Research Article
Integral Majorization Theorem for Invex Functions
M. Adil Khan,
1
Adem KJlJçman,
2
and N. Rehman
3
1
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
2
Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM),
43400 Serdang, Selangor, Malaysia
3
Department of Mathematics and Statistics, Allama Iqbal Open University, H-8, Islamabad, Pakistan
Correspondence should be addressed to Adem Kılıc ¸man; akilic@upm.edu.my
Received 18 December 2013; Accepted 11 February 2014; Published 13 March 2014
Academic Editor: S. D. Purohit
Copyright © 2014 M. Adil Khan et al. Tis 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.
We obtain some general inequalities and establish integral inequalities of the majorization type for invex functions. We give
applications to relative invex functions.
1. Introduction
Let ,:[,]→ R be two decreasing real functions. Ten
the function is said to majorize if
∫
()≥∫
() for ∈[,],
∫
()=∫
()
(1)
(see [1, p. 417] and [2, p. 324]).
Te following result is known as majorization theorem for
integrals (see [1, p. 417] and [2, p. 325]).
Teorem 1 (see [1, 2]). Let ,:[,]→ be two decreasing
real functions, where ⊂ R is an interval. Te function
majorizes if and only if the inequality
∫
( ())≤∫
(())
(2)
holds for all continuous convex functions :→ 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 defnition of
invex function.
Defnition 2. Let :→ R be a diferentiable function on
the interval , and let :×→ R be a function of two
variables. Te function is said to be -invex if, for all ,∈
,
()≥ ()+
()(,); (3)
see [5, pp. 1]. is called invex if is -invex for some .
Clearly, each diferentiable convex function :→ R
is an -invex function with (,) = − for , ∈
. It is known that a diferentiable function is invex if and
only if each stationary point is a global minimum point [5].
Tis fact was the motivation to introduce invex functions in
optimization theory [6].
Let Θ: R
2
→ R be an arbitrary function vanishing at
points of the form (0,), ∈ R. It is easy to verify that if a
diferentiable function satisfes the condition
Θ(
(),
0
(,))≥0
implies ()− ()≥0 for ,∈
(4)
for some functions
0
, then is invex.
In particular each pseudoconvex function is invex [5, pp.
3-4]. In fact, it is sufcient to consider Θ(,)= for ,∈
R and
0
(,)=− for ,∈.
In the multidimensional case ( ≥ 1), we have the
following defnition.
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 149735, 4 pages
http://dx.doi.org/10.1155/2014/149735