Abstract

In this article, we first presented a new integral identity concerning differentiable mappings defined on m-invex sets. By using the notion of generalized relative semi-(r; m, p, q, h₁, h₂)-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Ostrowski type inequalities are established. It is pointed out that some new special cases can be deduced from main results of the article.

1. Introduction

The subsequent inequality is known as Ostrowski inequality which gives an upper bound for the approximation of the integral average by the value at point

Theorem 1.1. Let be a mapping differentiable on and let with If for all then

(1.1)

Ostrowski inequality is playing a very important role in all the fields of mathematics, especially in the theory of approximations. Thus such inequalities were studied extensively by many researches and numerous generalizations, extensions and variants of them for various kind of functions like bounded variation, synchronous, Lipschitzian, monotonic, absolutely, continuous and n-times differentiable mappings etc. appeared in a number of papers ^{1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 29, 30, 31, 36, 37, 38, 40, 43, 45}. In numerical analysis many quadrature rules have been established to approximate the definite integrals ^{14, 25, 27, 28, 32, 35, 39, 41}. Ostrowski inequality provides the bounds for many numerical quadrature rules.

Let us recall some special functions and evoke some basic definitions as follows.

Definition 1.2. The incomplete beta function is defined for as

Definition 1.3. ⁴⁴ A set is said to be a relative convex (-convex) set, if and only if, there exists a function such that,

(1.2)

Definition 1.4. ⁴⁴ A function f is said to be a relative convex (-convex) function on a relative convex (-convex) set if and only if, there exists a function such that,

(1.3)

Definition 1.5. ⁴ A set is said to be invex with respect to the mapping if for every and

Notice that every convex set is invex with respect to the mapping but the converse is not necessarily true ^{4, 42}.

Definition 1.6. ³⁴ The function f defined on the invex set is said to be preinvex with respect if for every and we have that

The concept of preinvexity is more general than convexity since every convex function is preinvex with respect to the mapping but the converse is not true.

Definition 1.7. ²⁵ Let be a non-negative function and The function f on the invex set K is said to be h-preinvex with respect to if

(1.4)

for each and where

Definition 1.8. ⁴¹ Let be a positive function, We say that is -convex, if is non-negative and for all and one has

(1.5)

Definition 1.9. ³⁹ Let be a non-negative function, we say that is a tgs-convex function on if the inequality

(1.6)

holds for all and We say that is tgs-concave if is tgs-convex.

Definition 1.10. ²⁸ A function: is said to be m-MT-convex, if is positive and for and with satisfies the following inequality

(1.7)

Definition 1.11. ³³ Let be an open m-invex set with respect to and are continuous functions. A function is said to be generalized -preinvex with respect to if

(1.8)

is valid for all and with some fixed If the inequality (1.8) reverses, then f is said to be generalized -preincave on

Definition 1.12. ¹¹ A set is named as m-invex with respect to the mapping for some fixed if holds for each and any

Remark 1.13. In Definition 1.12, under certain conditions, the mapping could reduce to For example when then the m-invex set degenerates an invex set on

We are in position to introduce the notion of generalized relative semi-(r; m, p, q, h₁, h₂)-preinvex mappings.

Definition 1.14. Let be an open m-invex set with respect to the mapping Suppose and are continuous. A mapping is said to be generalized relative semi-(r; m, p, q, h₁, h₂)-preinvex, if

(1.9)

holds for all and for any fixed and some fixed where

is the weighted power mean of order r for positive numbers and

Remark 1.15. In Definition 1.14, if we choose and then we get Definition 1.11. If we choose and then we get -preinvex function ^{15, 17}.

Remark 1.16. For r = p = q = 1, let us discuss some special cases in Definition 1.14 as follows.

(I) If taking for then we get generalized relative semi-(m, s)-Breckner-preinvex mappings.

(II) If taking then we get generalized relative semi-(m, P)-preinvex mappings.

(III) If taking for then we get generalized relative semi-(m, s)-Godunova-Levin-Dragomir-preinvex mappings.

(IV) If taking then we get generalized relative semi-(m, h)-preinvex mappings.

(V) If taking then we get generalized relative semi-(m, tgs)-preinvex mappings.

(VI) If taking then we get generalized relative semi-m-MT-preinvex mappings.

It is worth to mention here that to the best of our knowledge all the special cases discussed above are new in the literature.

Let see the following example of a generalized relative semi-(r; m, p, q, h₁, h₂)-preinvex mappings which is not convex.

Example 1.17. Let taking for all for any fixed and Consider the mapping as follows

Define a bifunction by

Then f is generalized relative semi--preinvex mapping for any fixed and for all But is not preinvex with respect to and also it is not convex (consider and ).

Motivated by the above literatures, the main objective of this article is to establish some new estimates on generalizations of Ostrowski type inequalities associated with differentiable generalized relative semi-(r; m, p, q, h₁, h₂)-preinvex mappings on m-invex sets. It is pointed out that some new special cases will be deduced from main results of the article.

2. Main Results

In this section, in order to prove our main results regarding some generalizations of Ostrowski type inequalities for differentiable generalized relative semi-(r; m, p, q, h₁, h₂)-preinvex mappings, we need the following new integral identity.

Lemma 2.1. Let be a continuous function. Suppose be an open m-invex subset with respect to for some fixed and Assume that

be a differentiable mapping on and Then for any two complex numbers and the following integral identity holds:

(2.1)

Proof. Integrating by parts and changing the variable of definite integrals yield

Proof. Integrating by parts and changing the variable of definite integrals yield

This completes the proof of our lemma.

Remark 2.2. In Lemma 2.1, if we choose and we get ( ⁷, Lemma 8).

Throughout this paper we denote

Corollary 2.3. With the assumption in Lemma 2.1, we have for any that

(2.2)

Remark 2.4. If we take in (2.2), then we get

(2.3)

Remark 2.5. If we take and in (2.1), then we get

(2.4)

Remark 2.6. If we take in (2.2), then we get generalized Montgomery's identity, i.e.

(2.5)

Using relation (2.1), the following results can be obtained for the corresponding version for power of the first derivative.

Theorem 2.7. Let and Let be an open m-invex subset with respect to for some fixed Suppose and are continuous functions and are two complex numbers. Assume that be a differentiable function on where If is generalized relative semi-(r; m, p₁, p₂, h₁, h₂)-preinvex mappings, then the following inequality holds:

(2.6)

where

and

Proof. Suppose that and From Lemma 2.1, generalized relative semi-(r; m, p₁, p₂, h₁, h₂)-preinvexity of Hölder inequality, Minkowski inequality, properties of the modulus and using the elementary inequality where we have

So, the proof of this theorem is complete.

We point out some special cases of Theorem 2.7.

Corollary 2.8. In Theorem 2.7 for we have the following inequality for generalized relative semi-(r; m, p₁, p₂, h₁, h₂)-preinvex mappings:

(2.7)

Corollary 2.9. In Theorem 2.7 for and we have the following inequality for generalized relative semi-(r;m,p₁,p₂,h)-preinvex mappings:

(2.8)

Corollary 2.10. In Theorem 2.7 for we have the following inequality for generalized relative semi-(r;m,p₁,p₂,s)-Breckner-preinvex mappings:

(2.9)

Corollary 2.11. In Theorem 2.7 for we have the following inequality for generalized relative semi-(r;m,p₁,p₂,-s)-Godunova-Levin-Dragomir-preinvex mappings:

(2.10)

Corollary 2.12. In Theorem 2.7 for we have the following inequality for generalized relative semi-(r;m,p₁,p₂,tgs)-preinvex mappings:

(2.11)

where is incomplete beta function and

Corollary 2.13. In Theorem 2.7 for we have the following inequality for generalized relative semi-(r;m,p₁,p₂)-MT-preinvex mappings:

(2.12)

Theorem 2.14. Let and Let be an open m-invex subset with respect to for some fixed Suppose and are continuous functions and are two complex numbers. Assume that be a differentiable function on where If is generalized relative semi-(r; m, p₁, p₂, h₁, h₂)-preinvex mappings, then the following

(2.13)

where

and are defined as in Theorem 2.7.

Proof. Suppose that and From Lemma 2.1, generalized relative semi-(r; m, p₁, p₂, h₁, h₂)-preinvexity of the well-known power mean inequality, Minkowski inequality, properties of the modulus and using the elementary inequality where we have

So, the proof of this theorem is complete.

We point out some special cases of Theorem 2.14.

Corollary 2.15. In Theorem 2.14 for we have the following inequality for generalized relative semi-(r; m, p₁, p₂, h₁, h₂)-preinvex mappings:

(2.14)

Corollary 2.16. In Theorem 2.14 for and we have the following inequality for generalized relative semi-(r; m, p₁, p₂, h)-preinvex mappings:

(2.15)

Corollary 2.17. In Theorem 2.14 for we have the following inequality for generalized relative semi-(r; m, p₁, p₂, s)-Breckner-preinvex mappings:

(2.16)

Corollary 2.18. In Theorem 2.14 for we have the following inequality for generalized relative semi-(r; m, p₁, p₂, s)-Godunova-Levin-Dragomir-preinvex mappings:

(2.17)

Corollary 2.19. In Theorem 2.14 for we have the following inequality for generalized relative semi-(r; m, p₁, p₂, tgs)-preinvex mappings:

(2.18)

Corollary 2.20. In Theorem 2.14 for we have the following inequality for generalized relative semi-(r; m, p₁, p₂)-MT-preinvex mappings:

(2.19)

Remark 2.21. Using our Theorems 2.7 and 2.14 for different values of for some suitable continuous functions and complex numbers we can get some new Ostrowski type inequalities associated with generalized relative semi-(r; m, p₁, p₂, h₁, h₂)-preinvex mappings.

Remark 2.22. Applying our Theorems 2.7 and 2.14, we can deduce some new inequalities using special means associated with generalized relative semi-(r; m, p₁, p₂, h₁, h₂) -preinvex mappings.

3. Conclusions

In this article, we first presented a new integral identity concerning differentiable mappings defined on m-invex set. By using the notion of generalized relative semi-(r; m, p₁, p₂, h₁, h₂)-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Ostrowski type inequalities are established. It is pointed out that some new special cases are deduced from main results of the article. Motivated by this new interesting class of generalized relative semi-(r; m, p₁, p₂, h₁, h₂)-preinvex mappings we can indeed see to be vital for fellow researchers and scientists working in the same domain. We conclude that our methods considered here may be a stimulant for further investigations concerning Ostrowski, Hermite-Hadamard and Simpson type integral inequalities for various kinds of preinvex functions involving local fractional integrals, fractional integral operators, Caputo k-fractional derivatives, q-calculus, (p, q)-calculus, time scale calculus and conformable fractional integrals.

Acknowledgments

The first author would like to thanks University Ismail Qemali of Vlora for its financial support.

References