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Farid et al. Journal of Inequalities and Applications (2015) 2015:283 DOI 10.1186/s13660-015-0808-z

R E S E A R C H Open Access

New mean value theorems andgeneralization of Hadamard inequality viacoordinated m-convex functionsG Farid1, M Marwan2 and Atiq Ur Rehman1*

*Correspondence:[email protected] Institute of InformationTechnology, Kamra Road, Attock,43600, PakistanFull list of author information isavailable at the end of the article

AbstractWe derive new mean value theorems for functionals associated with Hadamardinequality for convex functions on the coordinates. We present some Hadamard typeinequalities and related results form-convex functions on the coordinates.

Keywords: convex functions;m-convex functions; convex functions on coordinates;Hadamard inequality; mean value theorems

1 IntroductionWe start with the very basic concept of a convex function that has been seen as importantever since it was defined.

Definition . A real valued function f : I →R, where I is an interval inR, is called convexif

f(λx + ( – λ)y

) ≤ λf (x) + ( – λ)f (y), ()

where λ ∈ [, ], and x, y ∈ I .

One cannot deny the importance of convex functions. It is used by many mathemati-cians in many fields of mathematics such as functional analysis, mathematical statistics,and complex analysis (see for example [–] and references therein). In the field of in-equalities, convex functions play a unique role. Most of the new inequalities till now de-fined are results and consequences of (). The most cardinal and classical inequality forconvex functions is stated in the following.

Theorem . Let f : I →R be convex function and a, b ∈ I with a < b, then

f(

a + b

)≤

b – a

∫ b

af (x) dx ≤ f (a) + f (b)

. ()

This famous integral inequality can be traced back to the papers presented by Hermite[] and Hadamard []. Researchers have used inequality () several times for giving gen-eralization and modification of Hadamard type inequalities using different modification

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of convex functions. For refinements, counterparts, and generalizations see for example[, –].

Recently, many authors have considered a convex function on the coordinates to give theHadamard inequality on the coordinates and its different variants. Also they have givenmany results associated with it (e.g. see [–]).

Definition . Let � := [a, b] × [c, d] ⊂ R with a < b and c < d. A function f : � → R

is called convex on the coordinates if the partial mapping fy : [a, b] → R, fy(u) := f (u, y)and fx : [c, d] → R, fx(v) := f (x, v) are convex, where they are defined for all y ∈ [c, d] andx ∈ [a, b].

We will keep the notation � = [a, b] × [c, d] throughout this paper.Recall that a mapping f : � → R is convex in � if, for (x, y), (u, v) ∈ � and α ∈ [, ],

the following inequality holds:

f(α(x, y) + ( – α)(u, v)

) ≤ αf (x, y) + ( – α)f (u, v).

It can be seen that every convex mapping f : � →R is convex on the coordinates but theconverse is not true. Dragomir gave the Hadamard inequality for a rectangle in the planefor convex functions on the coordinates (see []).

Theorem . Suppose that f : � → R is convex on the coordinates on �. Then one hasthe following inequalities:

f(

a + b

,c + d

)

≤

[

b – a

∫ b

af(

x,c + d

)dx +

d – c

∫ d

cf(

a + b

, y)

dy]

≤ (b – a)(d – c)

∫ b

a

∫ d

cf (x, y) dx dy

≤

[

b – a

∫ b

af (x, c) dx +

b – a

∫ b

af (x, d) dx +

d – c

∫ d

cf (a, y) dy

+

d – c

∫ d

cf (b, y) dy

]

≤

[f (a, c) + f (a, d) + f (b, c) + f (b, d)

].

In [], Toader defined the concept of m-convexity, an intermediate between the usualconvexity and star shape properties.

Definition . The function f : [, b] → R is said to be m-convex, where m ∈ [, ], if forevery x, y ∈ [, b] and t ∈ [, ] we have

f(tx + m( – t)y

) ≤ tf (x) + m( – t)f (y). ()

In [] using inequality () and () Dragomir and Toader gave the following Hadamardtype inequality for m-convex functions.


Theorem . Let f : [,∞) → R be m-convex function with m ∈ (, ] and ≤ a < b. Iff ∈ L[a, b], then one has the following inequalities:

f(

a + b

)≤

b – a

∫ b

a

f (x) + mf ( xm )

dx

≤ m +

[f (a) + f (b)

+ m

( f ( am ) + f ( b

m )

)]. ()

They also gave the following related results to the Hadamard type inequality for m-convex functions.

Theorem . Let f : [,∞) →R be an m-convex function with m ∈ (, ]. If ≤ a < b < ∞and f ∈ L[a, b], then one has the inequality

b – a

∫ b

af (x) dx ≤ min

{ f (a) + mf ( bm )

,

f (b) + mf ( am )

}.

Theorem . Let f : [,∞) →R be an m-convex function with m ∈ (, ]. If ≤ a < b < ∞and f is differentiable on (,∞), then one has the inequality

f (mb)m

–b – a

f ′(mb) ≤

b – a

∫ b

af (x) dx

≤ (b – ma)f (d) – (a – mb)f (a)(b – a)

.

In this paper, new mean value theorems of Cauchy type for functionals associated withnonnegative differences of the Hadamard inequality on the coordinates are proved. Gen-eralized results related to the Hadamard inequality for m-convex functions on the coor-dinates are also given.

2 Mean value theoremsWe know that if a function f is twice differentiable on an interval I then it is convex onI if and only if its second order derivative is nonnegative. If a function f (X) := f (x, y) hascontinuous second order partial derivatives on interior of � then it is convex on � if theHessian matrix

Hf (X) =

⎛

⎝∂f (X)

∂x∂f (X)∂y∂x

∂f (X)∂x∂y

∂f (X)∂y

⎞

⎠

is nonnegative definite, that is, vHf (X)vτ is nonnegative for all real nonnegative vector v(see [], p.).

It is easy to see that f : � →R is coordinated convex on � iff

f ′′x (y) =

∂f (x, y)∂y and f ′′

y (x) =∂f (x, y)

∂x

are nonnegative for all interior points (x, y) in �.


For a real valued function f : � →R we define

H(f ) =

[

b – a

∫ b

af(

x,c + d

)dx +

d – c

∫ d

cf(

a + b

, y)

dy]

– f(

a + b

,c + d

). ()

One can note that H(f ) ≥ if f is convex on the coordinates in �.To give the mean value theorems of Cauchy type, we need the following lemma.

Lemma . Let f : � →R be a function such that

m ≤ ∂f (x, y)∂x ≤ M and m ≤ ∂f (x, y)

∂y ≤ M

for all interior points (x, y) in �.Consider the functions g, h : � →R defined as

g(x, y) =

max{M, M}(x + y) – f (x, y)

and

h(x, y) = f (x, y) –

min{m, m}(x + y).

Then g and h are coordinated convex in �.

Proof Since

∂g(x, y)∂x = max{M, M} –

∂f (x, y)∂x ≥

and

∂g(x, y)∂y =

∂f (x, y)∂y – min{m, m} ≥

for all interior points (x, y) in �, g is convex on the coordinates in �.Similarly one can prove that h is convex on the coordinates in �. �

Theorem . Let f : � → R be a function, which has continuous partial derivatives ofsecond order in � and q(x, y) := x + y. Then there exist (η, ξ) and (η, ξ) in the interiorof � such that

H(f ) =

∂f (η, ξ)∂x H(q)

and

H(f ) =

∂f (η, ξ)∂x H(q),

provided that H(q) = .


Proof Since f has continuous partial derivatives of second order in compact set �, thereexist real numbers m, m, M, and M such that

m ≤ ∂f (x, y)∂x ≤ M and m ≤ ∂f (x, y)

∂y ≤ M.

Now consider functions g defined in Lemma .. As g is convex on the coordinates in �,

H(g) ≥ ,

that is,

H(

max{M, M}q – f (x, y))

≥ .

From this we get

H(f ) ≤ max{M, M}H(q). ()

On the other hand, for the function h, one has

min{m, m}H(q) ≤ H(f ). ()

As H(q) = , combining inequalities () and (), we get

min{m, m} ≤ H(f )H(q)

≤ max{M, M}.

Then there exist (η, ξ) and (η, ξ) in the interior of � such that

H(f )H(q)

=∂f (η, ξ)

∂x andH(f )H(q)

=∂f (η, ξ)

∂y .

Hence the required result follows. �

3 Hadamard type inequalities for m-convex function on two coordinatesIn this section we give Hadamard type inequalities for m-convex functions on two coor-dinates. First of all we give the definitions of m-convex functions in two coordinates.

Definition . Let � = [, b] × [, d] ⊂ [,∞), then a function f : � → R, will be calledm-convex on the coordinates if the partial mappings fy : [, b] → R, fy(u) := f (u, y), andfx : [, d] →R, fx(v) := f (x, v), are m-convex on [, b] and [, d], respectively.

In the following we give the Hadamard type inequality for m-convex functions on thecoordinates.

Theorem . Let � = [, b] × [, d] ⊂ [,∞) with b, d > and f : � → R be m-convexon the coordinates in � with m ∈ (, ]. If fx ∈ L[, d] and fy ∈ L[, b], ≤ a < b, ≤ c < d.


Then we have

b – a

∫ b

af(

x,c + d

)dx +

d – c

∫ d

cf(

a + b

, y)

dy

≤ (b – a)(d – c)

∫ b

a

∫ d

c

(f (x, y) + m

(f(

x,ym

)+ f

(xm

, y)))

dy dx

≤ (m + )

[f (a, c) + f (b, c) + f (a, d) + f (b, d) + m

(f(

am

, c)

+ f(

bm

, c)

+ f(

am

, d)

+ f(

bm

, d)

+ f(

a,cm

)+ f

(a,

dm

)+ f

(b,

cm

)+ f

(b,

dm

))

+ m(

f(

am

,cm

)+ f

(bm

,cm

)+ f

(am

,dm

)+ f

(bm

,dm

))].

Proof Since mapping f : � → R is m-convex on the coordinates, the functions fx and fy

are m-convex on [, d] and [, b], respectively. Using () for the function fy we have

fy

(a + b

)≤

b – a

∫ b

a

( fy(x) + mfy( xm )

)dx

≤ m +

[fy(a) + fy(b)

+ m

fy( am ) + fy( b

m )

],

that is,

f(

a + b

, y)

≤ b – a

∫ b

a

( f (x, y) + mf ( xm , y)

)dx

≤ m +

[f (a, y) + f (b, y)

+ m

f ( am , y) + f ( b

m , y)

]. ()

From this one has

d – c

∫ d

cf(

a + b

, y)

dy

≤ (b – a)(d – c)

∫ b

a

∫ d

c

( f (x, y) + mf ( xm , y)

)dy dx

≤ m + (d – c)

∫ d

c

[f (a, y) + f (b, y)

+ m

f ( am , y) + f ( b

m , y)

]dy. ()

Similarly using () for the function fx we have

b – a

∫ b

af(

x,c + d

)dx

≤ (b – a)(d – c)

∫ b

a

∫ d

c

( f (x, y) + mf (x, ym )

)dy dx

≤ m + (b – a)

∫ b

a

[f (x, c) + f (x, d)

+ m

f (x, cm ) + f (x, d

m )

]dx. ()


By adding () and (), we get

b – a

∫ b

af(

x,c + d

)dx +

d – c

∫ d

cf(

a + b

, y)

dy

≤ (b – a)(d – c)

∫ b

a

∫ d

c

(f (x, y) + m

(f(

x,ym

)+ f

(xm

, y)))

dy dx

≤ m +

[

b – a

∫ b

a

(f (x, c) + f (x, d)

+ m

f (x, cm ) + f (x, d

m )

)dx

+

d – c

∫ d

c

(f (a, y) + f (b, y)

+ m

f ( am , y) + f ( b

m , y)

)dy

]. ()

For fixed y using the m-convexity of fy we have

f (x, y) ≤ f (x, y) + mf ( xm , y)

. ()

Performing the average integral over the interval [a, b] and using () we get

b – a

∫ b

af (x, y) dx ≤

b – a

∫ b

a

( f (x, y) + mf ( xm , y)

)dx

≤ m +

[f (a, y) + f (b, y)

+ m

f ( am , y) + f ( b

m , y)

]. ()

Similarly for fixed x using the m-convexity of fx one has

d – c

∫ d

cf (x, y) dy ≤

d – c

∫ d

c

( f (x, y) + mf (x, ym )

)dy

≤ m +

[f (x, c) + f (x, d)

+ m

f (x, cm ) + f (x, d

m )

]. ()

Considering () for y = c, d, () for x = a, b, then () for y = cm , d

m , () for x = am , b

m tomultiply later with m. Adding all these inequalities, we obtain

b – a

∫ b

a

(f (x, c) + f (x, d)

+ m

f (x, cm ) + f (x, d

m )

)dx

+

d – c

∫ d

c

(f (a, y) + f (b, y)

+ m

f ( am , y) + f ( b

m , y)

)dy

≤

[

b – a

∫ b

a

( f (x, c) + mf ( xm , c) + f (x, d) + mf ( x

m , d)

+ mf (x, c

m ) + mf ( xm , c

m ) + f (x, dm ) + mf ( x

m , dm )

)dx

+

d – c

∫ d

c

( f (a, y) + mf (a, ym ) + f (b, y) + mf (b, y

m )

+ mf ( a

m , y) + mf ( am , y

m ) + f ( bm , y) + mf ( b

m , ym )

)dy

]


≤

[f (a, c) + f (b, c) + f (a, d) + f (b, d) + m

(f(

am

, c)

+ f(

bm

, c)

+ f(

am

, d)

+ f(

bm

, d)

+ f(

a,cm

)+ f

(a,

dm

)+ f

(b,

cm

)+ f

(b,

dm

))

+ m(

f(

am

,cm

)+ f

(bm

,cm

)+ f

(am

,dm

)+ f

(bm

,dm

))]. ()

Now combining inequalities in () and (), we get the last two inequalities of the theo-rem. �

Theorem . Let � = [, b] × [, d] ⊂ [,∞) with b, d > and f : � → R be m-convexon the coordinate in � with m ∈ (, ]. If fx ∈ L[, d] and fy ∈ L[, b], then

f(

a + b

,c + d

)≤

b – a

∫ b

a

( f (x, c+d ) + mf ( x

m , c+d )

)dx

+

c – d

∫ d

c

( f ( a+b , y) + mf ( a+b

, ym )

)dy. ()

Proof By using () for y = c+d

f(

a + b

,c + d

)≤

b – a

∫ b

a

( f (x, c+d ) + mf ( x

m , c+d )

)dx. ()

Applying the first inequality in () for fx on [c, d] we have

f(

x,c + d

)≤

c – d

∫ d

c

( f (x, y) + mf (x, ym )

)dy.

Put x = a+b we get

f(

a + b

,c + d

)≤

c – d

∫ d

c

( f ( a+b , y) + mf ( a+b

, ym )

)dy. ()

By adding () and () we get (). �

Remark . By putting m = in Theorem . and Theorem . and combining we getinequalities in Theorem ..

Theorem . Let f , fx, and fy be defined as in Theorem .. Then one has the followinginequality:

(b – a)(d – c)

∫ b

a

∫ d

cf (x, y) dy dx

≤ min

{

b – a

∫ b

a

( f (x, c) + mf (x, dm )

)dx,

b – a

∫ b

a

( f (x, d) + mf (x, cm )

)dx

}


+ min

{

d – c

∫ d

c

( f (a, y) + mf ( bm , y)

)dy,

d – c

∫ d

c

( f (b, y) + mf ( am , y)

)dy

}. ()

Proof Since mapping f : � → R is m-convex on the coordinates, the functions fx and fy

are m-convex on [, d] and [, b], respectively. Thus we have by Theorem .

b – a

∫ b

afy(x) dx ≤ min

{ fy(a) + fy( bm )

,

fy(b) + fy( am )

},

that is,

b – a

∫ b

af (x, y) dx ≤ min

{ f (a, y) + f ( bm , y)

,

f (b, y) + f ( am , y)

}.

Performing the average integral over the interval [c, d]

(b – a)(d – c)

∫ b

a

∫ d

cf (x, y) dx

≤ min

{

d – c

∫ d

c

( f (a, y) + f ( bm , y)

)dy,

d – c

∫ d

c

( f (b, y) + f ( am , y)

)dy

}. ()

Similarly for fx one has

(b – a)(d – c)

∫ b

a

∫ d

cf (x, y) dx

≤ min

{

b – a

∫ b

a

( f (x, c) + f (x, dm )

)dx,

b – a

∫ b

a

( f (x, d) + f (x, cm )

)dx

}. ()

For the desired result we add inequalities () and (). �

Theorem . Let fx : [, d] ⊂ [,∞) → R, fx(u) = f (x, u) and fy : [, b] ⊂ [,∞) → R,fy(v) = f (v, y) be partial m-convex mappings with m ∈ (, ]. If ≤ a < b < ∞, ≤ c < d < ∞,also if fx ∈ L[, d] and fy ∈ L[, b] with fy and fx are differentiable on (,∞), then we havethe following inequality:

m

(

b – a

∫ b

af (x, md) dx +

d – c

∫ d

cf (mb, y) dy

)

–

(d – cb – a

∫ b

a

∂f (x, md)∂y

dx +b – ad – c

∫ d

c

∂f (mb, y)∂x

dy)

≤ (b – a)(d – c)

∫ b

a

∫ d

cf (x, y) dy dx ≤

(b – a)(d – c)

×[

(d – mc)∫ b

af (x, d) dx + (b – ma)

∫ d

cf (b, y) dy

–(

(c – md)∫ b

af (x, c) dx + (a – mb)

∫ d

cf (a, y) dy

)]. ()


Proof Since the functions fx and fy are m-convex on [, d] and [, b], respectively, and thefunctions fx and fy are differentiable on (,∞), applying Theorem . we have

fx(md)m

–d – c

f ′x(md) ≤

d – c

∫ d

cfx(y) dy

≤ (d – mc)fx(d) – (c – md)fx(c)(d – c)

and

fy(mb)m

–b – a

f ′y (mb) ≤

b – a

∫ b

afy(x) dy

≤ (b – ma)fy(b) – (a – mb)fy(a)(b – a)

.

This gives us

f (x, md)m

–d – c

∂f (x, md)

∂y≤

d – c

∫ d

cf (x, y) dy

≤ (d – mc)f (x, d) – (c – md)f (x, c)(d – c)

()

and

f (mb, y)m

–b – a

∂f (mb, y)

∂x≤

b – a

∫ b

af (x, y) dx

≤ (b – ma)f (b, y) – (a – mb)f (a, y)(b – a)

. ()

Integrating () over [a, b] and () over [c, d] we get

b – a

∫ b

a

f (x, md)m

dx –d – c

(b – a)

∫ b

a

∂f (x, md)∂y

dx

≤ (b – a)(d – c)

∫ b

a

∫ d

cf (x, y) dy dx

≤ b – a

∫ b

a

((d – mc)f (x, d) – (c – md)f (x, c)

(d – c)

)dx ()

and

d – c

∫ d

c

f (mb, y)m

dy –b – a

(d – c)

∫ d

c

∂f (mb, y)∂x

dy

≤ (b – a)(d – c)

∫ b

a

∫ d

cf (x, y) dy dx

≤ d – c

∫ d

c

((b – ma)f (b, y) – (a – mb)f (a, y)

(b – a)

)dy, ()

respectively.Adding () and () we get (). �


Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally.

Author details1COMSATS Institute of Information Technology, Kamra Road, Attock, 43600, Pakistan. 2Government Degree College,Peshawar, Pakistan.

AcknowledgementsThe research of the first and third authors is partially funded by the COMSATS Institute of Information Technology,Islamabad, Pakistan.

Received: 17 July 2015 Accepted: 1 September 2015

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