International Journal of Analysis and Applications
ISSN 2291-8639
Volume 5, Number 2 (2014), 185-190
http://www.etamaths.com

GENERALIZATION OF INTEGRAL INEQUALITIES FOR

PRODUCT OF CONVEX FUNCTIONS

M. A. LATIF

Abstract. In this paper, generalizations of some inequalities for product of

convex functions are given.

1. Introduction

A function f : [a,b] → R, with [a,b] ⊂ R, is said to be convex if whenever, x,
y ∈ [a,b] and t ∈ [0, 1] the following inequality holds:

f (tx + (1 − t) y) ≤ tf (x) + (1 − t) f (y) .

This definition has its origin in Jensen’s result from [1] and has opened up a very
useful and multi-disciplinary domain of mathematics, namely, convex analysis. A
largely applied inequality for convex functions, due to its geometrical significance, is
the Hermite-Hadamard’s inequality which has generated a wide range of directions
for extensions and rich mathematical literature.

Hermite-Hadamrd’s inequality is stated as follows:
A convex function satisfies:

(1.1) f

(
a + b

2

)
≤

1

b−a

∫ b
a

f (x) dx ≤
f (a) + f (b)

2
.

In a recent paper, Pachpatte [2] established the following inequalities for product
of convex functions which can be derived from Hermite-Hadamard’s inequality:

Theorem 1. [2] Let f and g be real valued, nonnegative and convex functions on
[a,b]. Then

(1.2)
3

2
·

1

(b−a)2
∫ b
a

∫ b
a

∫ 1
0

f (tx + (1 − t) y) g (tx + (1 − t) y) dtdxdy

≤
1

b−a

∫ b
a

f (x) g (x) dx +
1

8

[
M (a,b) + N (a,b)

(b−a)2

]

2010 Mathematics Subject Classification. Primary 46C05, 46C99; Secondary 26D15, 26D20.
Key words and phrases. convex function, Hermite-Hadamard’s inequality, Pachpatte’s inequal-

ities, Jensen’s inequality.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

185



186 M. A. LATIF

and

(1.3)
3

b−a

∫ b
a

∫ 1
0

f

(
tx + (1 − t)

a + b

2

)
g

(
tx + (1 − t)

a + b

2

)
dtdx

≤
1

b−a

∫ b
a

f (x) g (x) dx +
1

4
·

1 + b−a
b−a

[M (a,b) + N (a,b)] ,

where

M (a,b) = f (a) g (a) + f (b) g (b)

N (a,b) = f (a) g (b) + f (b) g (a) .

The inequalities (1.2) and (1.3) are valid when the length of the interval [a,b]
does not exceed 1. The inequality (1.2) is sharp for linear functions defined on
[0, 1], while the inequality (1.3) does not have the same property.

In [3], Cristescu improved these inequalities by eliminating the condition b−a ≤ 1
and derived the inequalities which are sharp for the whole class of linear functions
defined on [0, 1]. The main result from [3] is the following:

Theorem 2. [3] Let f and g be real valued, nonnegative and convex functions on
[a,b]. Then

(1.4)
3

2
·

1

(b−a)2
∫ b
a

∫ b
a

∫ 1
0

f (tx + (1 − t) y) g (tx + (1 − t) y) dtdxdy

≤
1

b−a

∫ b
a

f (x) g (x) dx +
1

8
[M (a,b) + N (a,b)]

and

(1.5)
3

b−a

∫ b
a

∫ 1
0

f

(
tx + (1 − t)

a + b

2

)
g

(
tx + (1 − t)

a + b

2

)
dtdx

≤
1

b−a

∫ b
a

f (x) g (x) dx +
1

2
[M (a,b) + N (a,b)] ,

where M (a,b) and N (a,b) are defined in Theorem 1.

The main aim of this paper is to generalize the inequalities (1.4) and (1.5).

2. Main Results

Let I be an interval of R and let f : I → R be a convex functions on I, h :
[a,b] → R be continuous function such that h ([a,b]) ⊂ I and p : [a,b] → R be a
positive integrable function a, b ∈ R with a < b. Then the Jenesen’s inequality

f

(∫ b
a
p (x) h (x) dx∫ b
a
p (x) dx

)
≤
∫ b
a
p (x) f (h (x)) dx∫ b

a
p (x) dx

holds.
Assume that f and p are as above. Let us denote

P =

∫ b
a

p(x)dx, h̄ =
1

P

∫ b
a

p(x)h(x)dx.



GENERALIZATION OF INTEGRAL INEQUALITIES 187

The following is the Hermite-Hdamard type inequality in this case:

(2.1) f
(
h̄
)
≤

1

P

∫ b
a

f (h(x))p(x)dx ≤
f (h(a)) + f (h(b))

2
.

We now state the following lemma which is very useful in this section:

Lemma 1. Let [a,b] ⊂ R and f : [a,b] → R be a function and h : [a,b] → R be a
continuous function such that h ([a,b]) ⊂ [a,b]. Then the following statements are
equivalent

(1) function f is convex on [a,b]
(2) For every x, y ∈ [a,b], the function γ : [0, 1] → R defined by

γ (t) = f (th (x) + (1 − t) h (y))

is convex on [0, 1] for any positive real number λ.

Proof. It is a direct consequence of the convexity of the function f. �

Now we state and prove the main result of this section which will generalize the
Theorem 2.

Theorem 3. Let f and g be real valued, nonnegative and convex functions on [a,b].
Let h : [a,b] → R be continuous function such that h ([a,b]) ⊂ [a,b]and p : [a,b] → R
be a positive integrable function. Then

(2.2)

3

2P2

∫ b
a

∫ b
a

∫ 1
0

p (x) p (y) f (th (x) + (1 − t) h (y)) g (th (x) + (1 − t) h (y)) dtdxdy

≤
1

P

∫ b
a

f (h (x)) g (h (x)) p (x) dx +
1

8

[
M

′
(a,b) + N

′
(a,b)

]
,

where

M
′
(a,b) = f(h(a))g(h(a)) + f(h(b))g(h(b))

and

N
′
(a,b) = f(h(a))g(h(b)) + f(h(b))g(h(a)).

Proof. Since both functions f and g are convex, for every two points x,y ∈ [a,b]
and t ∈ [0, 1], the following inequalities are valid

f (th (x) + (1 − t) h (y)) ≤ tf (h (x)) + (1 − t) f (h (y))

and

g (th (x) + (1 − t) h (y)) ≤ tg (h (x)) + (1 − t) g (h (y))
Multiplying these inequalities side by side, we obtain

(2.3) f (th (x) + (1 − t) h (y)) g (th (x) + (1 − t) h (y))

≤ t2f (h (x)) g (h (x)) + (1 − t)2 f (h (y)) g (h (y))
+ t (1 − t) [f (h (x)) g (h (y)) + f (h (y)) g (h (x))] .

Due to Lemma 1 and known properties of convex functions, both sides of the
inequality (2.3) are integrable. Multiplying both sides of (2.3) by p (x) p (y) and



188 M. A. LATIF

integrating both sides of the inequality (2.3) with respect to t over [0, 1], with
respect to x and y over [a,b], we get

(2.4)∫ b
a

∫ b
a

∫ 1
0

p (x) p (y) f (th (x) + (1 − t) h (y)) g (th (x) + (1 − t) h (y)) dtdxdy

≤
2

3
P

∫ b
a

f (h (x)) g (h (x)) p(x)dx

+
1

3

(∫ b
a

f (h (x)) p(x)dx

)(∫ b
a

g (h (x)) p(x)dx

)
.

The convexity property of f and g allow us to use right side of the inequality (2.1)
and thus the above inequality (2.4) takes the form:

(2.5)∫ b
a

∫ b
a

∫ 1
0

p (x) p (y) f (th (x) + (1 − t) h (y)) g (th (x) + (1 − t) h (y)) dtdxdy

≤
2

3
P

∫ b
a

f (h (x)) g (h (x)) p(x)dx

+
P2

12
[f (h(a)) + f (h(b))] [g (h(a)) + g (h(b))]

=
2

3
P

∫ b
a

f (h (x)) g (h (x)) p(x)dx

+
P2

12

[
M

′
(a,b) + N

′
(a,b)

]
.

Multiplying both sides of the inequality (2.5) by 3
2P 2

, we get the desired result.
This completes the proof of the theorem. �

Theorem 4. Let f and g be real valued, nonnegative and convex functions on [a,b].
Let h : [a,b] → R be continuous function such that h ([a,b]) ⊂ [a,b]and p : [a,b] → R
be a positive integrable function. Then

(2.6)
3

P

∫ b
a

∫ 1
0

p(x)f
(
th (x) + (1 − t) h̄

)
g
(
th (x) + (1 − t) h̄

)
≤

1

P

∫ b
a

f(h(x))g(h(x))p(x)dx +
1

2

[
M

′
(a,b) + N

′
(a,b)

]
,

where

M
′
(a,b) = f(h(a))g(h(a)) + f(h(b))g(h(b))

and

N
′
(a,b) = f(h(a))g(h(b)) + f(h(b))g(h(a)).

Proof. Again by the convexity of the functions f and g, we have

f
(
th (x) + (1 − t) h̄

)
≤ tf(h(x)) + (1 − t) f

(
h̄
)

and

g
(
th (x) + (1 − t) h̄

)
≤ tg(h(x)) + (1 − t) g

(
h̄
)



GENERALIZATION OF INTEGRAL INEQUALITIES 189

Multiplying the above two inequalities side by side, we get

(2.7) f
(
th (x) + (1 − t) h̄

)
g
(
th (x) + (1 − t) h̄

)
≤ t2f(h(x))g(h(x)) + t (1 − t)

[
f(h(x))g(h̄) + g(h(x))f(h̄)

]
+ (1 − t)2 f(h̄)g(h̄).

Multiplying both sides of (2.7), by similar arguments as in obtaining (2.4) and using
the Jensen’s inequality, we have

(2.8)

∫ b
a

∫ 1
0

p(x)f
(
th (x) + (1 − t) h̄

)
g
(
th (x) + (1 − t) h̄

)
≤

1

3

∫ b
a

f(h(x))g(h(x))p(x)dx +
1

6

∫ b
a

[
f(h(x))g(h̄) + g(h(x))f(h̄)

]
p(x)dx

+
1

3

∫ b
a

f(h̄)g(h̄)p(x)dx

≤
1

3

∫ b
a

f(h(x))g(h(x))p(x)dx+
2

3P

(∫ b
a

f(h(x))p(x)dx

)(∫ b
a

g(h(x))p(x)dx

)
.

An application of the inequality (2.1), gives us

(2.9)

∫ b
a

∫ 1
0

p(x)f
(
th (x) + (1 − t) h̄

)
g
(
th (x) + (1 − t) h̄

)
≤

1

3

∫ b
a

f(h(x))g(h(x))p(x)dx +
P

6
[f(h(a)) + f(h(b))] [g(h(a)) + g(h(b))]

=
1

3

∫ b
a

f(h(x))g(h(x))p(x)dx +
P

6

[
M

′
(a,b) + N

′
(a,b)

]
.

Multiplying both sides of (2.9) by 3
P

, we get the desired inequality and hence the
proof of the theorem is complete. �

Remark 1. If in Theorem 3 and Theorem 4, p(x) = 1 and h(x) = x, x ∈ [a,b],
then P = b − a, h̄ = a+b

2
, M

′
(a,b) = M(a,b) and N

′
(a,b) = N(a,b). Then the

inequalities (2.2) and (2.6) reduce to the inequalities (1.4) and (1.5) respectively.
This also shows that our results generalize those results proved in Theorem 2.

References

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Math. B., 16 (1905), 49–69.

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tion, 6(E) (2003), [ONLINE: http://rgmia.vu.edu.au/v6(E).html].

[3] G. Cristescu, Improved integral inequalities for product of convex functions, Journal of In-

equalities in Pure and Applied Mathematics, 6(2005), Issue 2, Article 35.
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Publishers, Dordrecht / Boston / London, 2002.

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ities and Applications, RGMIA Monographs, Victoria University, 2000. [ONLINE:

http://rgmia.vu.edu.au/monographs/]



190 M. A. LATIF

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College of Science, Department of Mathematics, University of Hail, Hail 2440, Saudi

Arabia