Farid et al. Journal of Inequalities and Applications (2015) 2015:283
DOI 10.1186/s13660-015-0808-z
RESEARCH
Open Access
New mean value theorems and
generalization of Hadamard inequality via
coordinated m-convex functions
G Farid1 , M Marwan2 and Atiq Ur Rehman1*
*
Correspondence:
[email protected]
1
COMSATS Institute of Information
Technology, Kamra Road, Attock,
43600, Pakistan
Full list of author information is
available at the end of the article
Abstract
We derive new mean value theorems for functionals associated with Hadamard
inequality for convex functions on the coordinates. We present some Hadamard type
inequalities and related results for m-convex functions on the coordinates.
Keywords: convex functions; m-convex functions; convex functions on coordinates;
Hadamard inequality; mean value theorems
1 Introduction
We start with the very basic concept of a convex function that has been seen as important
ever since it was defined.
Definition . A real valued function f : I → R, where I is an interval in R, is called convex
if
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 mathematicians in many fields of mathematics such as functional analysis, mathematical statistics,
and complex analysis (see for example [–] and references therein). In the field of inequalities, convex functions play a unique role. Most of the new inequalities till now defined are results and consequences of (). The most cardinal and classical inequality for
convex 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
a
b
f (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 generalization and modification of Hadamard type inequalities using different modification
© 2015 Farid et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Farid et al. Journal of Inequalities and Applications (2015) 2015:283
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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 the
Hadamard inequality on the coordinates and its different variants. Also they have given
many 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] and
x ∈ [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 the
converse is not true. Dragomir gave the Hadamard inequality for a rectangle in the plane
for convex functions on the coordinates (see []).
Theorem . Suppose that f :  → R is convex on the coordinates on  . Then one has
the following inequalities:
f
a+b c+d
,


b d 

c+d
a+b

dx +
, y dy
f x,
f
≤
 b–a a

d–c c

b d

≤
f (x, y) dx dy
(b – a)(d – c) a c
b
b
d




f (x, c) dx +
f (x, d) dx +
f (a, y) dy
≤
 b–a a
b–a a
d–c c
d

+
f (b, y) dy
d–c c

≤ f (a, c) + f (a, d) + f (b, c) + f (b, d) .

In [], Toader defined the concept of m-convexity, an intermediate between the usual
convexity and star shape properties.
Definition . The function f : [, b] → R is said to be m-convex, where m ∈ [, ], if for
every 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 Hadamard
type inequality for m-convex functions.
Farid et al. Journal of Inequalities and Applications (2015) 2015:283
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Theorem . Let f : [, ∞) → R be m-convex function with m ∈ (, ] and  ≤ a < b. If
f ∈ L [a, b], then one has the following inequalities:
a+b
f

f (x) + mf ( mx )
dx

a
a
f ( m ) + f ( mb )
m +  f (a) + f (b)
≤
+m
.




≤
b–a
b
()
They also gave the following related results to the Hadamard type inequality for mconvex 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
a
f (a) + mf ( mb ) f (b) + mf ( ma )
f (x) dx ≤ min
,
.


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) b – a –
f (mb) ≤
m

b–a
≤
b
f (x) dx
a
(b – ma)f (d) – (a – mb)f (a)
.
(b – a)
In this paper, new mean value theorems of Cauchy type for functionals associated with
nonnegative differences of the Hadamard inequality on the coordinates are proved. Generalized results related to the Hadamard inequality for m-convex functions on the coordinates are also given.
2 Mean value theorems
We know that if a function f is twice differentiable on an interval I then it is convex on
I if and only if its second order derivative is nonnegative. If a function f (X) := f (x, y) has
continuous second order partial derivatives on interior of  then it is convex on  if the
Hessian matrix
⎛
Hf (X) =
∂  f (X)
⎝ ∂x
∂ f (X)
∂x∂y
⎞
∂  f (X)
∂y∂x ⎠
∂  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
fx (y) =
∂  f (x, y)
∂y
and
fy (x) =
∂  f (x, y)
∂x
are nonnegative for all interior points (x, y) in  .
Farid et al. Journal of Inequalities and Applications (2015) 2015:283
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For a real valued function f :  → R we define
H(f ) =
b d 


c+d
a+b
dx +
, y dy
f x,
f
 b–a a

d–c c

a+b c+d
,
.
–f


()
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)
≤ M
∂x
and m ≤
∂  f (x, y)
≤ M
∂y
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
∂  f (x, y)
∂  g(x, y)
=
max{M
,
M
}
–
≥


∂x
∂x
and
∂  g(x, y) ∂  f (x, y)
=
– min{m , m } ≥ 
∂y
∂y
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 of
second order in  and q(x, y) := x + y . Then there exist (η , ξ ) and (η , ξ ) in the interior
of  such that
H(f ) =
 ∂  f (η , ξ )
H(q)

∂x
H(f ) =
 ∂  f (η , ξ )
H(q),

∂x
and
provided that H(q) = .
Farid et al. Journal of Inequalities and Applications (2015) 2015:283
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Proof Since f has continuous partial derivatives of second order in compact set  , there
exist real numbers m , m , M , and M such that
m ≤
∂  f (x, y)
≤ M
∂x
and
m ≤
∂  f (x, y)
≤ M .
∂y
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 )
≤ max{M , M }.
H(q)
Then there exist (η , ξ ) and (η , ξ ) in the interior of  such that
H(f ) ∂  f (η , ξ )
=
H(q)
∂x
and
H(f ) ∂  f (η , ξ )
=
.
H(q)
∂y
Hence the required result follows.
3 Hadamard type inequalities for m-convex function on two coordinates
In this section we give Hadamard type inequalities for m-convex functions on two coordinates. 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 called
m-convex on the coordinates if the partial mappings fy : [, b] → R, fy (u) := f (u, y), and
fx : [, 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 the
coordinates.
Theorem . Let = [, b] × [, d] ⊂ [, ∞) with b, d >  and f : → R be m-convex
on the coordinates in with m ∈ (, ]. If fx ∈ L [, d] and fy ∈ L [, b],  ≤ a < b,  ≤ c < d.
Farid et al. Journal of Inequalities and Applications (2015) 2015:283
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Then we have
d 
c+d
a+b
dx +
, y dy
f x,
f

d–c c

a
b d
y

x
f (x, y) + m f x,
+f
,y
dy dx
≤
(b – a)(d – c) a c
m
m
(m + )
a
b
f (a, c) + f (b, c) + f (a, d) + f (b, d) + m f
,c + f
,c
≤

m
m
b
c
d
c
d
a
,d + f
, d + f a,
+ f a,
+ f b,
+ f b,
+f
m
m
m
m
m
m
a c
b c
a d
b d
+ m f
,
+f
,
+f
,
+f
,
.
m m
m m
m m
m m

b–a
b
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 ( mx )
dx

fy ( ma ) + fy ( mb )
m +  fy (a) + fy (b)
≤
+m
,



that is,
f
b
f (x, y) + mf ( mx , y)

a+b
,y ≤
dx

b–a a

f ( ma , y) + f ( mb , y)
m +  f (a, y) + f (b, y)
≤
+m
.



()
From this one has

d–c
d
f
c
a+b
, y dy

b d
f (x, y) + mf ( mx , y)

dy dx
(b – a)(d – c) a c

d
f ( a , y) + f ( mb , y)
f (a, y) + f (b, y)
m+
+m m
dy.
≤
(d – c) c


≤
()
Similarly using () for the function fx we have
c+d
dx
f x,

a
b d
f (x, y) + mf (x, my )

≤
dy dx
(b – a)(d – c) a c

b
f (x, mc ) + f (x, md )
f (x, c) + f (x, d)
m+
+m
dx.
≤
(b – a) a



b–a
b
()
Farid et al. Journal of Inequalities and Applications (2015) 2015:283
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By adding () and (), we get
d c+d
a+b

f x,
f
dx +
, y dy

d–c c

a
b d
x
y

+f
,y
dy dx
f (x, y) + m f x,
≤
(b – a)(d – c) a c
m
m
b
f (x, mc ) + f (x, md )
f (x, c) + f (x, d)
m+

≤
+m
dx

b–a a


d
f ( a , y) + f ( mb , y)
f (a, y) + f (b, y)

+m m
dy .
+
d–c c



b–a
b
()
For fixed y using the m-convexity of fy we have
f (x, y) + mf ( mx , y)
.

f (x, y) ≤
()
Performing the average integral over the interval [a, b] and using () we get

b–a
b
a

f (x, y) dx ≤
b–a
b
a
f (x, y) + mf ( mx , y)
dx

f ( ma , y) + f ( mb , y)
m +  f (a, y) + f (b, y)
≤
+m
.



()
Similarly for fixed x using the m-convexity of fx one has

d–c
d
f (x, y) dy ≤
c

d–c
d
c
f (x, y) + mf (x, my )
dy

≤
f (x, mc ) + f (x, md )
m +  f (x, c) + f (x, d)
+m
.



Considering () for y = c, d, () for x = a, b, then () for y =
multiply later with m. Adding all these inequalities, we obtain
b
f (x, mc ) + f (x, md )
f (x, c) + f (x, d)
+m
dx


a
d
f ( a , y) + f ( mb , y)
f (a, y) + f (b, y)

+
+m m
dy
d–c c


b
f (x, c) + mf ( mx , c) + f (x, d) + mf ( mx , d)


≤
 b–a a

c
x c
d
f (x, m ) + mf ( m , m ) + f (x, m ) + mf ( mx , md )
dx
+m

d
f (a, y) + mf (a, my ) + f (b, y) + mf (b, my )

+
d–c c

a
a y
b
f ( , y) + mf ( m , m ) + f ( m , y) + mf ( mb , my )
dy
+m m


b–a
c d
, ,
m m
() for x =
()
a b
,
m m
to
Farid et al. Journal of Inequalities and Applications (2015) 2015:283

a
b
≤
f (a, c) + f (b, c) + f (a, d) + f (b, d) + m f
,c + f
,c

m
m
a
b
c
d
c
d
+f
,d + f
, d + f a,
+ f a,
+ f b,
+ f b,
m
m
m
m
m
m
a c
b c
a d
b d
,
+f
,
+f
,
+f
,
.
+ m f
m m
m m
m m
m m
Page 8 of 11
()
Now combining inequalities in () and (), we get the last two inequalities of the theorem.
Theorem . Let = [, b] × [, d] ⊂ [, ∞) with b, d >  and f : → R be m-convex
on the coordinate in with m ∈ (, ]. If fx ∈ L [, d] and fy ∈ L [, b], then
a+b c+d
,
f


Proof By using () for y =
f
a+b c+d
,


≤
b
) + mf ( mx , c+d
)
f (x, c+d


dx

a
d a+b
f (  , y) + mf ( a+b
, y)

 m
dy.
+
c–d c


≤
b–a
()
c+d


b–a
b
a
f (x, c+d
) + mf ( mx , c+d
)


dx.

()
Applying the first inequality in () for fx on [c, d] we have
d
f (x, y) + mf (x, my )

c+d
≤
dy.
f x,

c–d c

Put x =
a+b

f
we get
a+b c+d
,


≤

c–d
d
c
f ( a+b
, y) + mf ( a+b
, y)

 m
dy.

By adding () and () we get ().
()
Remark . By putting m =  in Theorem . and Theorem . and combining we get
inequalities in Theorem ..
Theorem . Let f , fx , and fy be defined as in Theorem .. Then one has the following
inequality:
b d

f (x, y) dy dx
(b – a)(d – c) a c
b
f (x, c) + mf (x, md )

dx,
≤ min
b–a a

b
f (x, d) + mf (x, mc )

dx
b–a a

Farid et al. Journal of Inequalities and Applications (2015) 2015:283
Page 9 of 11
d
f (a, y) + mf ( mb , y)

dy,
+ min
d–c c

d
f (b, y) + mf ( ma , y)

dy .
d–c c

()
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
fy (x) dx ≤ min
a
fy (a) + fy ( mb ) fy (b) + fy ( ma )
,
,


that is,

b–a
a
b
f (a, y) + f ( mb , y) f (b, y) + f ( ma , y)
,
.
f (x, y) dx ≤ min


Performing the average integral over the interval [c, d]
b d

f (x, y) dx
(b – a)(d – c) a c
d
d
f (a, y) + f ( mb , y)
f (b, y) + f ( ma , y)


dy,
dy .
≤ min
d–c c

d–c c

()
Similarly for fx one has
b d

f (x, y) dx
(b – a)(d – c) a c
b
b
f (x, c) + f (x, md )
f (x, d) + f (x, mc )


dx,
dx .
≤ min
b–a a

b–a a

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 have
the following inequality:
b
d



f (x, md) dx +
f (mb, y) dy
m b – a a
d–c c
b
∂f (x, md)
b – a d ∂f (mb, y)
 d–c
dx +
dy
–
 b–a a
∂y
d–c c
∂x
b d


≤
f (x, y) dy dx ≤
(b – a)(d – c) a c
(b – a)(d – c)
b
d
× (d – mc)
f (x, d) dx + (b – ma)
f (b, y) dy
– (c – md)
a
a
b
c
f (x, c) dx + (a – mb)
c
d
f (a, y) dy .
()
Farid et al. Journal of Inequalities and Applications (2015) 2015:283
Page 10 of 11
Proof Since the functions fx and fy are m-convex on [, d] and [, b], respectively, and the
functions fx and fy are differentiable on (, ∞), applying Theorem . we have

fx (md) d – c –
f (md) ≤
m
 x
d–c
≤
d
fx (y) dy
c
(d – mc)fx (d) – (c – md)fx (c)
(d – c)
and
fy (mb) b – a 
–
f (mb) ≤
m
 y
b–a
≤
b
fy (x) dy
a
(b – ma)fy (b) – (a – mb)fy (a)
.
(b – a)
This gives us
f (x, md) d – c ∂f (x, md)

–
≤
m

∂y
d–c
≤
d
f (x, y) dy
c
(d – mc)f (x, d) – (c – md)f (x, c)
(d – c)
()
and
f (mb, y) b – a ∂f (mb, y)

–
≤
m

∂x
b–a
≤
b
f (x, y) dx
a
(b – ma)f (b, y) – (a – mb)f (a, y)
.
(b – a)
()
Integrating () over [a, b] and () over [c, d] we get
b
d–c
f (x, md)
∂f (x, md)
dx –
dx
m
(b – a) a
∂y
a
b d

f (x, y) dy dx
≤
(b – a)(d – c) a c
b

(d – mc)f (x, d) – (c – md)f (x, c)
≤
dx
b–a a
(d – c)

b–a
b
()
and
d
b–a
f (mb, y)
∂f (mb, y)
dy –
dy
m
(d
–
c)
∂x
c
c
b d

≤
f (x, y) dy dx
(b – a)(d – c) a c
d

(b – ma)f (b, y) – (a – mb)f (a, y)
≤
dy,
d–c c
(b – a)

d–c
d
respectively.
Adding () and () we get ().
()
Farid et al. Journal of Inequalities and Applications (2015) 2015:283
Page 11 of 11
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally.
Author details
1
COMSATS Institute of Information Technology, Kamra Road, Attock, 43600, Pakistan. 2 Government Degree College,
Peshawar, Pakistan.
Acknowledgements
The 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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