NEW INTEGRAL INEQUALITIES IN QUANTUM ...

International Journal of Analysis and Applications ISSN 2291-8639 Volume 7, Number 1 (2015), 50-58 http://www.etamaths.com

NEW INTEGRAL INEQUALITIES IN QUANTUM CALCULUS KAMEL BRAHIM1,∗ , SABRINA TAF2 AND BOCHRA NEFZI1

Abstract. In this paper, we study the q-analogue of Klamkin-McLenaghan’s and Grueb-Reinboldt’s inequalities then we use the Riemann-Liouville fractional q-integral to get some new integral results.

1. introduction Let us consider (1.1) 1 T (f, g; a, b) = b−a

b

Z

f (x)g(x)dx − a

1 b−a

Z

!

b

f (x)dx a

1 b−a

where f and g are two integrable functions on [a, b], and ! Z b Z b 1 1 Tq (f, g; a, b) = f (x)g(x)dq x − f (x)dq x b−a a b−a a

1 b−a

!

b

Z

g(x)dx a

!

b

Z

g(x)dq x a

where f and g are two functions defined on [a, b]q . The well-known Gr¨ uss integral inequality can be stated as follows (see [10, 15]): |T (f, g; a, b)| ≤

(1.2)

1 (M − m)(N − n), 4

provided that f and g are two integrable functions on [a, b] such that (1.3)

0 < m ≤ f (x) ≤ M < ∞, 0 < n ≤ g(x) ≤ N < ∞, , x ∈ [a, b].

The constant 41 is best possible. Gauchman gave the q-integral Gr¨ uss inequality as follows (see [8]): 1 (M − m)(N − n), 4 In [12], the authors proved the following Klamkin-McLenaghan inequality (1.4) !2 r r !2 n n n n n X X X X X M m 2 2 wk ak wk bk − wk ak bk ≤ − wk ak bk wk b2k , n N |Tq (f, g; a, b)| ≤

k=1

k=1

k=1

k=1

k=1

2010 Mathematics Subject Classification. 35A23. Key words and phrases. Gr¨ uss inequality, fractional q-integral, Klamkin-McLenaghan inequality, Grueb-Reinboldt inequality. c

2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

50

NEW INTEGRAL INEQUALITIES IN QUANTUM CALCULUS

51

ak M where 0 < m k = 1, . . . , n. N ≤ bk ≤ n < ∞, wk > 0, In [9], the authors proved the following Grueb-Reinboldt inequality !2 n n n 2 X X X (M N + mn) wk ak bk (1.5) wk a2k wk b2k ≤ 4nmN M k=1

k=1

k=1

In [6], Dragomir and Diamond proved that (1.6)

|T (f, g; a, b)| ≤

1 1 (M − m)(N − n) √ · · 4 b−a mM nN

Z

b

f (x)dx · a

1 b−a

Z

b

g(x)dx, a

and (1.7) |T (f, g; a, b)| ≤

√

M−

√

s Z b Z b √ √ 1 1 f (x)dx · g(x)dx. m N− n b−a a b−a a

In recent years, many researches have studies (1.1) and number of inequalities appeared in literature (see [1, 2, 3, 4, 5, 14, 18]). The main objective of this paper is to establish some new q-fractional integral inequalities of Klamkin-McLenaghan and Grueb-Reinboldt type. This paper is organized as follows: in section 2, we present some preliminary results and notation. In section 3, we state the q-analogue of Klamkin-McLenaghan and Grueb-Reinboldt inequalities, then we establish some new q-fractional integral inequalities.

2. Basic Definitions For the convenience of the reader, we provide in this section a summary of the mathematical notations and definitions used in this paper (see [7, 13, 16]). We write for a, b ∈ C and q ∈ (0, 1), (a; q)∞ =

∞ Y

(1 − aq k ),

(a − b)(α) = aα

k=0

( ab ; q)∞ (q α ab ; q)∞

.

The q-Jackson integral from 0 to a is defined by (see [11]) Z a ∞ X (2.1) f (x)dq x = (1 − q)a f (aq n )q n , 0

n=0

provided the sum converges absolutely. The q-Jackson integral in a generic interval [a, b] is given by (see [11]) Z b Z b Z a (2.2) f (x)dq x = f (x)dq x − f (x)dq x. a

0

0

n

In the case a = bq , we can write Z b n−1 X (2.3) f (x)dq x = (1 − q)b f (bq k )q k . a

k=0

52

BRAHIM, TAF AND NEFZI

The fractional q-integral of the Riemann-Liouville type is (see [16]) Z x 1 α Jq f (x) = (x − qt)(α−1) f (t)dq t; α > 0 Γq (α) 0 ∞ X xα = (1 − q) (2.4) (1 − q n+1 )(α−1) f (xq n )q n . Γq (α) n=0 where Γq (α) =

1 1−q

Z 0

1

u 1−q

α−1 eq (qu)dq u,

and

eq (t) =

∞ Y

(1 − q k t).

k=0

The q-fractional integration has the following semi-group property for α, β ∈ R+ (Jqβ Jqα f )(x) = (Jqα+β f )(x). For the expression (2.4), when f (x) = xλ , we get another expression that will be used later: Γq (λ + 1) α+λ x . Jqα (xλ ) = Γ(α + λ + 1) Finally, for b > 0 and a = bq n , n = 1, 2, ..., ∞, we write [a, b]q = {bq k : 0 ≤ k ≤ n},

(2.5)

[0, b]q = {bq k : k ∈ N}.

3. Main results Theorem 1. Let f and g be two functions defined on [a, b]q satisfying the condition 0 < m ≤ f (t) ≤ M < ∞, 0 < n ≤ g(t) ≤ N < ∞, t ∈ [a, b]q

(3.1)

Then one has the inequality (3.2) s Z b Z b √ √ √ √ 1 1 f (x)dq x · g(x)dq x. M− m N− n |Tq (f, g; a, b)| ≤ b−a a b−a a The following Lemma is used to prove Theorem 1: Lemma 1. Let h and l are two functions defined on [a, b]q such that (3.3)

0 < m1 ≤ h(t) ≤ M1 < ∞, 0 < n1 ≤ l(t) ≤ N1 < ∞, t ∈ [a, b]q .

Then, we have (3.4) Z b

2

Z

Z

2

l (x)dq x −

h (x)dq x a

b

a

2

b

h(x)l(x)dq x a

r ≤

M1 − n1

r

m1 N1

!2 Z

b

Z h(x)l(x)dq x

a

Proof. From the condition (3.8), we have p p p m1 q k ≤ h(bq k ) q k ≤ M1 q k , and n1

p p p q k ≤ l(bq k ) q k ≤ N1 q k .

a

b

l2 (x)dq x


53

Using the Klamkin-McLenaghan inequality (1.4), we obtain !2 n−1 n−1 n−1 X X X 2 k k 2 k k k k k h (bq )q l (bq )q − h(bq )l(bq )q k=0

k=0

k=0

r ≤ From (2.3), we get Z b Z b l2 (x)dq x h2 (x)dq x

m1 N1

!2 n−1 X

k

k

h(bq )l(bq )q

k

k=0

n−1 X

l2 (bq k )q k .

k=0

2

b

Z −

h(x)l(x)dq x a

a

a

r

M1 − n1

r ≤

M1 − n1

r

m1 N1

!2 Z

b

Z h(x)l(x)dq x

b

l2 (x)dq x

a

a

Lemma 1 is thus proved.

Proof of Theorem 1: Using the Cauchy-Schwartz inequality for double integrals, we have Z bZ b 1 |Tq (f, g; a, b)| = (f (x) − f (y))(g(x) − g(y))d xd y q q 2(b − a)2 a a Z bZ b 21 Z bZ b 1 2 2 ≤ (f (x) − f (y)) d xd y · (g(x) − g(y)) d xd y q q q q 2(b − a)2 a a a a Z b Z b 2 1 2 = 4 (b − a) f (x)d x − f (x)d x q q 2(b − a)2 a a Z b 2 21 Z b × (b − a) g 2 (x)dq x − g(x)dq x a

= (3.5)×

1 b−a

Z

1 b−a

Z

a

b

2

1 b−a

Z

1 b−a

Z

f (x)dq x − a b

g 2 (x)dq x −

a

2 21

b

f (x)dq x a

2 12

b

g(x)dq x

.

a

By Lemma 1, we get b

Z

1 b−a

f (x)dq x − a

≤

(3.6)

√

!2

b

Z

1 b−a

2

f (x)dq x a

√

M−

2 m

1 b−a

Z

b

f (x)dq x a

and 1 b−a (3.7)

Z

b 2

g (x)dq x − a

≤

1 b−a √

Z

N−

!2

b

g(x)dq x a

√

2 n

1 b−a

Z

b

g(x)dq x a

54


From (3.5), (3.6) and (3.7), we deduce the desired inequality (3.9). Theorem 1 is thus proved. Corollary 1. Let f and g be two functions defined on [0, b]q satisfying the condition (3.8)

0 < m ≤ f (t) ≤ M < ∞, 0 < n ≤ g(t) ≤ N < ∞, t ∈ [0, b]q

Then one has the inequality sZ b Z b √ √ √ 1 √ (3.9) |Tq (f, g; 0, b)| ≤ M− m N− n f (x)dq x · g(x)dq x. b 0 0 Proof. By taking a = bq n in the previous theorem and by tending n to ∞ we obtain the result. Theorem 2. Let f and g be two functions defined on [a, b]q satisfying the condition (3.1). Then one has the inequality Z b Z b 1 (M − m)(N − n) 1 √ f (x)dq x · g(x)dq x (3.10) |Tq (f, g; a, b)| ≤ · b−a a b−a a 4 nmN M Lemma 2. [17] Let f and g are two functions defined on [a, b]q satisfying the condition (3.8). Then we have Z b 2 Z b Z b (M N + mn)2 2 2 (3.11) f (x)dq x g (x)dq x ≤ f (x)g(x)dq x 4nmN M a a a Proof of Theorem 2: Using Lemma 2, we get Z (b − a) a

b

(M + m)2 f (x)dq x ≤ 4mM 2

Z

2

b

f (x)dq x

,

a

Hence (3.12) 2 2 Z b Z b Z b 1 1 1 (M − m)2 f 2 (x)dq x − f (x)dq x ≤ f (x)dq x . b−a a b−a a 4mM b−a a Similarly, we have (3.13) 2 2 Z b Z b Z b 1 1 (N − n)2 1 g 2 (x)dq x − g(x)dq x ≤ g(x)dq x . b−a a b−a a 4nN b−a a From (3.5), (3.12) and (3.13), we deduce the desired inequality (3.14). Theorem 2 is thus proved. Corollary 2. Let f and g be two functions defined on [0, b]q satisfying the condition (3.8). Then one has the inequality Z b Z b (M − m)(N − n) √ (3.14) |Tq (f, g; 0, b)| ≤ · f (x)dq x · g(x)dq x 4b2 nmN M 0 0 Theorem 3. Let f and g be two positive functions defined on [0, ∞) satisfying the condition (3.15)

0 < m ≤ f (τ ) ≤ M < ∞, 0 < n ≤ g(τ ) ≤ N < ∞, τ ∈ [0, t], t > 0.


55

Then for all α > 0, b > 0 we have bα α α α (3.16) Γq (α + 1) Jq (f (b)g(b)) − Jq f (b)Jq g(b) ≤

√

M−

√

√ √ m N− n

q bα Jqα f (b)Jqα g(b) Γq (α + 1)

The following Lemma is used to prove Theorem 3: Lemma 3. Let h and l be two positive functions on [0, ∞) such that (3.17)

0 < m1 ≤ h(τ ) ≤ M1 < ∞, 0 < n1 ≤ l(τ ) ≤ N1 < ∞, τ ∈ [0, t], t > 0.

Then for all α > 0, b > 0, we have (3.18) 2 Jqα h2 (b)Jqα l2 (b) − Jqα (h(b)l(b)) ≤

r

M1 − n1

r

m1 N1

!2 Jqα (h(b)l(b))Jqα l2 (b).

Proof. Using the Klamkin-McLenaghan inequality (1.4), we obtain ∞ X

h2 (bq k )(1 − q k+1 )(α−1) q k

k=0

−

∞ X

l2 (bq k )(1 − q k+1 )(α−1) q k

k=0

∞ X

!2 k

k

h(bq )l(bq )(1 − q

k+1 (α−1) k

)

q

k=0

r ≤

M1 − n1

r

m1 N1

!2

∞ X

h(bq k )l(bq k )(1 − q k+1 )(α−1) q k

k=0

∞ X

l2 (bq k )(1 − q k+1 )(α−1) q k .

k=0

From (2.4), we obtain Jqα h2 (b)Jqα l2 (b)

2 α − Jq (h(b)l(b)) ≤

r

M1 − n1

r

m1 N1

!2


Jqα (h(b)l(b))Jqα l2 (b).

Proof of Theorem 3: Define (3.19)

Q(τ, ρ) = (f (τ ) − f (ρ))(g(τ ) − g(ρ)) (α−1)

(α−1)

(b−qρ) Multiplying (3.19) by (b−qτ ) Γ2 (α) and double integrating with respect to q τ and ρ from 0 to b, we get Z bZ b 1 (3.20) (b − qτ )(α−1) (b − qρ)(α−1) Q(τ, ρ)dq τ dq ρ Γ2q (α) 0 0 bα = 2 J α (f (b)g(b)) − 2Jqα f (b)Jqα g(b). Γq (α + 1) q

56


On the other hand, using the Cauchy-Schwartz inequality we get 1 Z bZ b (b − qτ )(α−1) (b − qρ)(α−1) Q(τ, ρ)dq τ dq ρ 2 Γq (α) 0 0 21 Z bZ b 1 (α−1) (α−1) 2 ≤ (b − qτ ) (b − qρ) (f (τ ) − f (ρ)) d τ d ρ q q Γ2q (α) 0 0 21 Z bZ b 1 (α−1) (α−1) 2 (b − qτ ) (b − qρ) (g(τ ) − g(ρ)) d τ d ρ × . q q Γ2q (α) 0 0 Then, 1 Z bZ b (α−1) (α−1) (3.21) 2 (b − qτ ) (b − qρ) Q(τ, ρ)dq τ dq ρ Γq (α) 0 0 1 1 2 2 2 2 bα bα α 2 α α 2 α ≤ 2 J f (b) − Jq f (b) × J g (b) − Jq g(b) Γq (α + 1) q Γq (α + 1) q Using Lemma 3, we get (3.22)

2 bα Jqα f 2 (b) − Jqα f (b) ≤ Γq (α + 1)

2 bα Jqα g 2 (b) − Jqα g(b) ≤ Γq (α + 1)

√

√

2 m

√

2

M−

bα Jqα f (b) Γq (α + 1)

and (3.23)

√

N−

n

bα Jqα g(b) . Γq (α + 1)

Using (3.20), (3.22) and (3.23), we deduce the desired inequality (3.16). Theorem 3 is thus proved. Theorem 4. Let f and g be two positive functions defined on [0, ∞) satisfying the condition (3.15). Then for all α > 0, b > 0, we have (3.24) (M − m)(N − n) α bα α α α ≤ √ Jq f (b)Jqα g(b). J (f (b)g(b)) − J f (b)J g(b) q q Γq (α + 1) q 4 mM nN

To prove Theorem 4 we need the following result: Lemma 4. Let h and l be two positive functions on [0, ∞) such that the condition (3.17). Then for all α > 0, b > 0, we have Jqα h2 (b)Jqα l2 (b) ≤

(3.25)

(M N + mn)2 α (Jq h(b)l(b))2 4nmN M

Proof. Using Grueb-Reinboldt inequality (1.5), we obtain ! ∞ ! ∞ X X h2 (bq k )(1 − q k+1 )α−1 q k l2 (bq k )(1 − q k+1 )α−1 q k k=0

k=0 2

(3.26)

≤

(M N + mn) 4nmN M

∞ X k=0

!2 h(bq k )l(bq k )(1 − q k+1 )α−1 q k


57

Which implies that (3.27)

(Jqα h2 (b))(Jqα l2 (b)) (M N + mn)2 ≤ (Jqα h(b)l(b))2 4nmN M


Proof of Theorem 4: Using Lemma 4, we get bα (M + m)2 α Jqα f 2 (b) ≤ (Jq f (b))2 Γq (α + 1) 4mM thus, (3.28)

bα (M − m)2 α Jqα f 2 (b) − (Jqα f (b))2 ≤ (Jq f (b))2 Γq (α + 1) 4mM

By the same, we obtain (3.29)

bα (N − n)2 α Jqα g 2 (b) − (Jqα g(b))2 ≤ (Jq g(b))2 Γq (α + 1) 4nN

Using (3.20), (3.28) and (3.29), we deduce the desired inequality (3.24). Theorem 4 is thus proved. References [1] Anber. A and Dahmani. Z., New integral results using P´ olya-Szeg¨ o inequalities, ACTA et Comm. Univ. Tari de Math. 17 (2) (2013), 171-178. [2] Brahim. K and Taf. S., On some fractional q-integral inequalities, Malaya Journal of Matematik 3(1)(2013), 21-26. [3] Brahim. K and Taf. S., Some Fracional Intehral Inequalities In Quantum Calculus, Journal of Fractional Calculus and Application. 4(2)(2013), 245-250. [4] Dahmani. Z., Tabahrit. L and Taf.S., New generalisation of Gr¨ uss inequality using RiemannLiouville fractional integrals, Bull. Math. Anal. Appl. 2(2010), 93-99. [5] Dragomir. S.S., Some integral inequalities of Gr¨ uss type, Indian J.Pure Apll. Math. 31(2000),397-415. [6] Dragomir. S.S. and Diamond. N.T., Integral inequalities of Gr¨ uss type via P´ olya-Szeg¨ o and Shisha-Mond results,East Asian Math. J.19 (2003), 27-39. [7] Gasper. G and Rahman. R., Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. [8] Gauchman. H., Integral inequalities in q-calculus, Comput. Math. Appl. 47 (2-3)(2004), 281300. [9] Grueb. W and Rheinboldt. W., On generalisation of an inequality of L.V. Kantrovich, Proc. Amer. Math. Soc., 10 (1959), 407-415. R [10] Gr¨ uss. G., Uber das maximum des absoluten Betrages von 1/(b − a) ab f (x)g(x)dx − 1/(b − R R a)2 ab f (x)dx ab g(x)dx, Math.Z.39(1)(1935), 215-226. [11] Jackson. F.H., On a q-Definite Integrals. Quarterly Journal of Pure and Applied Mathematics 41, 1910, 193-203. [12] Klamkin. D.S and McLenaghan. J.E., An ellipse inequality, Math. Mag., 50 (1977), 261-263. [13] Kac. V.G and Cheung. P., Quantum Calculus, Universitext, Springer-Verlag, New York, (2002). [14] Mercer. A.McD and Mercer. P., New proofs of the Gr¨ uss inequality, Aust. J. Math. Anal. Apll. 1 (2) (2004), Article ID 12. [15] Mitrinovi´ c. D.S., Peˇ cari´ c. J.E., Fink. A.M., Classical and New Inequalities in Analysis, in: Mathematics and its Applications, Vol. 61, Kluwer Acadmic Publishers, Dordrecht, The Netherlands, 1993. [16] Rajkovi´ c. P.M and Marinkonvi´ c. S.D., Fractional integrals and derivatives in q-calculus, Applicable analysis and discrete mathematics, 1 (2007), 311-323.

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[17] Rajkovi´ c. P., Marinkonvi´ c. S.D and Stankovi´ c. M.S., The inequalities for some types of qintegrals, ArXiv. Math. CA, 8 May 2006. ˇ [18] Sarikaya. M.Z, Aktan. N and Yildirim. H., On weighted Cebyˇ sev-Gr¨ uss type inequalities on time scales, J. Math. Inequal. 2(2008), 185-195. 1 Departement 2 Department

of Mathematics, Faculty of Science of Tunis, Tunisia

of Mathematics, Faculty SEI, UMAB University of Mostaganem, Alge-

ria ∗ Corresponding

author