Abstract
A new class of mappings that includes the class of Lipschitzian mappings is introduced. For this kind of mappings, new integral inequalities of Hadamard’s type are obtained. Our results are extensions of many previous contributions related to integral inequalities for Lipschitzian mappings.
1. Introduction
A function is said to be -Lipschitzian, where , iffor all . In [1, 2], some integral inequalities of Hadamard’s type involving -Lipschitzian functions were derived. In particular, it was shown that if is -Lipschitzian in , , then
In [3], some Hadamard-type and Bullen-type inequalities were obtained for the same class of functions. We recall below the main results established in [3]. Let be -Lipschitzian in .(i)Let with . Let , , andwhere denotes the signum function. Then,
Notice that if and , then (5) reduces to (2). Moreover, if and , then (5) reduces to (3).(ii)Let , and , where . Let , and
Then,
Notice that if and , then (7) reduces to the Bullen-type inequality
Moreover, if and , then (7) reduces to the Bullen-type inequality (see [4])
For other works related to integral inequalities for Lipschitzian functions, see, for example, [5–11] and the references therein.
Motivated by the above mentioned results, in this paper, we obtain some Hadamard-type integral inequalities for a new class of functions which includes the class of Lipschitzian functions.
2. The Class of Generalized Lipschitzian Functions
Let be a given function and be fixed.for all .(i) is -Lipschitzian(ii)For all ,which shows that is -Lipschitzian.
Definition 1. We say that is-Lipschitzian, where, if
Proposition 2. Let . The following statements are equivalent:
Proof. Suppose that is -Lipschitzian. Taking in (10), (ii) follows. Suppose now that satisfies (11). Then, for all ,
Remark 3. It can be easily seen that if is-Lipschitzian, then is -Lipschitzian, for all .
Let us denote by the set of functions such that there exists for which is -Lipschitzian. Moreover, we denote by the set of functions such that there exist and for which is -Lipschitzian. The following example shows that
Example 1. Letbe the function defined by
One observes that is not continuous at , which shows that . On the other hand, one hasfor all , which shows that is -Lipschitzian. Hence, .
3. Inequalities of Hadamard’s Type
We first fix some notations that will be used throughout this section. Let be -Lipschitzian, where , , , and . We denote by the quantity defined by
Theorem 4. Let ( is a positive integer), , and be such that . Then,
Proof. Since , one has
Hence, by the assumption on , one obtainswhich yields
On the other hand, an elementary calculation shows that
Therefore, combining (20) with (21), (17) follows.
We investigate below some particular cases of Theorem 4.
Corollary 5. Let and . Then,
Proof. In Theorem 4, let , and . Then, by (17), one obtains (22).
Corollary 6. Let . Then,
Proof. In Corollary 5, let and . Then, by (22), one obtains (23).
Corollary 7. Let . Then,
Proof. In Corollary 5, let . Then, by (22), one obtains (24).
Corollary 8. Let . Then,
Proof. In Corollary 7, let . Then, by (24), one obtains (25).
Corollary 9. We have
Proof. Taking in (25), one obtains (26).
Corollary 10. We have
Proof. In Corollary 7, let . Then, by (24), one obtains (27).
Corollary 11. Let , that is, is -Lipschitzian. Then,
Proof. By (16), for , one has
(14)(52)
Then, by (27), one obtains (28).
Remark 12. Corollary 11 shows that the inequality (2) which was obtained in [1, 2] for the class of -Lipschitzian functions still holds for -Lipschitzian functions.
Corollary 13. Let . Then,
Proof. In Corollary 5, taking and , (29) follows.
Corollary 14. We have
Proof. In (29), taking , (30) follows.
Corollary 15. Let, that is,is-Lipschitzian. Then,
Proof. In (30), taking , (31) follows.
Corollary 16. Let , and be such that . Then,
Proof. In Theorem 4, let . Then, by (17), one obtains (32).
Corollary 17. Let . Then,
Proof. In Corollary 16, let and . Then, using (32), (33) follows.
Corollary 18. We have
Proof. In (33), taking , (34) follows.
Corollary 19. Let, that is,is-Lipschitzian. Then,
Proof. In (34), taking , (35) follows.
Corollary 20. We have
Proof. In (33), taking , (36) follows.
Corollary 21. Let, that is,is-Lipschitzian. Then,
For the next result, let be the set of functions satisfying the following conditions:
() .
() , for all .
Notice that under the above conditions, the function is invertible. We denote by its inverse.
Theorem 22. Let. Then, for all,
Proof. Taking , one obtains
On the other hand, for , one has
Combining (39) with (40), it holds that
Next, using the assumption on , one obtainsfor all , which yields
Finally, combining (41) and (43), (38) follows.
We investigate below some particular cases of Theorem 22.
Corollary 23. Let. Then,
Proof. Taking in (38), (44) follows.
Corollary 24. Let. Then, for all,
Proof. Let
It can be easily seen that and
Hence, by Theorem 22, one has
On the other hand,
Finally, combining (48) with (49), (45) follows.
Corollary 25. Let. Then,
Proof. Taking in (45), (50) follows.
Corollary 26. Let . Let, that is, is-Lipschitzian. Then, for all ,
Proof. Taking in (45), (51) follows.
Corollary 27. Let . Ifis-Lipschitzian, then,
Proof. Taking in (51), (52) follows.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors contributed equally to this work.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No. RGP-1435-034.