Abstract
The objective of this paper is to establish -analogue of some well-known inequalities in analysis, namely, Poincaré-type inequalities, Sobolev-type inequalities, and Lyapunov-type inequalities. Our obtained results may serve as a useful source of inspiration for future works in quantum calculus.
1. Introduction and Preliminaries
Mathematical inequalities play a crucial role in the development of various branches of mathematics as well as other disciplines of science. In particular, integral inequalities involving the function and its gradient provide important tools in the proof of regularity of solutions to differential and partial differential equations, stability, boundedness, and approximations. One of these categories of inequalities is the Poincaré-type inequality. Namely, if is a bounded (or bounded at least in one direction) domain of , then, there exists a constant such that for all ,
For a smooth bounded domain , the best constant satisfying the above inequality is equal to , where is the first eigenvalue of in , and is the Laplacian operator (see, e.g., [1–5]). Due to the importance of Poincaré inequality in the qualitative analysis of partial differential equations and also in numerical analysis, numerous contributions dealing with generalizations and extensions of this inequality appeared in the literature (see, e.g., [6–17] and the references therein). Another important inequality involving the function and its gradient is the Sobolev inequality (see [18, 19]). Namely, if is a smooth function of compact support in , then where is a dimensionless constant and denotes the gradient of . For further results related to Sobolev-type inequalities and their applications, see, for example, [20–26].
Lyapunov’s inequality is one of the important results in analysis. It was shown that this inequality is very useful in the study of spectral properties of differential equations, namely, stability of solutions, eigenvalues, and disconjugacy criteria. More precisely, consider the second order differential equation under the Dirichlet boundary conditions where . Obviously, the trivial function is a solution to (3)–(4). Lyapunov’s inequality provides a necessary criterion for the existence of a nontrivial solution. Namely, if is a nontrivial solution to (3)–(4), then (see Lyapunov [27] and Borg [28])
Since the appearance of the above result, numerous contributions related to Lyapunov-type inequalities have been published (see, e.g., [18, 29–32] and the references therein).
On the other hand, because of its usefulness in several areas of physics (thermostatistics, conformal quantum mechanics, nuclear and high energy physics, black holes, etc.), the theory of quantum calculus received a considerable attention by many researchers from various disciplines (see, e.g., [33–35]).
In this paper, motivated by the abovementioned contributions, our goal is to derive -analogs of some Poincaré-type inequalities, Sobolev-type inequalities, and Lyapunov-type inequalities. Notice that only the one dimensional case is considered in this work.
We recall below some notions and properties related to -calculus (see, e.g., [36–51] and the references therein).
We first fix . Let be the set of positive natural numbers, i.e., , and .
Definition 1. The -derivative of a function () is defined by
Remark 2. Using L’Hospital’s rule, one obtains which shows that for all .
Remark 3. It can be easily seen that
Lemma 4 (see [45]). Let . Then
Definition 5. The -integral of a function is defined by and
Remark 6. Obviously, if , then
Lemma 7 (see [39]). Let , , and . Then (i)(ii)For all , (iii).
Lemma 8 (see [45]). Let . Then (i)(ii)
Remark 9. Notice that in general, for ,
Namely, following [40], consider the function defined by
Then, one has
Therefore, an elementary calculation shows that
Hence, one has
We have the following integration by parts rule.
Lemma 10 (see [45]). Let, . Then
Let us introduce the set
Let , , i.e., and , i.e.,
Let be such that , i.e., for some and for some . In this case, for , by Definition 5, one has
which is a finite sum. Hence, one deduces the following property.
Lemma 11. Let , where , . Let be such that . Then
2. Poincaré and Sobolev Type Inequalities
Let be fixed.
Theorem 12. Let and . Let be such that Then
Proof. Let , where . Notice that since , then . By property (i) of Lemma 8, one has Since , it holds that
Next, by property (i) of Lemma 7, one obtains
Again, using property (i) of Lemma 8, and the fact that , one obtains
Hence, by Lemma 11, one deduces that
Combining (28) with (30), it holds that
On the other hand, by Hölder’s inequality (see property (iii) of Lemma 7), one has
Notice that
Therefore,
Combining (31) with (34), one deduces that which yields
i.e.,
Notice that since , the above inequality is also true for . Hence, by property (ii) of Lemma 7, one deduces that
Finally, (25) follows from (33) and (38).
Remark 13. Inequality (25) is the one dimensional -analog of the Poincaré-type inequality derived by Pachpatte [11].
Theorem 14. Let and , . Let be such that Then
Proof. From (37) and (70), one has
Multiplying (41) by (42), one obtains
Next, using the inequality , , one deduces that
On the other hand, by Hölder’s inequality (see property (iii) of Lemma 7), for , one has
Hence, combining (44) with (45), it holds that
Since the above inequality holds for all , by property (ii) of Lemma 7, one deduces that
Finally, (40) follows from (33) and (47).
Remark 15. Inequality (40) is the one dimensional -analog of the Poincaré-type inequality derived by Pachpatte [10].
Theorem 16. Let, , and, . Let, be such thatThen
Proof. Let , where . From (31), one has
On the other hand, by Hölder’s inequality (see property (iii) of Lemma 7) and (33), one has
Hence, by (50), one deduces that which yields
Next, using the discrete version of Hölder’s inequality, one obtains
On the other hand, by Hölder’s inequality (see property (iii) of Lemma 7) and (33), one has
Therefore, by (54), one deduces that
Since the above inequality is true for all (recall that ), by property (ii) of Lemma 7, and using (33), one deduces that which yields
Next, using Hölder’s inequality with exponents and (notice that by assumption), one obtains which yields
Furthermore, the discrete Hölder’s inequality shows that
Hence, by (60), one deduces that
Finally, combining (58) with (62), (49) follows.
Remark 17. Inequality (49) is the one dimensional -analog of the Poincaré-type inequality derived by Pachpatte [12].
Theorem 18. Let , . Let be such that Then
Proof. Let , where . By Lemma 4, property (i) of Lemma 8, and using the boundary conditions, one has and
Combining (65) with (66), it holds that
Using Hölder’s inequality, one obtains
Since the above inequality holds for all , using property (ii) of Lemma 7, integrating over , and using (33), (64) follows.
Remark 19. Inequality (64) is the one dimensional -analog of the Sobolev-type inequality derived by Pachpatte [11].
3. Lyapunov-Type Inequalities
We fix and , . Consider the second order -difference equation under the boundary conditions where and . We suppose that there exists a constant such that
Obviously, from (71), one has . Hence, is a trivial solution to (69) and (70). The following theorem provides a necessary condition for the existence of a nontrivial solution to (69) and (70) satisfying .
Theorem 20. Suppose that is a solution to (69) and (70) satisfying
Then
Proof. Let , where . Since , using property (i) of Lemma 8, one has
By Hölder’s inequality (see property (iii) of Lemma 7) and (33), one obtains which yields
Similarly, since , one has which implies that (see Lemma 11)
Since , then is a finite sum (see (22)). Hence, we can apply Hölder’s inequality to get
Multiplying (76) by (79), one obtains
i.e.,
Using the inequality , , it holds that
i.e., (recall that and )
Consider now the function
By (69), one has
Multiplying (85) by and integrating over , one obtains
On the other hand, using the integration by parts rule (see Lemma 10) and the boundary conditions (70), one has
Hence, by (86) and the definition of , one deduces that
Next, using (71), one obtains
Furthermore, by (83) and property (ii) of Lemma 7, one deduces that
Therefore, by Hölder’s inequality, it holds that
Next, we claim that
Indeed, suppose that . By Definition 5, one obtains which yields
In particular, for , one has
i.e.,
Since , one deduces that , which contradicts (72). This proves (92). Now, dividing (91) by , it holds that which yields (73).
Using the inequality one deduces from Theorem 20 the following result.
Corollary 21. Suppose that is a solution to (69) and (70) satisfying (72). Then
Consider now the second order -difference equation
under the boundary conditions (70), where and satisfies (71). Notice that (100) is a special case of (69) with . Hence, by Theorem 20 and Corollary 21, one deduces the following results.
Corollary 3.2. Suppose that is a solution to (100) and (70) satisfying (72). Then
Corollary 22. Suppose that is a solution to (100) and (70) satisfying (72). Then
Remark 23. Inequality (102) with () is the -analogue of Lyapunov inequality (5) with and.
4. Conclusion
Integral inequalities involving the function and its gradient are very useful in the study of existence, uniqueness, and qualitative properties of solutions to ordinary and partial differential equations. Motivated by the importance of -calculus in applications, integral inequalities involving the function and its -derivative are obtained. Namely, we derived the -analogue of some Poincaré-type inequalities and Sobolev-type inequalities. We also established the -analogue of some Lyapunov-type inequalities. We hope that our results will serve as a useful inspiration for future works in the context of -calculus.
Data Availability
No data were used in 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–237.