# Integral Majorization Theorem for Invex Functions

Authors: M. Adil Khan; Adem Kılıçman; N. Rehman
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We obtain some general inequalities and establish integral inequalities of the majorization type for invex functions. We give applications to relative invex functions.


---

## Body

## 1. Introduction

Letx
,
y
:
[
a
,
b
]
→
ℝ be two decreasing real functions. Then the function x is said tomajorize
y if
(1)
∫
a
s
‍
x
(
t
)
d
t
≥
∫
a
s
‍
y
(
t
)
d
t
for
s
∈
[
a
,
b
]
,
∫
a
b
‍
x
(
t
)
d
t
=
∫
a
b
‍
y
(
t
)
d
t
(see [1, p. 417] and [2, p. 324]).The following result is known asmajorization theorem for integrals (see [1, p. 417] and [2, p. 325]).Theorem 1 (see [1, 2]).
Letx
,
y
:
[
a
,
b
]
→
I be two decreasing real functions, where I
⊂
ℝ is an interval. The function x majorizes y if and only if the inequality
(2)
∫
a
b
‍
ϕ
(
y
(
t
)
)
d
t
≤
∫
a
b
‍
ϕ
(
x
(
t
)
)
d
t
holds for all continuous convex functions ϕ
:
I
→
ℝ such that the integrals exist.Some authors have investigated the weighted versions of (2) (see [1, 3, 4]).In our main results we will use the following definition of invex function.Definition 2.
Letϕ
:
I
→
ℝ be a differentiable function on the interval I, and let η
:
I
×
I
→
ℝ be a function of two variables. The function ϕ is said to be η-invex if, for all x
,
y
∈
I,
(3)
ϕ
(
x
)
≥
ϕ
(
y
)
+
ϕ
′
(
y
)
η
(
x
,
y
)
;
see  [5, pp. 1]. ϕ is called invex if ϕ is η-invex for some η.Clearly, each differentiable convex functionϕ
:
I
→
ℝ is an η-invex function with η
(
x
,
y
)
=
x
-
y for x
,
y
∈
I. It is known that a differentiable function is invex if and only if each stationary point is a global minimum point [5]. This fact was the motivation to introduce invex functions in optimization theory [6].LetΘ
:
ℝ
2
→
ℝ be an arbitrary function vanishing at points of the form (
0
,
b
), b
∈
ℝ. It is easy to verify that if a differentiable function ϕ satisfies the condition
(4)
Θ
(
ϕ
′
(
y
)
,
η
0
(
x
,
y
)
)
≥
0
implies
ϕ
(
x
)
-
ϕ
(
y
)
≥
0
for
x
,
y
∈
I
for some functions η
0, then ϕ is invex.In particular each pseudoconvex function is invex [5, pp. 3-4]. In fact, it is sufficient to consider Θ
(
a
,
b
)
=
a
b for a
,
b
∈
ℝ and η
0
(
x
,
y
)
=
x
-
y for x
,
y
∈
I.In the multidimensional case(
n
≥
1
), we have the following definition.Definition 3.
Let〈
·
,
·
〉 be an inner product on ℝ
n. Let η
:
I
n
×
I
n
→
ℝ
n be a function of two variables. One has the following.(i)
A differentiable functionF
:
I
n
→
ℝ is said to be η-invex if, for all x, y
∈
I
n,
(5)
F
(
x
)
-
F
(
y
)
≥
〈
∇
F
(
y
)
,
η
(
x
,
y
)
〉
,
where ∇ denotes the gradient [5, pp. 1].
(ii)
A differentiable functionF
:
I
n
→
ℝ is said to be η-pseudo-invex if, for all x
,
y
∈
I
n,
(6)
〈
∇
F
(
y
)
,
η
(
x
,
y
)
〉
≥
0
implies
F
(
x
)
-
F
(
y
)
≥
0
(see [6]).
(iii)
A differentiable functionF
:
I
n
→
ℝ is said to be η-quasi-invex if, for all x
,
y
∈
I
n,
(7)
F
(
x
)
-
F
(
y
)
≤
0
implies
〈
∇
F
(
y
)
,
η
(
x
,
y
)
〉
≤
0
(see [6]).A differentiable real functionF
:
I
n
→
ℝ is said to be invex (resp., pseudoinvex, quasi-invex), if F is η-invex (resp., η-pseudo-invex, η-quasi-invex) for some functions η
:
I
n
×
I
n
→
ℝ
n.For applications of invex functions in optimization and mathematical programming, see [5–12] and for some recent results of majorization discrete results for invex functions see [13].In this paper, we extend integral version of majorization theorem from convex functions to invex ones. We also give some applications to relative invex functions.


## 2. Main Results

In the following theorem we obtain an inequality which we will use in our other results.Theorem 4.
Letϕ
:
I
→
ℝ be an η-invex function on the interval I, where η
:
I
×
I
→
ℝ is a continuous function, and let f
,
g
,
p
:
[
a
,
b
]
→
I be integrable functions with p
(
t
)
≥
0 for t
∈
[
a
,
b
]. Then
(8)
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
-
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
≥
∫
a
b
‍
p
(
t
)
ϕ
′
(
g
(
t
)
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
.Proof.
If we takex
→
f
(
t
), y
→
g
(
t
) in (3), we obtain
(9)
ϕ
(
f
(
t
)
)
-
ϕ
(
g
(
t
)
)
≥
ϕ
′
(
g
(
t
)
)
η
(
f
(
t
)
,
g
(
t
)
)
.
Multiplying (9) by p
(
t
)
≥
0 and integrating with respect to t, we deduce (8).The following weighted integral majorization theorem holds.Theorem 5.
Letϕ
:
I
→
ℝ be an η-invex function on the interval I, where η
:
I
×
I
→
ℝ is a continuous function, and let f
,
g
,
p
:
[
a
,
b
]
→
I be integrable functions with p
(
t
)
>
0 for t
∈
[
a
,
b
]. Moreover if ϕ
′
(
g
(
t
)
) and η
(
f
(
t
)
,
g
(
t
)
) are increasing (decreasing) on [
a
,
b
] and
(10)
∫
a
b
‍
p
(
t
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
=
0
,
then
(11)
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
≥
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
.Proof.
We know that the Chebyshev’s inequality is(12)
1
∫
a
b
‍
p
(
t
)
d
t
∫
a
b
‍
p
(
t
)
F
(
t
)
G
(
t
)
d
t
≥
1
∫
a
b
‍
p
(
t
)
d
t
∫
a
b
‍
p
(
t
)
F
(
t
)
d
t
1
∫
a
b
‍
p
(
t
)
d
t
∫
a
b
‍
p
(
t
)
G
(
t
)
d
t
,
where F
,
G
:
[
a
,
b
]
→
ℝ are the functions of same monotonicity and p
:
[
a
,
b
]
→
ℝ
+ is any integrable function.
By assumption the functionsϕ
′
(
g
(
t
)
), η
(
f
(
t
)
,
g
(
t
)
)
:
[
a
,
b
]
→
ℝ have the same monotonicity. Therefore, by applying Chebyshev's inequality (12) in the right hand side of (8), we have
(13)
∫
a
b
‍
p
(
t
)
ϕ
′
(
g
(
t
)
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
≥
1
∫
a
b
‍
p
(
t
)
d
t
∫
a
b
‍
p
(
t
)
ϕ
′
(
g
(
t
)
)
d
t
×
∫
a
b
‍
p
(
t
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
.
Also (10) holds, so from (8) and (13) we have
(14)
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
-
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
≥
∫
a
b
‍
p
(
t
)
ϕ
′
(
g
(
t
)
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
≥
1
∫
a
b
‍
p
(
t
)
d
t
∫
a
b
‍
p
(
t
)
ϕ
′
(
g
(
t
)
)
d
t
×
∫
a
b
‍
p
(
t
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
=
0
;
then we deduce the desired result (11).Remark 6.
In Theorem5 the assumption (10) is a strong condition for p, η. This can be relaxed if a monotonicity property for the η-invex function ϕ is assumed.Theorem 7.
Let all the assumptions of Theorem5 hold, but instead of (10) one has the assumption that
(15)
∫
a
b
‍
p
(
t
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
≥
0
,
and ϕ
:
I
→
ℝ is an η-invex increasing function on the interval I; then (11) holds.Proof.
The proof is similar to the proof of Theorem5 but by (15) and monotonicity of ϕ we have
(16)
∫
a
b
‍
p
(
t
)
ϕ
′
(
g
(
t
)
)
d
t
≥
0
,
∫
a
b
‍
p
(
t
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
≥
0
.
Using these in (13) we have
(17)
∫
a
b
‍
p
(
t
)
ϕ
′
(
g
(
t
)
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
≥
0
.Theorem 8.
Letϕ
:
I
→
ℝ be an η-invex function on the interval I, where η
:
I
×
I
→
ℝ is a continuous function, and let f
,
g
,
p
:
[
a
,
b
]
→
I be integrable functions with p
(
t
)
≥
0 for t
∈
[
a
,
b
]. One has the following.(i)
Ifϕ
′
∘
g is increasing on [
a
,
b
], ∫
a
x
‍
p
(
t
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
≤
0 for all x
∈
[
a
,
b
] and ∫
a
b
‍
p
(
t
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
=
0, then
(18)
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
≥
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
.
(ii)
Ifϕ
′
∘
f is decreasing on [
a
,
b
], ∫
a
x
‍
p
(
t
)
η
(
g
(
t
)
,
f
(
t
)
)
d
t
≥
0 for all x
∈
[
a
,
b
] and ∫
a
b
‍
p
(
t
)
η
(
g
(
t
)
,
f
(
t
)
)
d
t
=
0, then
(19)
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
≤
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
.Proof.
Let setG
(
x
)
=
∫
a
x
‍
p
(
t
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t; then G
(
x
)
≤
0 for all x
∈
[
a
,
b
], and G
(
a
)
=
G
(
b
)
=
0. If ϕ
′
∘
g is increasing on [
a
,
b
], then
(20)
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
-
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
≥
∫
a
b
‍
p
(
t
)
ϕ
′
(
g
(
t
)
)
η
(
f
(
t
)
,
g
(
t
)
)
d
t
=
∫
a
b
‍
ϕ
′
(
g
(
t
)
)
d
G
(
t
)
=
ϕ
′
(
g
(
t
)
)
G
(
t
)
|
a
b
-
∫
a
b
‍
G
(
t
)
d
ϕ
′
(
f
(
t
)
)
=
-
∫
a
b
‍
G
(
t
)
d
ϕ
′
(
g
(
t
)
)
≥
0
.
This proves (18).
Similarly, settingF
(
x
)
=
∫
a
x
‍
p
(
t
)
η
(
g
(
t
)
,
f
(
t
)
)
d
t, then F
(
x
)
≥
0 for all x
∈
[
a
,
b
], and F
(
a
)
=
F
(
b
)
=
0.
Now ifϕ
′
∘
f is decreasing on [
a
,
b
], then by using Theorem 4 we have
(21)
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
-
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
≥
∫
a
b
‍
p
(
t
)
ϕ
′
(
f
(
t
)
)
η
(
g
(
t
)
,
f
(
t
)
)
d
t
=
∫
a
b
‍
ϕ
′
(
f
(
t
)
)
d
F
(
t
)
=
ϕ
′
(
f
(
t
)
)
F
(
t
)
|
a
b
-
∫
a
b
‍
F
(
t
)
d
ϕ
′
(
f
(
t
)
)
=
-
∫
a
b
‍
F
(
t
)
d
ϕ
′
(
f
(
t
)
)
≥
0
.
This proves (19).The following extension of majorization theorem for relative invex function can be given.Theorem 9.
Letf
,
g
,
p
:
[
a
,
b
]
→
ℝ be integrable functions with p being positive function. Suppose ϕ
,
ψ
:
ℝ
→
ℝ are such that ψ is a strictly increasing function and ϕ
∘
ψ
-
1 is η-invex function on ℝ, where η
:
ℝ
×
ℝ
→
ℝ is a continuous function. One has the following.(i)
If(
ϕ
∘
ψ
-
1
)
′
∘
ψ
∘
g and η
(
ψ
(
f
(
t
)
)
,
ψ
(
g
(
t
)
)
) are increasing (decreasing) on [
a
,
b
], and ∫
a
b
‍
p
(
t
)
η
(
ψ
(
f
(
t
)
)
,
ψ
(
g
(
t
)
)
)
d
t
=
0, then
(22)
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
≥
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
.
(ii)
If(
ϕ
∘
ψ
-
1
)
′
∘
ψ
∘
g is increasing on [
a
,
b
], ∫
a
x
‍
p
(
t
)
η
(
ψ
(
f
(
t
)
)
,
ψ
(
g
(
t
)
)
)
d
t
≤
0 for all x
∈
[
a
,
b
], and∫
a
b
‍
p
(
t
)
η
(
ψ
(
f
(
t
)
)
,
ψ
(
g
(
t
)
)
)
d
t
=
0, then
(23)
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
≥
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
.
(iii)
If(
ϕ
∘
ψ
-
1
)
′
∘
ψ
∘
f is decreasing on [
a
,
b
], ∫
a
x
‍
p
(
t
)
η
(
ψ
(
g
(
t
)
)
,
ψ
(
f
(
t
)
)
)
d
t
≥
0 for all x
∈
[
a
,
b
], and ∫
a
b
‍
p
(
t
)
η
(
ψ
(
g
(
t
)
)
,
ψ
(
f
(
t
)
)
)
d
t
=
0, then
(24)
∫
a
b
‍
p
(
t
)
ϕ
(
g
(
t
)
)
d
t
≤
∫
a
b
‍
p
(
t
)
ϕ
(
f
(
t
)
)
d
t
.Proof.
The proof of this theorem is similar to the proof of Theorems5 and 8.Remark 10.
By usingη
(
x
,
y
)
=
x
-
y in Theorems 4, 5, 7, 8, and 9 we recover the integral majorization results for convex functions given in [1, 4, 14].



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