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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 149735, 4 pages
http://dx.doi.org/10.1155/2014/149735
Research Article

Integral Majorization Theorem for Invex Functions

1Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
2Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
3Department of Mathematics and Statistics, Allama Iqbal Open University, H-8, Islamabad, Pakistan

Received 18 December 2013; Accepted 11 February 2014; Published 13 March 2014

Academic Editor: S. D. Purohit

Copyright © 2014 M. Adil Khan et al. This 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.

Abstract

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 be two decreasing real functions. Then the function is said to majorize if (see [1, p. 417] and [2, p. 324]).

The following result is known as majorization theorem for integrals (see [1, p. 417] and [2, p. 325]).

Theorem 1 (see [1, 2]). Let be two decreasing real functions, where is an interval. The function majorizes if and only if the inequality holds for all continuous convex functions 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. Let be a differentiable function on the interval , and let be a function of two variables. The function is said to be -invex if, for all , see  [5, pp. 1]. is called invex if is -invex for some .

Clearly, each differentiable convex function is an -invex function with for . 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].

Let be an arbitrary function vanishing at points of the form , . It is easy to verify that if a differentiable function satisfies the condition for some functions , then is invex.

In particular each pseudoconvex function is invex [5, pp. 3-4]. In fact, it is sufficient to consider for and for .

In the multidimensional case , we have the following definition.

Definition 3. Let be an inner product on . Let be a function of two variables. One has the following.(i)A differentiable function is said to be -invex if, for all , , where denotes the gradient [5, pp. 1].(ii)A differentiable function is said to be -pseudo-invex if, for all , (see [6]).(iii)A differentiable function is said to be -quasi-invex if, for all , (see [6]).

A differentiable real function is said to be invex (resp., pseudoinvex, quasi-invex), if is -invex (resp., -pseudo-invex, -quasi-invex) for some functions .

For applications of invex functions in optimization and mathematical programming, see [512] 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. Let be an -invex function on the interval , where is a continuous function, and let be integrable functions with for . Then

Proof. If we take , in (3), we obtain Multiplying (9) by and integrating with respect to , we deduce (8).

The following weighted integral majorization theorem holds.

Theorem 5. Let be an -invex function on the interval , where is a continuous function, and let be integrable functions with for . Moreover if and are increasing (decreasing) on and then

Proof. We know that the Chebyshev’s inequality is where are the functions of same monotonicity and is any integrable function.
By assumption the functions , have the same monotonicity. Therefore, by applying Chebyshev's inequality (12) in the right hand side of (8), we have
Also (10) holds, so from (8) and (13) we have then we deduce the desired result (11).

Remark 6. In Theorem 5 the assumption (10) is a strong condition for , . This can be relaxed if a monotonicity property for the -invex function is assumed.

Theorem 7. Let all the assumptions of Theorem 5 hold, but instead of (10) one has the assumption that and is an -invex increasing function on the interval ; then (11) holds.

Proof. The proof is similar to the proof of Theorem 5 but by (15) and monotonicity of we have Using these in (13) we have

Theorem 8. Let be an -invex function on the interval , where is a continuous function, and let be integrable functions with for . One has the following.(i)If is increasing on , for all and , then (ii)If is decreasing on , for all and , then

Proof. Let set ; then for all , and . If is increasing on , then This proves (18).
Similarly, setting , then for all , and .
Now if is decreasing on , then by using Theorem 4 we have This proves (19).

The following extension of majorization theorem for relative invex function can be given.

Theorem 9. Let be integrable functions with being positive function. Suppose are such that is a strictly increasing function and is -invex function on , where is a continuous function. One has the following.(i)If and are increasing (decreasing) on , and , then (ii)If is increasing on , for all , and, then (iii)If is decreasing on , for all , and , then

Proof. The proof of this theorem is similar to the proof of Theorems 5 and 8.

Remark 10. By using in Theorems 4, 5, 7, 8, and 9 we recover the integral majorization results for convex functions given in [1, 4, 14].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the anonymous referees for careful checking of the details and for helpful comments that improved the paper. The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the GP-IBT Grant Scheme having project no. GP-IBT/2013/9420100.

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