁, y₃ β1−αlogf

x₃, y₁

1−α 1−β

logf x₃, y₃ x₃−x₂

x₃−x₁ y₃−y₂ y₃−y₁logf

x₁, y₁

x₃−x₂ x₃−x₁

y₂−y₁ y₃−y₁ logf

x₁, y₃ x2−x1

x₃−x₁ y₃−y₂ y₃−y₁ logf

x₃, y₁

x2−x1

x₃−x₁ y₂−y₁ y₃−y₁logf

x₃, y₃ ,

2.13

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and we can write

logf^x²^y²^y³^x³ x1, y1

f^y¹^x²^y²^x³ x1, y3

f^x¹^y²^x²^y³ x3, y1

f^x¹^y¹^y²^x² x3, y3

f^x¹^y³^x³^y¹ x2, y2

f^x²^y³^y²^x³

x1, y1

f^y¹^x³^x²^y² x1, y3

f^x¹^y³^x²^y² x3, y1

f^x¹^y²^y¹^x² x3, y3

f^x¹^y¹^x³^y³ x2, 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 → R be coordinated convex (concave) on Δ. If f is a twice diﬀerentiable onΔ^◦, thenfis sub(super)harmonic onΔ^◦.

Proof. Sincefis coordinated convex onΔthen the partial mappingsf_y:a, b → R,f_yu fu, yandfx : c, d → R,fxv fx, v, are convex for ally ∈ c, dand x ∈ a, b.

Equivalently, sincefis diﬀerentiable we can write

0≤f_x ∂²f

∂²y 2.15

for ally∈c, d, and

0≤f_y ∂²f

∂²x 2.16

for allx∈a, b, which imply that

f_xf_y ∂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. Letf : Δ ⊆ R² → R be a coordinated convex (concave) function on Δ. 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 and the maximum principal the required result holdssee14, page 264.

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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

x,y→lima1,a2supf x, y

≤ lim

x,y→a1,a2infg x, 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

logf x, y

dy dx

≤log ⁴

fa, 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

ψ β

C^βA^βB−1 ln

A^βB, 0≤β≤1 3.2

is convex for allβ∈0,1. Moreover, ₁

0

ψ β

dβ≤ ψ0 ψ1

2 , 3.3

for allA, B, C >1.

Proof. Sinceψis twice diﬀerentiable for allβ∈0,1withA, B, C >1, we note that for all 0<

β₁≤β₂<1,ψβ1≤ψβ2, which shows thatψis increasing and thusψis nonnegative which

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is equivalent to saying thatψ is increasing and henceψ is convex. Now, using inequality 1.1, we get

₁

0

C^βA^βB−1 ln

A^βBdβ ₁

0

ψ β

dβ≤ 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

f x, y

dx 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

C^βA^βB−1 ln

A^βBdβ, 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

3.7

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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

f

αa 1−αb, βc 1−β

d

≤f^αβa, cf^β1−αb, cf^α^1−βa, df^1−α^1−βb, d f^αβa, cf^βb, cf^−αβb, cf^αa, df^−αβa, d

×fb, df^−βb, df^−αb, df^αβb, 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

f

d dα 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

f

d dα dβ

≤fb, d ₁

0

₁

0

fa, d fb, d

αfb, c fb, d

β

dα dβ

fb, d 1

0

B^αdα 1

0

C^βdβ

fb, d B−1

lnB

C−1 lnC

.

3.10

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3IfB1, then ₁

0

₁

0

f

d dα dβ

≤fb, d ₁

0

₁

0

A^αβC^βdα dβ

fb, d ₁

0

A^αC−1 lnA^αCdα

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

f

d dα dβ

≤fb, d ₁

0

₁

0

A^αβB^αdα dβ

fb, d ₁

0

A^βB−1 ln

A^βBdβ

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

f

d dα dβ

≤fb, d ₁

0

₁

0

A^αβB^αC^βdα dβfb, d ₁

0

C^βdβfb, dC−1 lnC.

3.13

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6IfAC1, then ₁

0

₁

0

f

d dα dβ

≤fb, d ₁

0

₁

0

B^αdαfb, dB−1 lnB.

3.14

7IfBC1, then ₁

0

₁

0

f

d dα dβ

≤fb, d ₁

0

₁

0

A^αβB^αC^βdα dβ

fb, d ₁

0

₁

0

A^βα

dα dβ

fb, d ₁

0

A^α−1 lnA^αdα

−fb, dγln−lnA Ei1,−lnA

lnA .

3.15

8IfA, B, C >1,then

fb, d ₁

0

₁

0

C^β

"

A^βB−1 ln

A^βB

#

dβ. 3.16

Therefore, byLemma 3.2, we deduce that

fb, d ₁

0

₁

0

A^αβB^αC^βdα dβ≤ fb, d 2

B−1

lnBC· AB−1 lnAB

. 3.17

9IfA, B, C /1, we have

fb, d ₁

0

₁

0

C^β

"

A^βB−1 ln

A^βB

#

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 Maple Software.

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Corollary 3.5. InTheorem 3.3, if 1fx, y fx, then

1 b−a

_b

a

fxdx≤L

fa, fb

, 3.19

and for instance, iff1x e^x^p,p≥1 we deduce

1 b−a

_b

a

e^x^pdx≤L

e^a^p, e^b^p

. 3.20

2fx, y f1xf2y, then I≤L

f1a, f1b L

f2c, f2d

, 3.21

and for instance, iff₁x, y e^x^p^y^q, p, q≥1, we deduce

1 b−ad−c

_b

a

_d

c

e^x^p^y^qdx dy≤L

e^a^p, e^b^p L

e^c^p, e^d^p

. 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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