On Some Integral Inequalities in Quantum Calculus

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 Ω is a bounded (or bounded at least in one direction) domain of RN , then, there exists a constant C = CðΩÞ > 0 such that for all u ∈H0 ðΩÞ, ð


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 Ω is a bounded (or bounded at least in one direction) domain of ℝ N , then, there exists a constant C = CðΩÞ > 0 such that for all u ∈ H 1 0 ðΩÞ, For a smooth bounded domain Ω, the best constant C satisfying the above inequality is equal to λðΩÞ −1 , where λðΩÞ is the first eigenvalue of −Δ in H 1 0 ðΩÞ, and Δ is the Laplacian operator (see, e.g., [1][2][3][4][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][7][8][9][10][11][12][13][14][15][16][17] and the references therein). Another important inequality involving the function and its gradient is the Sobolev inequality (see [18,19]). Namely, if u is a smooth function of compact support in ℝ 2 , then where κ > 0 is a dimensionless constant and ∇u denotes the gradient of u. For further results related to Sobolevtype inequalities and their applications, see, for example, [20][21][22][23][24][25][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 f ∈ Cð½m 1 , m 2 Þ. Obviously, the trivial function ϑ ≡ 0 is a solution to (3)-(4). Lyapunov's inequality provides a necessary criterion for the existence of a nontrivial solution. Namely, if ϑ ∈ C 1 ð½m 1 , m 2 Þ 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][30][31][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][34][35]).
In this paper, motivated by the abovementioned contributions, our goal is to derive q-analogs of some Poincarétype inequalities, Sobolev-type inequalities, and Lyapunovtype inequalities. Notice that only the one dimensional case is considered in this work.

Journal of Function Spaces
Therefore, an elementary calculation shows that Hence, one has We have the following integration by parts rule.

Poincaré and Sobolev Type Inequalities
Let q ∈ ð0, 1Þ be fixed. Then Proof Since ϑð0Þ = 0, it holds that Next, by property (i) of Lemma 7, one obtains Again, using property (i) of Lemma 8, and the fact that ϑðTÞ = 0, 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 3 Journal of Function Spaces (iii) of Lemma 7), one has Notice that Therefore, Combining (31) with (34), one deduces that which yields i.e., Notice that since ϑðTÞ = 0, the above inequality is also true for σ = 0. Hence, by property (ii) of Lemma 7, one deduces that Finally, (25) follows from (33) and (38). (25) is the one dimensional q-analog of the Poincaré-type inequality derived by Pachpatte [11].

Journal of Function Spaces
Then Proof. Let t = q σ T, where σ ∈ ℕ. 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 σ ∈ ℕ 0 (recall that t = q σ T), by property (ii) of Lemma 7, and using (33), one deduces that which yields Next, using Hölder's inequality with exponents 2mðp − 1Þ/2mðp − 1Þ − p and 2mðp − 1Þ/p (notice that 2mðp − 1Þ/p > 1 by assumption), one obtains which yields Furthermore, the discrete Hölder's inequality shows that
Remark 19. Inequality (64) is the one dimensional q-analog of the Sobolev-type inequality derived by Pachpatte [11].

Lyapunov-Type Inequalities
We fix q ∈ ð0, 1Þ and T ∈ Λ q , T > 0. Consider the second order q-difference equation under the boundary conditions where a, f ∈ Cð½0, TÞ and φ : ℝ → ℝ. We suppose that there exists a constant L φ > 0 such that Obviously, from (71), one has φð0Þ = 0. Hence, ϑ ≡ 0 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 ϑðtÞ ≠ 0, 0 < t < T.

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 q-calculus in applications, integral inequalities involving the function and its q-derivative are obtained. Namely, we derived the q-analogue of some Poincaré-type inequalities and Sobolev-type inequalities. We also established the q -analogue of some Lyapunov-type inequalities. We hope that our results will serve as a useful inspiration for future works in the context of q-calculus.

Data Availability
No data were used in this study.

Conflicts of Interest
The authors declare no conflict of interest.