Integral Inequalities for Some New Classes of Convex Functions
Muhammad Aslam Noor^{*}, Khalida Inayat Noor, Muhammad Uzair Awan
Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan
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To cite this article:
Muhammad Aslam Noor, Khalida Inayat Noor, Muhammad Uzair Awan. Integral Inequalities for Some New Classes of Convex Functions. American Journal of Applied Mathematics. Special Issue: Proceedings of the 1st UMT National Conference on Pure and Applied Mathematics (1st UNCPAM 2015). Vol. 3, No. 3-1, 2015, pp. 1-5. doi: 10.11648/j.ajam.s.2015030301.11
Abstract: In this paper, we introduce a new class of convex functions, which is called nonconvex functions. We show that this class unifies several previously known and new classes of convex functions. We derive several new inequalities of Hermite-Hadamard type for nonconvex functions. Some special cases are also discussed. Results proved in this paper continue to hold for these special cases.
Keywords: Convex, Hermite-Hadamard’s Inequalities, Convex Functions
1. Introduction
Theory of convexity has played an important role in the development of various branches of pure and applied sciences, see [2]. In recent years, the cconcepts of convex functions have been generalized in various directions using novel and innovative ideas, see[1,2,3,4,5,7,8,10,11,12,13,16]. Varosanec [11] introduced the class of -convex functions. This class of functions unifies various classes of convex functions and is being used to discuss several concepts in a unified manners.
Definition1.1[16]. Let a nonnegative function. We say that is -convex function, if is nonnegative and for all and .
(1)
The class of -convex functions along with classical convex functions unifies some other known classes of convex functions, that is, -Breckner convex functions [1], -convex functions [7], Godunova-Levin functions [8] and -Godunova-Levin functions [4].
It is known that theory of convex functions and theory of inequalities are very much interrelated to each other. One of the most extensively studied inequality in the literature is Hermite-Hadmard’s inequality. This inequality provides necessary and sufficient condition for a function to be convex. For some useful details on Hermite-Hadmard’s inequality, readers are referred to [3,4,5,6,7,9,10,11,12,13,14,15].
Inspired by the ongoing research, we in this paper, introduce a class of -convex functions, which is called nonconvex. This class also generalizes the class of -convex functions. This new class of noconvex functions is quite flexible and includes several previously known and new classes as special cases. We derive some Hermite-Hadmard type inequalities for this new class, which is main motivation of this paper. The interested readers may find its applications in pure and applied sciences.
We now define a new class of convex functions, which is called as -convex function.
Definition1.2. Let a nonnegative function. A function
is convex function, if is nonnegative and , and .
Remark. For in Definition 1.2, then, we have the definition of -convex functions, introduced and studied by Varosanec [16]. For , , , the class of -convex functions reduces to the class of , -Breckner convex functions [1], -Godunova-Levin functions [4] and -functions [7] respectively.
In the next section, we derive some Hermite-Hadamard type inequalities for the class of -convex functions. We also discuss some special cases.
To prove some of our main results, we need following result, which is proved using the technique of [15].
Lemma 1.4. Let be -convex functions.Then for any and , we have
Proof. As we know that , can be represented as , .
Thus which is the desired result.
2. Main Results
In this section we derive several integral inequalities of Hermite-Hadamard type for -convex functions, which is the main motivation of this paper.
Theorem 2.1. Let be -convex function. If , then, for , we have
Proof. Let be -convex, then
Integration of above inequality with respect to on , leads to the following step
Also
Integration of above inequality with respect to on , we have
This completes the proof.
Special Cases.
We now discuss some special cases.
i. If in Theorem 2.1 then, we have Theorem 6 [14].
ii. If in Theorem 2.1 then, we have Hermite-Hadamard type inequality for s-convex functions, see [6].
iii. If in Theorem 2.1 then, we have Hermite-Hadamard type inequality for s-Godunova-Levin convex functions, see [4].
iv. If in Theorem 2.1 then, we have Hermite-Hadamard type inequality for P functions, see [7].
We need the following result, which is proved essentially using the technique of [15].
Theorem 2.2. Let be -convex function on , and is non-negative, integrable and symmetric about , then
Proof. Using the given facts, we have
which is the desired result.
Note that, if s=1, then Theorem 2.2 reduces to Theorem 5 [15].
We now derive Hermite-Hadamard-Fejer version of inequality for the class of convex functions.
Theorem 2.3. Let be -convex functions and and is a non-negative, integrable and symmetric about , then, for , we have
Proof. Using the given facts and Lemma 1.4, we have
To prove the other part of the inequality, we consider
This completes the proof.
Theorem 2.4. Let be the product of two -convex functions, and , then, for
Proof. Since is -convex function, then, for , we have
For
we have
Integration of both sides of the above inequality with respect to on , leads to following step
which is the first part of the inequality.
Using the fact that is -convex function and a simple integrations gives us the second part of the inequality. This completes the proof.
Theorem 2.5. Let and , , be the functions such that , then, for and , we have
where
and .
Proof. We can write
Integration of both sides with respect to on , leads to the following step
This implies that
which is the left side of the inequality. Now we derive the other side of the inequality.
For this we have
Integration of both sides of above inequality with respect to on , leads to the following step
which is the desired result.
Remark. For suitable and appropriate choces of auxiliary function one can obtain several new and known Hermite-Hadamard type inequalities for other classes of convex functions. We leave this to the interested readers.
3. Conclusion
We have introduced and investigated some new classes of convex functions involving and auxiliary function h. Several Hermite-Hadamard type of inequalities are obtained. Our results represents refinement and improvement of previously known results. The ideas and techniques of this paper may inspire further research in this field. The interested readers are encouraged to find the application of these new classes of convex functions in pure and applied sciences. This is an other aspect of future research.
Acknowldgment
Authors would like to thank Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities. This research is supported by HEC NRPU project No: 20-1966/R&D/11-2553.
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