# On Some Integral Inequalities in Quantum Calculus

Authors: Mohamed Jleli; Bessem Samet
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The objective of this paper is to establishq-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.


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## Body

## 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, ifΩ is a bounded (or bounded at least in one direction) domain of ℝN, then, there exists a constant C=CΩ>0 such that for all u∈H01Ω,
(1)∫Ωux2dx≤C∫Ω‍∇ux2dx.For a smooth bounded domainΩ, the best constant C satisfying the above inequality is equal to λΩ−1, where λΩ is the first eigenvalue of −Δ in H01Ω, and Δ 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 u is a smooth function of compact support in ℝ2, then
(2)∫ℝ2u4xdx≤κ2∫ℝ2u2xdx∫ℝ2∇ux2dx,where κ>0 is a dimensionless constant and ∇u denotes the gradient of u. 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(3)−ϑ′′t=ftϑt,m1<t<m2,under the Dirichlet boundary conditions
(4)ϑmi=0,i=1,2,where f∈Cm1,m2. 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∈C1m1,m2 is a nontrivial solution to (3)–(4), then (see Lyapunov [27] and Borg [28])
(5)∫m1m2‍ftdt>4m1−m2.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 deriveq-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 toq-calculus (see, e.g., [36–51] and the references therein).We first fixq∈0,1. Let ℕ be the set of positive natural numbers, i.e., ℕ=1,2,3,⋯, and ℕ0=ℕ∪0.Definition 1.
Theq-derivative of a function ϑ∈C10,T(T>0) is defined by
(6)Dqϑt=ϑt−ϑqt1−qtif0<t≤T,ϑ′0ift=0.Remark 2.
Using L’Hospital’s rule, one obtains(7)limt→0+Dqϑt=ϑ′0,which shows that Dqϑ∈C0,T for all ϑ∈C10,T.Remark 3.
It can be easily seen that(8)limq→1−Dqϑt=ϑ′t,0≤t≤T.Lemma 4 (see [45]).
Letϑ,ρ∈C10,T. Then
(9)Dqϑρt=ϑqtDqρt+ρtDqϑt.Definition 5.
Theq-integral of a function ϑ∈C0,T is defined by
(10)∫0tϑξdqξ=1−qt∑σ=0∞qσϑqσt,0≤t≤T,and
(11)∫st‍ϑξdqξ=∫0t‍ϑξdqξ−∫0s‍ϑξdqξ,0<s≤t≤T.Remark 6.
Obviously, ifϑ∈C0,T, then
(12)∫0tϑξdqξ<∞,∫st‍ϑξdqξ<∞,0<s≤t≤T.Lemma 7 (see [39]).
Letϑ,ρ∈C0,T, 0≤t≤T, p>1 and p′=p/p−1. Then
(i)
∫0tϑξdqξ≤∫0t‍ϑξdqξ(ii)
For allσ∈ℕ0, ϑqσt≤ρqσt∫0tϑξdqξ≤∫0tρξdqξ(iii)
∫0Tϑξρξdqξ≤∫0Tϑξpdqξ1/p∫0Tρξp′dqξ1/p′.Lemma 8 (see [45]).
Letϑ∈C10,T. Then
(i)
∫stDqϑξdqξ=ϑt−ϑs,0≤s≤t≤T(ii)
Dq∫0tϑξdqξ=ϑt,0<t≤TRemark 9.
Notice that in general, for0<s<t≤T,
(13)∫stϑξdqξ≤∫stϑξdqξ.Namely, following [40], consider the function ϑ:0,1→ℝ defined by
(14)ϑξ=11−q4q−nξ−1−3qifqn+1≤ξ≤qn1+q2,n∈ℕ0,41−q−ξq−n+1−1ifqn1+q2≤ξ≤qn,n∈ℕ0,0ifξ=0.Then, one has(15)ϑqn=−1andϑqn1+q2=1,foralln∈ℕ0.Therefore, an elementary calculation shows that(16)∫1+q/21ϑξdqξ=−3+q2and∫1+q/21ϑξdqξ=1−q2.Hence, one has(17)∫1+q/21ϑξdqξ>∫1+q/21ϑξdqξ.We have the following integration by parts rule.Lemma 10 (see [45]).
Letϑi∈C10,T, i=1,2. Then(18)∫0Tϑ1ξDqϑ2ξdqξ=ϑ1ξϑ2ξξ=0T−∫0Tϑ2qξDqϑ1ξdqξ.Let us introduce the set(19)Λq=qn:n∈ℕ∪0.LetT∈Λq, T>0, i.e.,
(20)T=qk,forsomek∈ℕ,and Iq=0,T∩Λq, i.e.,
(21)Iq=qi+k:i∈ℕ0∪0.Lets,t∈Iq be such that 0<s<t, i.e., t=qi+k for some i∈ℕ0 and s=qi+k+j for some j∈ℕ. In this case, for ϑ∈C0,T, by Definition 5, one has
(22)∫stϑξdqξ=∫0tϑξdqξ−∫0sϑξdqξ=1−qt∑σ=0∞qσϑqσt−1−qs∑σ=0∞qσϑqσs=1−q∑σ=0∞tqσϑqσt−∑σ=0∞sqσϑqσs=1−q∑σ=0∞qσ+i+kϑqσ+i+k−∑σ=0∞qσ+i+k+jϑqσ+i+k+j=1−q∑n=i+k∞qnϑqn−∑n=i+k+j∞qnϑqn=1−q∑n=i+ki+k+j−1qnϑqn,which is a finite sum. Hence, one deduces the following property.Lemma 11.
Letϑ,ρ∈C0,T, where T∈Λq, T>0. Lets,t∈Iq be such that 0<s<t. Then
(23)∫stϑξdqξ≤∫st∣ϑξ∣dqξ.


## 2. Poincaré and Sobolev Type Inequalities

Letq∈0,1 be fixed.Theorem 12.
Letp>1 and T∈Λq,T>0. Let ϑ∈C10,T be such that
(24)ϑ0=ϑT=0.
Then(25)∫0Tϑξpdqξ≤T2p∫0TDqϑξpdqξ.Proof.
Lett=qσT, where σ∈ℕ. Notice that since T∈Λq, then t∈Iq. By property (i) of Lemma 8, one has
(26)ϑt−ϑ0=∫0tDqϑξdqξ.
Sinceϑ0=0, it holds that
(27)ϑt=∫0tDqϑξdqξ.Next, by property (i) of Lemma7, one obtains
(28)ϑt≤∫0tDqϑξdqξ.Again, using property (i) of Lemma8, and the fact that ϑT=0, one obtains
(29)−ϑt=∫tTDqϑξdqξ.Hence, by Lemma11, one deduces that
(30)ϑt≤∫tTDqϑξdqξ.Combining (28) with (30), it holds that
(31)ϑt≤12∫0TDqϑξdqξ.On the other hand, by Hölder’s inequality (see property (iii) of Lemma7), one has
(32)∫0TDqϑξdqξ≤∫0TDqϑξpdqξ1/p∫0T1dqξp−1/p.Notice that(33)∫0T1dqξ=1−qT∑n=0∞qn=T.Therefore,(34)∫0TDqϑξdqξ≤Tp−1/p∫0TDqϑξpdqξ1/p.Combining (31) with (34), one deduces that
(35)ϑt≤Tp−1/p2∫0TDqϑξpdqξ1/p,which yields
(36)ϑtp≤Tp−12p∫0TDqϑξpdqξ,i.e.,(37)ϑqσTp≤Tp−12p∫0TDqϑξpdqξ,σ∈ℕ.Notice that sinceϑT=0, the above inequality is also true for σ=0. Hence, by property (ii) of Lemma 7, one deduces that
(38)∫0Tϑξpdqξ≤Tp−12p∫0TDqϑξpdqξ∫0T1dqξ.Finally, (25) follows from (33) and (38).Remark 13.
Inequality (25) is the one dimensional q-analog of the Poincaré-type inequality derived by Pachpatte [11].Theorem 14.
Letp1,p2>1 and T∈Λq, T>0. Letϑ1,ϑ2∈C10,T be such that
(39)ϑi0=ϑiT=0,i=1,2.
Then(40)∫0Tϑ1ξp1ϑ2ξp2dqξ≤12T2p1+p2∫0TDqϑ1ξ2p1+Dqϑ2ξ2p2dqξ.Proof.
From (37) and (70), one has
(41)ϑ1qσT∣p1≤Tp1−12p1∫0TDqϑ1ξp1dqξ,σ∈ℕ0,(42)ϑ2qσTp2≤Tp2−12p2∫0TDqϑ2ξp2dqξ,σ∈ℕ0.Multiplying (41) by (42), one obtains
(43)ϑ1qσTp1ϑ2qσTp2≤Tp1+p2−22p1+p2∫0TDqϑ1ξp1dqξ∫0TDqϑ2ξp2dqξ.Next, using the inequality2AB≤A2+B2, A,B∈ℝ, one deduces that
(44)ϑ1qσTp1ϑ2qσTp2≤Tp1+p2−22p1+p2+1∫0TDqϑ1ξp1dqξ2+∫0TDqϑ2ξp2dqξ2.On the other hand, by Hölder’s inequality (see property (iii) of Lemma7), for i=1,2, one has
(45)∫0TDqϑiξpidqξ2≤T∫0TDqϑiξ2pidqξ.Hence, combining (44) with (45), it holds that
(46)ϑ1qσTp1ϑ2qσTp2≤Tp1+p2−12p1+p2+1∫0TDqϑ1ξ2p1+Dqϑ2ξ2p2dqξ.Since the above inequality holds for allσ∈ℕ0, by property (ii) of Lemma 7, one deduces that
(47)∫0Tϑ1ξp1ϑ2ξp2dqξ≤Tp1+p2−12p1+p2+1∫0TDqϑ1ξ2p1+∣Dqϑ2ξ∣dqξ∫0T1dqξ.Finally, (40) follows from (33) and (47).Remark 15.
Inequality (40) is the one dimensional q-analog of the Poincaré-type inequality derived by Pachpatte [10].Theorem 16.
Letp>1, m>p/2p−1, N∈ℕandT∈Λq, T>0. Letϑi∈C10,T, i=1,2,⋯,Nbe such that(48)ϑi0=ϑiT=0.
Then(49)∫0T∑i=1Nϑiξ2p/p−1dqξ2mp−1/p≤1NN42mT6m−1p−2m/p∑i=1N∫0TDqϑiξ4mdqξ.Proof.
Lett=qσT, where σ∈ℕ. From (31), one has
(50)∣ϑit∣≤12∫0T∣Dqϑiξ∣dqξ,i=1,2,⋯,N.On the other hand, by Hölder’s inequality (see property (iii) of Lemma7) and (33), one has
(51)∫0T∣Dqϑiξ∣dqξ2≤T∫0TDqϑiξ2dqξ.Hence, by (50), one deduces that
(52)ϑit2≤T4∫0TDqϑiξ2dqξ,i=1,2,⋯,N,which yields
(53)∑i=1Nϑit2p/p−1≤T4p/p−1∑i=1N∫0TDqϑiξ2dqξp/p−1.Next, using the discrete version of Hölder’s inequality, one obtains(54)∑i=1Nϑit2p/p−1≤T4p/p−1∑i=1N11/p−1∑i=1N∫0TDqϑiξ2dqξp/p−1=N1/p−1T4p/p−1∑i=1N∫0TDqϑiξ2dqξp/p−1.On the other hand, by Hölder’s inequality (see property (iii) of Lemma7) and (33), one has
(55)∫0TDqϑiξ2dqξ≤∫0T1dqξ1/p∫0TDqϑiξ2p/p−1dqξp−1/p=T1/p∫0TDqϑiξ2p/p−1dqξp−1/p.Therefore, by (54), one deduces that
(56)∑i=1Nϑit2p/p−1≤N1/p−1T4p/p−1T1/p−1∑i=1N∫0TDqϑit2p/p−1dqξ.Since the above inequality is true for allσ∈ℕ0 (recall that t=qσT), by property (ii) of Lemma 7, and using (33), one deduces that
(57)∫0T∑i=1Nϑiξ2p/p−1dqξ≤N1/p−1T24p/p−1∑i=1N∫0TDqϑit2p/p−1dqξ,which yields
(58)∫0T∑i=1Nϑiξ2p/p−1dqξ2mp−1/p≤N2m/pT242m∑i=1N∫0TDqϑit2p/p−1dqξ2mp−1/p.Next, using Hölder’s inequality with exponents2mp−1/2mp−1−p and 2mp−1/p (notice that 2mp−1/p>1 by assumption), one obtains
(59)∫0TDqϑit2p/p−1dqξ≤∫0T1dqξ2mp−1−p/2mp−1∫0TDqϑit4mdqξp/2mp−1=T2mp−1−p/2mp−1∫0TDqϑit4mdqξp/2mp−1,which yields
(60)∑i=1N∫0TDqϑit2p/p−1dqξ≤T2mp−1−p/2mp−1∑i=1N∫0TDqϑit4mdqξp/2mp−1.Furthermore, the discrete Hölder’s inequality shows that(61)∑i=1N∫0TDqϑit4mdqξp/2mp−1≤N2mp−1−p/2mp−1∑i=1N∫0TDqϑit4mdqξp/2mp−1.Hence, by (60), one deduces that
(62)∑i=1N∫0TDqϑit2p/p−1dqξ≤T2mp−1−p/2mp−1N2mp−1−p/2mp−1∑i=1N∫0TDqϑit4mdqξp/2mp−1.Finally, combining (58) with (62), (49) follows.Remark 17.
Inequality (49) is the one dimensional q-analog of the Poincaré-type inequality derived by Pachpatte [12].Theorem 18.
LetT∈Λq, T>0. Let ϑ∈C10,T be such that
(63)ϑ0=ϑT=0.
Then(64)∫0Tϑ2ξdqξ≤T2∫0Tϑqξ+ϑξ2dqξ1/2∫0TDqϑξ2dqξ1/2.Proof.
Lett=qσT, where σ∈ℕ. By Lemma 4, property (i) of Lemma 8, and using the boundary conditions, one has
(65)ϑ2t=∫0tϑqξ+ϑξDqϑξdqξand
(66)−ϑ2t=∫tTϑqξ+ϑξDqϑξdqξ.Combining (65) with (66), it holds that
(67)ϑ2t≤12∫0Tϑqξ+ϑξDqϑξdqξ.Using Hölder’s inequality, one obtains(68)ϑ2t≤12∫0Tϑqξ+ϑξ2dqξ1/2∫0TDqϑξ2dqξ1/2.Since the above inequality holds for allσ∈ℕ0, using property (ii) of Lemma 7, integrating over 0,T, and using (33), (64) follows.Remark 19.
Inequality (64) is the one dimensional q-analog of the Sobolev-type inequality derived by Pachpatte [11].


## 3. Lyapunov-Type Inequalities

We fixq∈0,1 and T∈Λq, T>0. Consider the second order q-difference equation
(69)−DqDqϑt/q+atDqϑt=ftφϑt,0<t<T,under the boundary conditions
(70)ϑ0=ϑT=0,where a,f∈C0,T and φ:ℝ→ℝ. We suppose that there exists a constant Lφ>0 such that
(71)φx≤Lφx,x∈ℝ.Obviously, from (71), one has φ0=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 ϑt≠0,0<t<T.Theorem 20.
Suppose thatϑ∈C10,T is a solution to (69) and (70) satisfying
(72)ϑt≠0,0<t<T.Then(73)1≤Lφ∫0TξT−ξ4fξdqξ+∫0TξT−ξ4aξ2dqξ1/2.Proof.
Lets=qσT, where σ∈ℕ. Since ϑ0=0, using property (i) of Lemma 8, one has
(74)ϑs=∫0sDqϑξdqξ.By Hölder’s inequality (see property (iii) of Lemma7) and (33), one obtains
(75)ϑs≤s∫0sDqϑξ2dqξ1/2,which yields
(76)ϑs2≤s∫0sDqϑξ2dqξ.Similarly, sinceϑT=0, one has
(77)−ϑs=∫sTDqϑξdqξ,which implies that (see Lemma 11)
(78)ϑs≤∫sTDqϑξdqξ.Sinces,T∈Iq, then ∫sTDqϑξdqξ is a finite sum (see (22)). Hence, we can apply Hölder’s inequality to get
(79)ϑs2≤T−s∫sTDqϑξ2dqξ.Multiplying (76) by (79), one obtains
(80)ϑs4≤sT−s∫0sDqϑξ2dqξ∫sTDqϑξ2dqξ,i.e.,(81)ϑs2≤sT−s∫0sDqϑξ2dqξ1/2∫sTDqϑξ2dqξ1/2.Using the inequality2AB≤A2+B2, A,B∈ℝ, it holds that
(82)ϑs2≤sT−s2∫0sDqϑξ2dqξ+∫sTDqϑξ2dqξ,i.e., (recall thats=qσT and ϑT=0)
(83)ϑqσT2≤qσTT−qσT2∫0TDqϑξ2dqξ,σ∈ℕ0.Consider now the function(84)wt=Dqϑt/q,0<t<T.By (69), one has
(85)−Dqwt+atDqϑtt=ftφϑt,0<t<T.Multiplying (85) by ϑt and integrating over 0,T, one obtains
(86)−∫0TϑξDqwξdqξ+∫0TaξϑξDqϑξdqξ=∫0Tfξφϑξϑξdqξ.On the other hand, using the integration by parts rule (see Lemma10) and the boundary conditions (70), one has
(87)−∫0TϑξDqwξdqξ=∫0TwqξDqϑξdqξ.Hence, by (86) and the definition of w, one deduces that
(88)∫0TDqϑξ2dqξ=∫0Tfξφϑξϑξdqξ−∫0T‍aξϑξDqϑξdqξ.Next, using (71), one obtains
(89)∫0TDqϑξ2dqξ≤Lφ∫0Tfξϑ2ξdqξ+∫0TaξϑξDqϑξdqξ.Furthermore, by (83) and property (ii) of Lemma 7, one deduces that
(90)∫0TDqϑξ2dqξ≤Lφ2∫0TDqϑξ2dqξ∫0TfξξT−ξdqξ+12∫0TaξξT−ξ1/4Dqϑξdqξ∫0TDqϑξ2dqξ1/2.Therefore, by Hölder’s inequality, it holds that(91)∫0TDqϑξ2dqξ≤Lφ2∫0TDqϑξ2dqξ∫0TfξξT−ξdqξ+12∫0Taξ2ξT−ξdqξ1/2∫0TDqϑξ2dqξ.Next, we claim that(92)∫0TDqϑξ2dqξ≠0.Indeed, suppose that∫0TDqϑξ2dqξ=0. By Definition 5, one obtains
(93)1−qT∑τ=0∞qτDqϑqτTDqϑqτT2=0,which yields
(94)DqϑqτT=0,τ∈ℕ0.In particular, forτ=1, one has
(95)DqϑT=0,i.e.,(96)ϑT−ϑqT=0.SinceϑT=0, one deduces that ϑqT=0, which contradicts (72). This proves (92). Now, dividing (91) by ∫0TDqϑξ2dqξ>0, it holds that
(97)1≤Lφ2∫0T∣fξ∣ξT−ξdqξ+12∫0Taξ2ξT−ξdqξ1/2,which yields (73).Using the inequality(98)ξT−ξ≤T24,0<ξ<T,one deduces from Theorem 20 the following result.Corollary 21.
Suppose thatϑ∈C10,T is a solution to (69) and (70) satisfying (72). Then
(99)1≤LφT4∫0T∣fξ∣dqξ+T2∫0Taξ2dqξ1/2.Consider now the second orderq-difference equation
(100)−DqDqϑt/q=ftφϑt,0<t<T,under the boundary conditions (70), where f∈C0,T and φ:ℝ→ℝ satisfies (71). Notice that (100) is a special case of (69) with a≡0. Hence, by Theorem 20 and Corollary 21, one deduces the following results.Corollary 3.2.
Suppose thatϑ∈C10,T is a solution to (100) and (70) satisfying (72). Then
(101)∫0TξT−ξ∣fξ∣dqξ≥2Lφ.Corollary 22.
Suppose thatϑ∈C10,T is a solution to (100) and (70) satisfying (72). Then
(102)∫0T∣fξ∣dqξ≥4LφT.Remark 23.
Inequality (102) with φx=x (Lφ=1) is the q-analogue of Lyapunov inequality (5) with m1=0andm2=T.


## 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 ofq-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.



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