q-Integral inequalities associated with some fractional q ... - Core

Agarwal et al. Journal of Inequalities and Applications (2015) 2015:345 DOI 10.1186/s13660-015-0860-8

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q-Integral inequalities associated with some fractional q-integral operators Praveen Agarwal1* , Silvestru Sever Dragomir2 , Jaekeun Park3 and Shilpi Jain4 *

Correspondence: [email protected] 1 Department of Mathematics, Anand International College of Engineering, Jaipur, 303012, Republic of India Full list of author information is available at the end of the article

Abstract In recent years fractional q-integral inequalities have been investigated by many authors. Therefore, the fractional q-integral inequalities have become one of the most powerful and far-reaching tools for the development of many branches of pure and applied mathematics. Here, we aim to establish some new fractional q-integral inequality by using fractional q-integral operators. Relevant connections of the results presented here with earlier ones are also pointed out. MSC: Primary 26D10; 26A33; secondary 26D15 Keywords: integral inequalities; Chebyshev functional; Riemann-Liouville fractional integral operator; Pólya-Szegö type inequalities; hypergeometric fractional integral operator

1 Introduction and preliminaries In recent years the study of fractional q-integral inequalities involving functions of independent variables has been an important research subject in mathematical analysis because the inequality technique is also one of the very useful tools in the study of special functions and theory of approximations. During the last two decades or so, several interesting and useful extensions of many of the fractional integral inequalities have been considered by several authors (see, for example, [–]; see also the very recent work []). The above-mentioned works have largely motivated our present study. For our purpose, we begin by recalling the well-known celebrated functional considered by Chebyshev [] and defined by  T(f , g) := b–a

a

b

 f (x)g(x) dx – b–a

b

f (x) dx a

 b–a

b

g(x) dx ,

(.)

a

where f (x) and g(x) are two integrable functions on [a, b]. If f (x) and g(x) are synchronous on [a, b], i.e.,

f (x) – f (y) g(x) – g(y) ≥ 

(.)

for any x, y ∈ [a, b], then T(f , g) ≥ . The functional (.) has attracted many researchers’ attention due to diverse applications in numerical quadrature, transform theory, probability and statistical problems. Among © 2015 Agarwal et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Agarwal et al. Journal of Inequalities and Applications (2015) 2015:345

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those applications, the functional (.) has also been employed to yield a number of integral inequalities (see, e.g., [–]; for a very recent work, see also []). In , Grüss [] proved the inequality T(f , g) ≤ (M – m)(N – n) , 

(.)

where f (x) and g(x) are two bounded functions, i.e., m ≤ f (x) ≤ M,

n ≤ g(x) ≤ N

(.)

for any m, M, n, N ∈ R and x, y ∈ [a, b]. Pólya and Szegö [] obtained the following inequality defined as b f  (x) dx a g  (x) dx  MN mn  ≤ , + b b mn MN ( a f (x) dx a g(x) dx)  b a

(.)

provided f , g satisfy (.) and m, n > . Similarly, Dragomir and Diamond proved that (see [], p., Eq. .) – n) T(f , g) ≤ (M – m)(N √  (b – a) mMnN

b

f (x) dx a

b

g(x) dx,

(.)

a

where f (x) and g(x) are two positive integrable functions so that  < m ≤ f (x) ≤ M < ∞,

 < n ≤ g(x) ≤ N < ∞

(.)

for a.e. x ∈ [a, b]. Recently, Anber and Dahmani [], by using the Riemann-Liouville fractional integral, presented some interesting integral inequalities of Pólya and Szegö type. Here, motivated essentially by the above work, we aim at establishing certain (presumably) new PólyaSzegö type q-inequalities associated with fractional q-integral operators. For our purpose, we need the following definitions and some properties. Definition  A real-valued function f (t) (t > ) is said to be in the space Cμn (n, μ ∈ R) if there exists a real number p > μ such that f (n) (t) = t p φ(t), where φ(t) ∈ C(, ∞). Here, for the case n = , we use a simpler notation Cμ = Cμ . Definition  Let (α) > , β and η be real or complex numbers. Then a q-analogue of α,β,η is given for | τt | <  by (see [], p., Eq. (.)) Saigo’s fractional integral Iq

t –β– Iqα,β,η f (t) := q (α)

t

(qτ /t; q)α– 

∞ (qα+β ; q)m (q–η ; q)m m=

m

(qα ; q)m (q; q)m

· q(η–β)m (–)m q–(  ) (τ /t – )m q f (τ ) dq τ .

(.)


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α,β,η

The integral operatorIq includes both the q-analogues of the Riemann-Liouville and Erdélyi-Kober fractional integral operators given by the following relationships:

Iqα f (t) : = Iqα,–α, f (t) t t α– = (qτ /t; q)α– f (τ ) dq τ q (α) 

(α > ;  < q < ),

(.)

and

Iqη,α f (t) : = Iqα,,η f (t) t –η– t = (qτ /t; q)α– τ η f (τ ) dq τ q (α) 

(α > ;  < q < ),

(.)

where (a; q)α is the q-shifted factorial. The q-shifted factorial (a; q)n is defined by

 (a; q)n := n– k=

 – aqk

(n = ), (n ∈ N),

(.)

where a, q ∈ C, and it is assumed that a = q–m (m ∈ N ). The q-shifted factorial for negative subscript is defined by (a; q)–n :=

 ( – aq– )( – aq– ) · · · ( – aq–n )

(n ∈ N ).

(.)

We also write (a; q)∞ :=

∞  – aqk

a, q ∈ C; |q| <  .

(.)

k=

It follows from (.), (.) and (.) that (a; q)n =

(a; q)∞ (aqn ; q)∞

(n ∈ Z),

(.)

which can be extended to n = α ∈ C as follows: (a; q)α =

(a; q)∞ (aqα ; q)∞

α ∈ C; |q| <  ,

(.)

where the principal value of qα is taken. For f (t) = t μ in (.), we get the known formula []

Iqα,β,η t μ :=

q (μ + )q (μ – β + η + ) μ–β x . q (μ – β + )q (μ + α + η + )

(.)

Lemma  (Choi and Agarwal []) Let  < q <  and f : [, ∞) → R be a continuous function with f (t) ≥  for all t ∈ [, ∞). Then we have the following inequalities:


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(i) The Saigo fractional q-integral operator of the function f (t) in (.)

Iqα,β,η f (t) ≥ 

(.)

for all α >  and β, η ∈ R with α + β >  and η < ; (ii) The q-analogue of Riemann-Liouville fractional integral operator of the function f (t) of order α in (.)

Iqα f (t) ≥ 

(.)

for all α > ; (iii) The q-analogue of Erdélyi-Kober fractional integral operator of the function f (t) in (.)

Iqη,α f (t) ≥ 

(.)

for all α >  and η ∈ R.

2 Certain fractional q-integral inequalities In this section, we establish certain Pólya-Szegö type integral inequalities for the synchronous functions involving the hypergeometric fractional integral operator (.), some of which are presumably (new) ones. For our purpose, we begin with providing the following lemma involving a q-analogue of Saigo’s fractional integral operator. Lemma  Let  < q < , u and v be two continuous and positive integrable functions on [, ∞) with  < m ≤ u(τ ) ≤ M < ∞,

 < n ≤ v(τ ) ≤ N < ∞

τ ∈ [, t], t >  .

(.)

Then the following inequality holds true: α,β,η

(Iq

α,β,η

{u (t)})(Iq

α,β,η

(Iq

{v (t)})

{u(t)}{v(t)})

 ≤ 

M N + m n

m n M N

 (.)

for all α > , and β, η ∈ R with α + β > , and η < . Proof From (.), for τ ∈ [, t], t > , we have u(τ ) m ≤ , v(τ ) N

(.)

which yields

N u(τ ) – m v(τ ) ≤ .

(.)

Analogously, we have u(τ ) n , ≤ M v(τ )

(.)


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from which one has

n u(τ ) – M v(τ ) ≤ .

(.)

Multiplying (.) and (.), we obtain (M N + m n )u(τ )v(τ ) ≥ M m v (τ ) + N n u (τ ).

(.)

Now, multiplying both sides of (.) by ∞ m t –β– (qα+β ; q)m (q–η ; q)m (η–β)m ·q (–)m q–(  ) (τ /t – )m (qτ /t; q)α– q α ; q) (q; q) q (α) (q m m m=

and taking q-integration of the resulting inequality with respect to τ from  to t with the aid of Definition , we get

(M N + m n )Iqα,β,η u(t)v(t) ≥ M m Iqα,β,η v (t) + N n Iqα,β,η u (t) .

(.)

√ Applying the AM-GM inequality, i.e., a + b ≥  ab, a, b ∈ R+ , we have

α,β,η  α,β,η  v (t) N n Iq u (t) . (M N + m n )Iqα,β,η u(t)v(t) ≥  M m Iq

(.)

This implies that after little simplification α,β,η

Iq

α,β,η

{u (t)}Iq α,β,η

{Iq

{v (t)}

{u(t)v(t)}}

 ≤ 

M N + m n

m n M N



This completes the proof of Lemma .

.

(.)

Theorem  Let  < q < , f and g be two positive integrable functions on [, ∞) and m, M, n, N be positive real numbers with inequality (.) holds. Then the following inequality holds true:

( – β + η) –β α,β,η f (t)g(t) t I q ( – β)( + α + η) ≤

(M – m)(N – n) α,β,η α,β,η Iq f (t) Iq g(t) √  mMnN

(.)

for all α > , and β, η ∈ R with α + β > , and η < . Proof Let f and g be two positive integrable functions on [, ∞). Then, for all τ , ρ ∈ (, t) with t > , we have A(τ , ρ) = f (τ ) – f (ρ) g(τ ) – g(ρ) ,

(.)

or, equivalently, A(τ , ρ) = f (τ )g(τ ) + f (ρ)g(ρ) – f (τ )g(ρ) – f (ρ)g(τ ).

(.)


Page 6 of 13

Now, multiplying both sides of (.) by ∞ m (qα+β ; q)m (q–η ; q)m (η–β)m t –β– ·q (–)m q–(  ) (τ /t – )m (qτ /t; q)α– q α q (α) (q ; q)m (q; q)m m=

and taking q-integration of the resulting inequality with respect to τ from  to t with the aid of Definition , we get t –β– q (α)

t

(qτ /t; q)α– 

∞ (qα+β ; q)m (q–η ; q)m m=

(qα ; q)m (q; q)m

m

· q(η–β)m (–)m q–(  ) (τ /t – )m q A(τ , ρ) dq τ

= Iqα,β,η f (t)g(t) +

( – β + η) t –β f (ρ)g(ρ) ( – β)( + α + η)

– g(ρ)Iqα,β,η f (t) – f (ρ)Iqα,β,η g(t) .

(.)

Again, multiplying both sides of (.) by ∞ m t –β– (qα+β ; q)m (q–η ; q)m (η–β)m (qρ/t; q)α– ·q (–)m q–(  ) (ρ/t – )m q α q (α) (q ; q)m (q; q)m m=

and taking q-integration of the resulting inequality with respect to ρ from  to t and using (.), we get t –(β+) q (α)

t

t

(qτ /t; q)α– (qρ/t; q)α– 



 ·q =

(η–β)m

m (–)m q–(  )

∞ (qα+β ; q)m (q–η ; q)m m=

(qα ; q)m (q; q)m

m (τ /t – )m q (ρ/t – )q A(τ , ρ) dq τ dq ρ

( – β + η) t –β Iqα,β,η f (t)g(t) – Iqα,β,η f (t)g(t) . ( – β)( + α + η)

By using the Cauchy-Schwarz inequality for double integrals, we have

∞ t –(β+) t t (qα+β ; q)m (q–η ; q)m (qτ /t; q)α– (qρ/t; q)α–  q (α)   (qα ; q)m (q; q)m m=

m · q(η–β)m (–) q ( ) (τ /t – )m q (ρ/t – )q A(τ , ρ) dq τ dq ρ

∞ (qα+β ; q)m (q–η ; q)m t –(β+) t t (qτ /t; q) (qρ/t; q) ≤ α– α– q (α)   (qα ; q)m (q; q)m m= 

m – m 



·q

(η–β)m

m (–)m q–(  )

t –(β+) +  q (α)

t

m  (τ /t – )m q (ρ/t – )q f (τ ) dq τ dq ρ

t

(qτ /t; q)α– (qρ/t; q)α– 



∞ (qα+β ; q)m (q–η ; q)m m=

(qα ; q)m (q; q)m

(.)


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 ·q

(η–β)m

m (–)m q–(  )

t –(β+) –  q (α)

t

m  (τ /t – )m q (ρ/t – )q f (τ ) dq τ dq ρ

t

(qτ /t; q)α– (qρ/t; q)α– 

∞ (qα+β ; q)m (q–η ; q)m



 m – m 

· q(η–β)m (–) q ( ) t –(β+) +  q (α)

t

m  (τ /t – )m q (ρ/t – )q g (τ ) dq τ dq ρ

t

(qτ /t; q)α– (qρ/t; q)α– 



·q

m (–)m q–(  )

t –(β+) –  q (α)

t

(qα ; q)m (q; q)m

m  (τ /t – )m q (ρ/t – )q g (τ ) dq τ dq ρ

t

(qτ /t; q)α– (qρ/t; q)α– 

∞ (qα+β ; q)m (q–η ; q)m m=

 (η–β)m

(qα ; q)m (q; q)m

m=



∞ (qα+β ; q)m (q–η ; q)m m=

(qα ; q)m (q; q)m

 ·q

(η–β)m

m (–)m q–(  )

 (τ /t

– )m q (ρ/t

– )m q g(τ )g(ρ) dq τ

dq ρ

.

(.)

Applying Definition , we get

∞ t –(β+) t t (qα+β ; q)m (q–η ; q)m (qτ /t; q)α– (qρ/t; q)α–  q (α)   (qα ; q)m (q; q)m m= 

·q

(η–β)m

m (–)m q–(  )

(τ /t

– )m q (ρ/t

– )m q A(τ , ρ) dq τ

dq ρ



α,β,η   ( – β + η) –β α,β,η  f (t) – Iq f (t) t Iq ≤ ( – β)( + α + η) 

  ( – β + η) t –β Iqα,β,η g  (t) – Iqα,β,η g(t) · . ( – β)( + α + η)

(.)

By applying Lemma , we get

( – β + η) t –β Iqα,β,η f  (t) ( – β)( + α + η)   M m α,β,η  Iq f (t) + ≤  m M =

(M + m) α,β,η  f (t) . Iq mM

(.)

After little simplification, we get

 ( – β + η) t –β Iqα,β,η f  (t) – Iqα,β,η f (t) ( – β)( + α + η)

 (M + m) –  Iqα,β,η f (t) ≤ mM

(.)


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or

 ( – β + η) t –β Iqα,β,η f  (t) – Iqα,β,η f (t) ( – β)( + α + η) ≤

(M – m) α,β,η  f (t) . Iq mM

(.)

Similarly, we get

 ( – β + η) t –β Iqα,β,η g  (t) – Iqα,β,η g(t) ( – β)( + α + η) ≤

(N – n) α,β,η  g(t) . Iq nN

(.)

Finally, by adding (.), (.), (.) and (.), side by side, we arrive at the desired result (.). In the sequel, we can present another inequality involving the q-fractional integral operator given in (.), asserted by the following lemma. Lemma  Let  < q < , u and v be two continuous and positive integrable functions on [, ∞) with (.) holds. Then the following inequality holds true: α,β,η

(Iq

γ ,δ,ζ

{u (t)})(Iq

{v (t)})

 ≤ γ ,δ,ζ α,β,η (Iq {u(t)}{v(t)})(Iq {u(t)}{v(t)}) 

M N + m n

m n M N

 (.)

for all α, γ > , and β, η, δ, ζ ∈ R with α + β > , γ + δ > , and η, ζ < . Proof To prove Lemma , we start from the condition m u(τ ) ≤ N v(τ )

τ ∈ [, t], t >  ,

(.)

we get m  v (τ ) ≤ u(τ )v(τ ) τ ∈ [, t], t >  . N

(.)

Now, multiplying both sides of (.) by ∞

(qγ +δ ; q)n (q–ζ ; q)n t –δ– (qρ/t; q)γ – (q (γ )) (qγ ; q)n (q; q)n n= n

· q(ζ –δ)n (–)n q–() (ρ/t – )nq

ρ ∈ (, t); t >  ,

and integrating with respect to ρ from  to t, we get m γ ,δ,ζ ,ν  γ ,δ,ζ ,ν

v (t) ≤ It u(t)v(t) . It N

(.)

Multiplying (.) and (.), we get the desired result (.). This completes the proof of Lemma .


Page 9 of 13

Theorem  Let  < q < , f and g be two positive integrable functions on [, ∞) and there exist positive real numbers m, n, M, N with inequality (.) holds. Then we have

( – δ + ζ ) ( – β + η) –δ α,β,η f (t)g(t) + t –β Iqγ ,δ,ζ f (t)g(t) ( – δ)( + γ + ζ ) t Iq ( – β)( + α + η)

γ ,δ,ζ α,β,η γ ,δ,ζ α,β,η f (t) Iq g(t) – Iq g(t) Iq f (t) – Iq ≤

(M – m)(N – n) α,β,η γ ,δ,ζ f (t) Iq g(t) Iq √  mMnN

(.)

for all α, γ > , and β, η, δ, ζ ∈ R with α + β > , γ + δ > , and η, ζ < . Proof Multiplying both sides of (.) by ∞

(qγ +δ ; q)n (q–ζ ; q)n t –δ– (qρ/t; q)γ – (q (γ )) (qγ ; q)n (q; q)n n= n

· q(ζ –δ)n (–)n q–() (ρ/t – )nq

ρ ∈ (, t); t >  ,

and integrating with respect to ρ from  to t, we get t –δ– t –β– q (γ )q (α) ·

t

t

(qρ/t; q)γ – (qτ /t; q)α– 



∞ ∞ (qα+β ; q)m (q–η ; q)m (qγ +δ ; q)n (q–ζ ; q)n m= n=

(qα ; q)m (q; q)m

(qγ ; q)n (q; q)n

m

· q(η–β)m (–)m q–(  ) (τ /t – )m q

n

· q(ζ –δ)n (–)n q–() (ρ/t – )nq A(τ , ρ) dq τ dq ρ =

( – δ + ζ ) ( – β + η) t –δ Iqα,β,η f (t)g(t) + t –β Iqγ ,δ,ζ f (t)g(t) ( – δ)( + γ + ζ ) ( – β)( + α + η)

(.) – Iqα,β,η f (t) Iqγ ,δ,ζ g(t) – Iqα,β,η g(t) Iqγ ,δ,ζ f (t) .

By using the Cauchy-Schwarz inequality for double integrals, we have t –δ– t –β– t t (qρ/t; q)γ – (qτ /t; q)α– q (γ )q (α)   ·


·q

(ζ –δ)n

(qα ; q)

n (–)n q–() (ρ/t

t –δ– t –β– ≤ q (γ )q (α) ·

m (q; q)m

t

(qγ ; q)

n (q; q)n

– )nq A(τ , ρ) dq τ

dq ρ

t

(qρ/t; q)γ – (qτ /t; q)α– 




m

· q(η–β)m (–)m q–(  ) (τ /t – )m q

(qα ; q)m (q; q)m n

(qγ ; q)n (q; q)n

· q(ζ –δ)n (–)n q–() (ρ/t – )nq f  (ρ) dq τ dq ρ

m

· q(η–β)m (–)m q–(  ) (τ /t – )m q


+ ·

t –δ– t –β– q (γ )q (α)

t

Page 10 of 13

t

(qρ/t; q)γ – (qτ /t; q)α– 




(qα ; q)m (q; q)m

(qγ ; q)n (q; q)n

m

· q(η–β)m (–)m q–(  ) (τ /t – )m q

n

· q(ζ –δ)n (–)n q–() (ρ/t – )nq f  (τ ) dq τ dq ρ t t t –δ– t –β– (qρ/t; q)γ – (qτ /t; q)α– – q (γ )q (α)   ·


(qα ; q)m (q; q)m

(qγ ; q)n (q; q)n

m

· q(η–β)m (–)m q–(  ) (τ /t – )m q



n – n

· q(ζ –δ)n (–) q ( ) (ρ/t – )nq f (τ )f (ρ) dq τ dq ρ

t –δ– t –β– · q (γ )q (α) ·

t

t

(qρ/t; q)γ – (qτ /t; q)α– 




(qα ; q)m (q; q)m

(qγ ; q)n (q; q)n

m

· q(η–β)m (–)m q–(  ) (τ /t – )m q

n

· q(ζ –δ)n (–)n q–() (ρ/t – )nq g  (ρ) dq τ dq ρ t t t –δ– t –β– + (qρ/t; q)γ – (qτ /t; q)α– q (γ )q (α)   ·


(qα ; q)

m (q; q)m

(qγ ; q)

n (q; q)n

m

· q(η–β)m (–)m q–(  ) (τ /t – )m q

n

· q(ζ –δ)n (–)n q–() (ρ/t – )nq g  (τ ) dq τ dq ρ t t t –δ– t –β– (qρ/t; q)γ – (qτ /t; q)α– – q (γ )q (α)   ·


·q

(ζ –δ)n

(qα ; q)m (q; q)m

n (–)n q–() (ρ/t

(qγ ; q)n (q; q)n

m

· q(η–β)m (–)m q–(  ) (τ /t – )m q

 – )nq g(τ )g(ρ) dq τ

dq ρ

.

(.)

Applying Definition , we get t –δ– t –β– t t (qρ/t; q)γ – (qτ /t; q)α– q (γ )q (α)   ∞ ∞ (qα+β ; q)m (q–η ; q)m (qγ +δ ; q)n (q–ζ ; q)n

m

· q(η–β)m (–)m q–(  ) (τ /t – )m q γ ; q) (q; q) (q n n m= n= (ζ –δ)n n –(n) n ·q (–) q (ρ/t – )q A(τ , ρ) dq τ dq ρ

( – β + η) ( – δ + ζ ) t –δ Iqα,β,η f  (t) + t –β Iqγ ,δ,ζ f  (t) ≤ ( – δ)( + γ + ζ ) ( – β)( + α + η) ·

(qα ; q)m (q; q)m


– Iqα,β,η

Page 11 of 13



γ ,δ,ζ  f (t) Iq f (t)

( – β + η) ( – δ + ζ ) γ ,δ,ζ  g (t) t –δ Iqα,β,η g  (t) + t –β It ( – δ)( + γ + ζ ) ( – β)( + α + η) 

. (.) – Iqα,β,η g(t) Iqγ ,δ,ζ g(t)

·

Applying Definition , we get

( – β + η) t –β Iqγ ,δ,ζ f  (t) – Iqγ ,δ,ζ f (t) Iqα,β,η f (t) ( – β)( + α + η) ≤

(M – m) γ ,δ,ζ α,β,η f (t) Iq f (t) , Iq mM

(.)

and

( – δ + ζ ) t –δ Iqα,β,η f  (t) – Iqγ ,δ,ζ f (t) Iqα,β,η f (t) ( – δ)( + γ + ζ ) ≤

(M – m) γ ,δ,ζ α,β,η f (t) Iq f (t) . Iq mM

(.)

Similarly, for the function g(t), we get

( – β + η) t –β Iqγ ,δ,ζ g  (t) – Iqγ ,δ,ζ g(t) Iqα,β,η g(t) ( – β)( + α + η) ≤

(M – m) γ ,δ,ζ α,β,η g(t) Iq g(t) , Iq mM

(.)

and

( – δ + ζ ) t –δ Iqα,β,η g  (t) – Iqγ ,δ,ζ g(t) Iqα,β,η g(t) ( – δ)( + γ + ζ ) ≤

(M – m) γ ,δ,ζ α,β,η g(t) Iq g(t) . Iq mM

(.)

Finally, in view of (.) to (.), we arrive at the desired result (.). This completes the proof of Theorem . Remark  It may be noted that the inequality in (.) when ζ = η reduces immediately to that in (.).

3 Special cases and concluding remarks By virtue of the unified nature of Saigo’s fractional q-integral operator (.), a large number of new and known integral inequalities involving q-analogues of the Riemann-Liouville and Erdélyi-Kober fractional integral operators are seen to follow as special cases of our main result. Indeed, by suitably specializing the values of parameters α, β, η (and γ , δ, ζ in addition of Theorem ), inequalities (.) and (.) in Theorems  and , respectively, would yield further Grüss type integral inequalities involving the above-mentioned integral operators.


Page 12 of 13

If we put β =  (and δ =  in addition in Theorem ), using (.), inequalities (.) and (.) gives the following results involving q-analogues of the Erdélyi-Kober fractional integral operators, which are believed to be new. Corollary  Let  < q < , f and g be two positive integrable functions on [, ∞) and m, M, n, N be positive real numbers with inequality (.) holds. Then the following inequality holds true: ( + η) η,α

(M – m)(N – n) η,α η,α (.) ( + α + η) Iq f (t)g(t) ≤ √mMnN Iq f (t) Iq g(t) for all α > , and η ∈ R with η < . Corollary  Let  < q < , f and g be two positive integrable functions on [, ∞) and there exist positive real numbers m, n, M, N with inequality (.) holds. Then we have ( + ζ ) η,α

( + η) ζ ,γ

( + γ + ζ ) Iq f (t)g(t) + ( + α + η) Iq f (t)g(t)

η,α – Iqη,α f (t) Iqζ ,γ g(t) – I,t g(t) Iqζ ,γ f (t) ≤

(M – m)(N – n) η,α ζ ,γ I,t f (t) Iq g(t) √  mMnN

(.)

for all α, γ > , and η, ζ ∈ R with η, ζ < . Similarly, if we set η =  and replace β by –α in Theorem  (and ζ =  and replace δ by –γ in addition in Theorem ), using (.), inequalities (.) and (.) gives the following results involving q-analogues of the Riemann-Liouville and Erdélyi-Kober fractional integral operators, which are also believed to be new. Corollary  Let  < q < , f and g be two positive integrable functions on [, ∞) and m, M, n, N be positive real numbers with inequality (.) holds. Then the following inequality holds true:

(M – m)(N – n) α α  α α ≤ f (t)g(t) (.) t I Iq f (t) Iq g(t) √ ( + α) q  mMnN for all α > . Corollary  Let  < q < , f and g be two positive integrable functions on [, ∞) and there exist positive real numbers m, n, M, N with inequality (.) holds. Then we have

  γ α α γ ( + γ ) t Iq f (t)g(t) + ( + α) t Iq f (t)g(t)

γ α γ α – Iq f (t) Iq g(t) – Iq g(t) Iq f (t) ≤

(M – m)(N – n) α γ Iq f (t) Iq g(t) √  mMnN

for all α, γ > .

(.)


Page 13 of 13

We conclude this paper by emphasizing, again, that our main result here, being of a very general nature, can be specialized to yield numerous interesting fractional integral inequalities including q-analogues of some known results (see, for example []).

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally in this article. They read and approved the final manuscript. Author details 1 Department of Mathematics, Anand International College of Engineering, Jaipur, 303012, Republic of India. 2 Department of Mathematics, College of Engineering and Science, Victoria University, P.O. Box 14428, Melbourne, MC 8001, Australia. 3 Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si, 356-706, Republic of Korea. 4 Department of Mathematics, Poornima College of Engineering, Sitapura, Jaipur, 302029, Republic of India. Acknowledgements The authors would like to express their deep thanks for the reviewers’ helpful comments. Received: 14 August 2015 Accepted: 8 October 2015 References 1. Anastassiou, G: q-Fractional inequalities. CUBO 13, 61-71 (2011) 2. Anastassiou, GA: Fractional Differentiation Inequalities. Springer, Dordrecht (2009) 3. Cerone, P, Dragomir, SS: A refinement of the Grüss inequality and applications. Tamkang J. Math. 38, 37-49 (2007) 4. Choi, J, Agarwal, P: Some new Saigo type fractional integral inequalities and their q-analogues. Abstr. Appl. Anal. 2014, Article ID 579260 (2014) 5. Choi, J, Agarwal, P: Certain fractional integral inequalities involving hypergeometric operators. East Asian Math. J. 30, 283-291 (2014) 6. Choi, J, Agarwal, P: Certain new pathway type fractional integral inequalities. Honam Math. J. 36, 437-447 (2014) 7. Choi, J, Agarwal, P: Certain integral transform and fractional integral formulas for the generalized Gauss hypergeometric functions. Abstr. Appl. Anal. 2014, Article ID 735946 (2014) 8. Mazouzi, S, Qi, F: On an open problem regarding an integral inequality. J. Inequal. Pure Appl. Math. 4(2), 31 (2003) 9. Mitrinović, DS, Peˇcarić, JE, Fink, AM: Classical and New Inequalities in Analysis. Mathematics and Its Applications (East European Series), vol. 61. Kluwer Academic, Dordrecht (1993) 10. Pachpatte, BG: On multidimensional Grüss type integral inequalities. J. Inequal. Pure Appl. Math. 3(2), 27 (2002) 11. Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999) 12. Zhu, C, Yang, W, Zhao, Q: Some new fractional q-integral Grüss-type inequalities and other inequalities. J. Inequal. Appl. 2012, Article ID 299 (2012) 13. Anber, A, Dahmani, Z: New integral results using Pólya-Szegö inequality. Acta Comment. Univ. Tartu Math. 17(2), 171-178 (2013) 14. Chebyshev, PL: Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites. Proc. Math. Soc. Charkov 2, 93-98 (1882) 15. Anastassiou, GA: Advances on Fractional Inequalities. Springer Briefs in Mathematics. Springer, New York (2011) 16. Belarbi, S, Dahmani, Z: On some new fractional integral inequalities. J. Inequal. Pure Appl. Math. 10(3), 86 (2009) (electronic) 17. Dahmani, Z, Mechouar, O, Brahami, S: Certain inequalities related to the Chebyshev’s functional involving a type Riemann-Liouville operator. Bull. Math. Anal. Appl. 3(4), 38-44 (2011) 18. Dragomir, SS: Some integral inequalities of Grüss type. Indian J. Pure Appl. Math. 31(4), 397-415 (2000) 19. Kalla, SL, Rao, A: On Grüss type inequality for hypergeometric fractional integrals. Matematiche 66(1), 57-64 (2011) 20. Lakshmikantham, V, Vatsala, AS: Theory of fractional differential inequalities and applications. Commun. Appl. Anal. 11, 395-402 (2007) ˇ 21. Ögünmez, H, Özkan, UM: Fractional quantum integral inequalities. J. Inequal. Appl. 2011, Article ID 787939 (2011) 22. Sulaiman, WT: Some new fractional integral inequalities. J. Math. Anal. 2(2), 23-28 (2011) 23. Wang, G, Agarwal, P, Chand, M: Certain Grüss type inequalities involving the generalized fractional integral operator. J. Inequal. Appl. 2014, Article ID 147 (2014) b b b 1 1 f (x)g(x) dx – (b–a) 24. Grüss, G: Über das Maximum des absoluten Betrages von b–a 2 a f (x) dx a g(x) dx. Math. Z. 39, a 215-226 (1935) 25. Pólya, G, Szegö, G: Aufgaben und Lehrsatze aus der Analysis. Band 1. Die Grundlehren der mathmatischen Wissenschaften, vol. 19. Springer, Berlin (1925) 26. Dragomir, SS, Diamond, NT: Integral inequalities of Grüss type via Pólya-Szegö and Shisha-Mond results. East Asian Math. J. 19(1), 27-39 (2003) 27. Garg, M, Chanchlani, L: q-Analogues of Saigo’s fractional calculus operators. Bull. Math. Anal. Appl. 3(4), 169-179 (2011) 28. Choi, J, Agarwal, P: Some new Saigo type fractional integral inequalities and their q-analogues. Abstr. Appl. Anal. 2014, Article ID 579260 (2014)