Research Article | Open Access
Integral Majorization Theorem for Invex Functions
We obtain some general inequalities and establish integral inequalities of the majorization type for invex functions. We give applications to relative invex functions.
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.
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 . This fact was the motivation to introduce invex functions in optimization theory .
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 ).(iii)A differentiable function is said to be -quasi-invex if, for all , (see ).
A differentiable real function is said to be invex (resp., pseudoinvex, quasi-invex), if is -invex (resp., -pseudo-invex, -quasi-invex) for some functions .
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
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).
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
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
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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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